GROUP ACTIONS ON CENTRAL SIMPLE ALGEBRAS: A GEOMETRIC APPROACH

Z. REICHSTEIN AND N. VONESSEN

Abstract. We study actions of linear algebraic groups on central simple algebras using algebro-geometric techniques. Suppose an algebraic group G acts on a central simple algebra A of degree n. We are interested in questions of the following type: (a) Do the G-fixed elements form a central simple subalgebra of A of degree n? (b) Does A have a G-invariant maximal subfield? (c) Does A have a splitting field with a G-action, extending the G-action on the center of A?

Somewhat surprisingly, we find that under mild assumptions on A and the actions, one can answer these questions by using techniques from birational invariant theory (i.e., the study of group actions on algebraic varieties, up to equivariant birational isomorphisms). In fact, group actions on central simple algebras turn out to be related to some of the central problems in birational invariant theory, such as the existence of sections, stabilizers in general position, affine models, etc. In this paper we explain these connections and explore them to give partial answers to questions (a)—(c).

Contents

1. Introduction 2 2. Preliminaries 5 3. The uniqueness of the associated variety 7 4. Brauer-Severi varieties 9 5. Algebraic actions 10 6. Proof of Theorem 1.4 13 7. The G-action on the center of A 14 8. Which geometric actions are algebraic? 16 9. Proof of Theorem 1.5 18 10. Proof of Theorem 1.7 20 11. More on G-splitting fields 21 12. An example: algebraic actions of unipotent groups 23 13. An example: the GL_{m}-action on UD(m, n), m ≥ n^{2} 24 14. An example: the GL_{2}-action on UD(2, 2) 25 15. An example: a finite group action on a cyclic algebra 27 Appendix A. Inner actions on division algebras 29 Appendix B. Regular actions on prime PI-algebras 31 References 32

Date: September 16, 2005. 2000 Mathematics Subject Classification. 14L30, 16K20, 16W22. Key words and phrases. Linear algebraic group, group action, central simple algebra,

division algebra.

Z. Reichstein was supported in part by an NSERC research grant. N. Vonessen gratefully acknowledges the support of the University of Montana and the

hospitality of the University of British Columbia during his sabbatical in 2002/2003, when part of this research was done.

1. Introduction

In this paper we study actions of linear algebraic groups G on central simple algebras A in characteristic zero. As usual, we will denote the center of A by Z(A) and the subalgebra of G-fixed elements of A by

A^{G} = {a ∈ A | g(a) = a ∀g ∈ G} .

We will be interested in questions such as the following:

(1.1) (a) Is A^{G} a central simple algebra of the same degree as A? (b) Does A have a G-invariant maximal subfield?

(c) Can the G-action on Z(A) be extended to a splitting field L, and if so, what is the minimal possible value of trdeg_{Z(A)} L?

Actions of finite groups on central simple algebras have been extensively studied in the 1970s and 80s in the context of group actions on noncommutative rings; for an overview see [M]. More recently, torus actions were considered in [RV_{1}] and [RV_{2}], and actions of solvable groups in [V_{3}], all by purely algebraic methods (cf. also [V_{1}, V_{2}]). Inner actions of compact groups were studied in [Sa]. The purpose of this paper is to introduce a geometric approach to the subject by relating it to “birational invariant theory”, i.e., to the study of group actions on algebraic varieties, up to birational isomorphism. In particular, we will see that the questions posed in (1.1) are related to some of the central problems in birational invariant theory, such as existence of affine models, quotients, stabilizers in general positions, sections, etc. (For an overview of birational invariant theory, see [PV, Chapters 1, 2, 7] and [P, Part 1].) To make the algebro-geometric techniques applicable, we always assume that the centers of our simple algebras are finitely generated field extensions of a fixed algebraically closed base field k of characteristic zero. All algebraic groups are assumed to be linear and defined over k.

Let G be an algebraic group and A be a finite-dimensional central simple algebra. Of course, we are primarily interested in studying G-actions on A which respect the structure of G as an algebraic (and not just an abstract) group. The following definition is natural in the geometric context.

It is well known that a finitely generated field extension of k can be interpreted as the field of rational functions k(X) on some irreducible variety X, where X is unique up to birational isomorphism. Similarly, a central simple algebra A of degree n is isomorphic (as a k-algebra) to the algebra k_{n}(X) of PGL_{n}-equivariant rational functions X −−→ M_{n}(k), where X is an irreducible variety with a generically free PGL_{n}-action. Here X is unique up to birational isomorphism of PGL_{n}-varieties. For details, see [RV_{4}, Theorem 7.8 and Section 8].

We will say that a G-action on a central simple algebra A = k_{n}(X) is geometric, if it is induced by a regular G-action on X, via

(1.2)  (gf)(x) = f(g^{−1}x)

for x ∈ X in general position. One can check that all rational functions gf: X −−→ M_{n}(k) lie in k_{n}(X) (i.e., are PGL_{n}-equivariant) if and only if the actions of G and PGL_{n} on X commute. So a regular G-action on X induces a G-action on A = k_{n}(X) precisely if X is a G × PGL_{n}-variety. To sum up:

1.3. Definition. An action of an algebraic group G on a central simple algebra A of degree n is said to be geometric if there is an irreducible G × PGL_{n}-variety X such that A is G-equivariantly isomorphic to k_{n}(X). We will call X the associated variety for this action.

The second part of the definition makes sense since the associated variety X is unique up to birational isomorphism (as a G × PGL_{n}-variety); see Corollary 3.2. Note that the PGL_{n}-action on X is necessarily generically free, since A ≃ k_{n}(X) is a central simple algebra of degree n; see Lemma 2.8. Conversely, any G × PGL_{n}-variety X, which is PGL_{n}-generically free, is the associated variety for the geometric action of G on the central simple algebra A = k_{n}(X) given by (1.2).

From an algebraic point of view it is natural to consider another class of actions, introduced in [V_{3}, §2] (and in the special case of torus actions in [RV_{2}, §5]). We shall call such actions algebraic; for a precise definition, see Section 5. The relationship between algebraic and geometric actions is discussed in Sections 5 and 8. In particular, every algebraic action is geometric; see Theorem 5.3.

We are now ready to address the questions posed in (1.1), in the context of geometric actions.

1.4. Theorem. Consider a geometric action of an algebraic group G on a central simple algebra A of degree n, with associated G × PGL_{n}-variety X.

(a) The fixed algebra A^{G} is a central simple algebra of degree n if and only if for x ∈ X in general position,

Stab_{G×PGL_{n}}(x) ⊆ G × {1} .

(b) The fixed algebra A^{G} contains an element with n distinct eigenvalues if and only if for every x ∈ X in general position there exists a torus T_{x} of PGL_{n} such that

Stab_{G×PGL_{n}}(x) ⊆ G × T_{x} .

We now turn to question (b) in (1.1).

1.5. Theorem. Consider a geometric action of an algebraic group G on a central simple algebra A of degree n, with associated G × PGL_{n}-variety X.

(a) A has a G-invariant maximal étale subalgebra if and only if there exists a G × PGL_{n}-equivariant rational map X −−→ PGL_{n}/N, where N is the normalizer of a maximal torus in PGL_{n} and G acts trivially on the homogeneous space PGL_{n}/N.

(b) If A has a G-invariant maximal étale subalgebra, then for every x ∈ X in general position there exists a maximal torus T_{x} of PGL_{n} such that

Stab_{G×PGL_{n}}(x) ⊆ G × N(T_{x}) .

Here N(T_{x}) denotes the normalizer of T_{x} in PGL_{n}.

(c) If the orbit Gx has codimension < n^{2} − n in X for x ∈ X in general position, then A has no G-invariant maximal étale subalgebras.

Here by an étale subalgebra of A we mean a subalgebra of A which is an étale algebra over Z(A); cf. 2.9. If A is a division algebra, the maximal étale subalgebras are just the maximal subfields.

The converse to Theorem 1.5(b) is false in general; see Proposition 15.3. Note that the points of the homogeneous space PGL_{n}/N parameterize the maximal tori in PGL_{n} (see the beginning of §9). The converse to part (b) is thus true if and only if the tori T_{x} can be chosen so that x −−→ T_{x} is a rational map. We also remark that Theorem 1.4(b) gives a necessary and sufficient condition for A to have a G-invariant maximal étale algebra of the form Z(A)[a], where a ∈ A^{G}; see Corollary 6.3.

Our final result addresses question (c) in (1.1). We begin with the following definition.

1.6. Definition. Suppose a group G acts on a central simple algebra A of degree n. We will say that A is G-split, if A is G-equivariantly isomorphic to M_{n}(Z(A)) = M_{n}(k) ⊗_{k} Z(A), where G acts via the second factor. We will say that a G-equivariant field extension L/ Z(A) is a G-splitting field for A if A ⊗_{Z(A)} L is G-split.

Note that if G acts trivially on A, then a G-splitting field is just a splitting field for A in the usual sense. Note also that a G-action on a split central simple algebra (i.e., a matrix algebra over a field) need not be G-split (cf. Example 6.2).

1.7. Theorem. Every geometric action of an algebraic group G on a central simple algebra A of degree n has a G-splitting field of the form L = k(X_{0}), where X_{0} is a G-variety and trdeg_{Z(A)}(L) = n^{2} − 1. Moreover, if G acts algebraically on A, then X_{0} can, in addition, be chosen to be affine.

In general, the value of trdeg_{Z(A)} L given in Theorem 1.7 is the smallest possible; see Proposition 13.1(b). If G is connected, we give a different construction of G-splitting fields in Section 11.

At the end of the paper we will present four examples illustrating our main results, Theorems 1.4, 1.5, and 1.7, and two appendices. Appendix A deals with inner actions on division algebras which need not be geometric, while Appendix B treats regular actions of algebraic groups (see Definition 5.1) on prime affine PI-algebras. Using Theorem 1.7, we show that such actions are “induced” by regular actions on commutative domains. Further results on geometric actions will appear in the paper [RV_{5}].

2. Preliminaries

2.1. Conventions. We work over a fixed algebraically closed base field k of characteristic zero. All algebras are k-algebras, and division algebras and central simple algebras are assumed to be finite-dimensional over their centers, which in turn are assumed to be finitely generated field extensions of k. All actions on algebras are by k-algebra automorphisms. Algebraic groups are always assumed to be linear algebraic groups over k, and G will always denote an algebraic group. Regular actions are meant to be regular over k; similarly for algebraic actions (see Definition 5.2). If K is a field, we shall denote the algebra of n × n matrices over K by M_{n}(K). If K = k, we will write M_{n} in place of M_{n}(k). We will sometimes view M_{n} as a k-algebra and sometimes as an algebraic variety, isomorphic to the affine space 𝔸^{n^{2}}.

2.2. G-varieties. By a G-variety X we mean an algebraic variety with a regular action of G. By a morphism X ⟶ Y of G-varieties, we mean a Gequivariant morphism. The notions of isomorphism, rational map, birational isomorphism, etc. of G-varieties are defined in a similar manner. As usual, given a G-action on X, we denote the orbit of x ∈ X by Gx and the stabilizer subgroup of x by Stab_{G}(x) ⊆ G. Throughout this paper we use [PV] as a reference for standard notions from invariant theory, such as rational and categorical quotients, stabilizers in general position, sections, etc.

2.3. Definition. We shall say that a G-action on X is (a) faithful if every 1 ≠ g ∈ G acts nontrivially on X,

(b) generically free if Stab_{G}(x) = {1} for x ∈ X in general position, and (c) stable if the orbit Gx is closed in X for x ∈ X in general position.

2.4. Lemma. Suppose the group G is either (a) finite or (b) diagonalizable. Then every faithful irreducible G-variety X is generically free.

Proof. (a) Since the G-action is faithful, X^{g} = {x ∈ X | gx = x} ≠ X for every 1 ≠ g ∈ G. Since each X^{g} is a closed subvariety of X, every point of the Zariski dense open subset X − ∪_{1≠g∈G} X^{g} has a trivial stabilizer in G.

Part (b) is an immediate corollary of a theorem of Richardson [Ri, Theorem 9.3.1]; see also [PV, Theorem 7.1]. □

The following example shows that, contrary to the assertion in [PV, Proposition 7.2], Lemma 2.4 fails if we only assume that the connected component of G is a torus. We shall return to this example in §14.

2.5. Example. Consider the natural linear action of the orthogonal group G = O_{2} on 𝔸^{2}. This action is faithful but not generically free: Stab_{G}(v) has order 2 for v ∈ k^{2} in general position. Indeed, for every non-isotropic vector v in k^{2}, there is a unique non-trivial element of O_{2}, leaving v invariant; this element is the orthogonal reflection in v. Note also O_{2} is a semidirect product of a one-dimensional torus with ℤ/2ℤ.

2.6. Lemma (Popov). Let G be a reductive group, X be an affine G-variety and V be a G-representation. Suppose the G-orbit of x ∈ X is closed in X and Stab(x) ⊆ Stab(v) for some v ∈ V. Then there exists a G-invariant morphism f: X ⟶ V such that f(x) = v.

Proof. In the case where Stab(x) = {1}, this lemma is stated and proved in [P, Theorem 1.7.12]. The same argument goes through in our slightly more general setting. □

2.7. Algebras of rational maps. If X is a PGL_{n}-variety, we will denote by RMaps_{PGL_{n}}(X, M_{n}) the k-algebra of PGL_{n}-equivariant rational maps f: X −−→ M_{n}, with addition and multiplication induced from M_{n}.

2.8. Lemma. Let Y be an irreducible PGL_{n}-variety. Then the following are equivalent:

(a) The PGL_{n}-action on Y is generically free.

(b) A = RMaps_{PGL_{n}}(Y, M_{n}) is a central simple algebra of degree n. If (a) and (b) hold then the center of A is RMaps_{PGL_{n}}(Y, k) = k(Y)^{PGL^{n}}. Here elements of k are identified with scalar matrices in M_{n}.

Proof. (b) ⇒ (a): Note that the center of A contains k(Y)^{PGL^{n}}. Choose f_{1}, …, f_{n^{2}} ∈ A which are linearly independent over k(Y)^{PGL^{n}}. By [Re, Lemma 7.4], f_{1}(y), …, f_{n^{2}}(y) are k-linearly independent in M_{n} for y ∈ Y in general position. Now consider the PGL_{n}-equivariant rational map

f = (f_{1}, …, f_{n^{2}}): Y −−→ (M_{n})^{n^{2}} .

For y ∈ Y in general position, Stab(f(y)) = {1}, so that also Stab(y) = {1}. Hence Y is PGL_{n}-generically free.

The implication (a) ⇒ (b) and the last assertion of the lemma are proved in [Re, Lemma 8.5] (see also [Re, Definition 7.3 and Lemma 9.1]). □

If the PGL_{n}-action on X is generically free, we will denote the central simple algebra RMaps_{PGL_{n}}(X, M_{n}) by k_{n}(X).

2.9. Maximal étale subalgebras. Let A be a central simple algebra of degree n. By an étale subalgebra of A we mean a subalgebra of A which is an étale algebra over Z(A), i.e., a finite direct sum of (separable) field extensions of Z(A). Note that since we are working in characteristic zero, the term “étale” could be replaced by “commutative semisimple”. We are interested in maximal étale subalgebras, i.e., étale subalgebras E of A satisfying the following equivalent conditions:

(a) dim_{Z(A)} E = deg(A),

(b) E is maximal among commutative subalgebras of A; cf. [Ro_{2}, Exercise 7.1.1]. Using the double centralizer theorem, one easily verifies that every étale subalgebra of A is contained in a maximal étale subalgebra, see, e.g., [J, Theorem 4.10 and Exercise 4.6.12] and [Ro_{2}, Exercise 7.1.2]. Of course, if A is a division algebra, then maximal étale subalgebras are just maximal subfields.

We will repeatedly use the following characterization of maximal étale subalgebras, which follows easily from [B, §V.7.2, Proposition 3].

2.10. Lemma. Let A be a central simple algebra of degree n with center K. Let a ∈ A. Then K[a] is a maximal étale subalgebra of A if and only if the eigenvalues of a are distinct. □

3. The uniqueness of the associated variety

Recall that given a generically free PGL_{n}-variety X, we write A = k_{n}(X) for the algebra of PGL_{n}-equivariant functions a: X −−→ M_{n}. A PGL_{n}- equivariant dominant rational map f: X′ −−→ X induces an embedding f^{∗}: A ↪ A′ of central simple algebras, where A′ = k_{n}(X′) and f(a) = a ∘ f: X′ −−→ M_{n}.

We now deduce a simple consequence of the functoriality of the maps X ↦ k_{n}(X) and f ↦ f^{∗}; see [RV_{4}, Theorem 1.2]. Recall that if X has a Gaction, which commutes with the PGL_{n}-action, then (1.2) defines a G-action on A = k_{n}(X), which we call geometric.

3.1. Lemma. Let X and X′ be G × PGL_{n}-varieties, which are PGL_{n}- generically free.

(a) If f: X′ −−→ X is a dominant rational map of G × PGL_{n}-varieties then the induced embedding f^{∗}: k_{n}(X) ↪ k_{n}(X′) of central simple algebras is G-equivariant.

(b) Every G-equivariant embedding j: k_{n}(X) ↪ k_{n}(X′) induces a dominant rational G × PGL_{n}-equivariant map j_{∗}: X′ −−→ X.

Proof. (a) By [RV_{4}, Theorem 1.2], since the diagram

\xymatrix{X′ \ar@{-->}[r]^{f} \ar[d]^{g} & X \ar[d]^{g} \\ X′ \ar@{-->}[r]^{f} & X}

commutes for every g ∈ G, so does the induced diagram

\xymatrix{k_{n}(X) \ar[r]^{f^{∗}} \ar[d]^{g^{−1}} & k_{n}(X′) \ar[d]^{g^{−1}} \\ k_{n}(X) \ar[r]^{f^{∗}} & k_{n}(X′).}

(b) Conversely, since the diagram

\xymatrix{k_{n}(X) \ar[r]^{j} \ar[d]^{g^{−1}} & k_{n}(X′) \ar[d]^{g^{−1}} \\ k_{n}(X) \ar[r]^{j} & k_{n}(X′)}

commutes, so does the induced diagram

\xymatrix{X′ \ar@{-->}[r]^{j_{∗}} \ar[d]^{g} & X \ar[d]^{g} \\ X′ \ar@{-->}[r]^{j_{∗}} & X}

□

3.2. Corollary. Given a geometric action of an algebraic group G on a central simple algebra A, the G × PGL_{n}-variety associated to this action is unique up to birational isomorphism.

Proof. Suppose two G×PGL_{n}-varieties X and X′ are both associated varieties for this action, i.e., k_{n}(X) and k_{n}(X′) are both G-equivariantly isomorphic to A. In other words, there are mutually inverse G-equivariant algebra isomorphism i: k_{n}(X) ⟶^{≃} k_{n}(X′) and j: k_{n}(X′) ⟶^{≃} k_{n}(X). Applying Lemma 3.1, i and j induce mutually inverse dominant G×PGL_{n}-equivariant rational map i_{∗}: X′ −−→^{≃} X and j_{∗}: X −−→^{≃} X′. We conclude that X and X′ are birationally isomorphic G × PGL_{n}-varieties. □

3.3. Example. Let G be a subgroup of PGL_{n}, and consider the conjugation action of G on A = M_{n}(k). We claim that the associated G × PGL_{n}-variety for this action is X = PGL_{n}, with G acting by translations on the right and PGL_{n} acting by translations on the left. More precisely, for (g, h) ∈ G × PGL_{n} and x ∈ X, (g, h) · x = hxg^{−1}. Consequently for f ∈ k_{n}(X),

(g · f)(x) = f((g, 1)^{−1} · x) = f(xg) ,

see (1.2). Note that since X is a single PGL_{n}-orbit, every PGL_{n}-equivariant rational map f: PGL_{n} −−→ M_{n} is necessarily regular. It is now easy to check that the k-algebra isomorphism

ϕ: k_{n}(X) = RMaps_{PGL_{n}}(PGL_{n}, M_{n}) ⟶^{≃} A = M_{n}

given by ϕ(f) = f(1) is G-equivariant.

3.4. Example. Let m ≥ 2, and consider the PGL_{n}-variety X = (M_{n})^{m}, where PGL_{n} acts by simultaneous conjugation, i.e., via

g · (a_{1}, …, a_{m}) = (ga_{1}g^{−1}, …, ga_{m}g^{−1}) .

Since m ≥ 2, this action is generically free. The associated division algebra k_{n}(X) is called the universal division algebra of m generic n × n-matrices and is denoted by UD(m, n). Identify the function field of X with k(x_{ij}^{(h)}),

where for each h = 1, …, m, x_{ij}^{(h)} are the n^{2} coordinate functions on copy number h of M_{n}, and identify the algebra of all rational maps X −−→ M_{n} with M_{n}(k(x_{ij}^{(h)})). Now we can think of UD(m, n) as the division subalgebra

of M_{n}(k(x_{ij}^{(h)})) generated by the m generic n × n matrices X^{(h)} = (x_{ij}^{(h)}), h = 1, …, m. Here X^{(h)} corresponds to projection (M_{n})^{m} ⟶ M_{n} given by (a_{1}, …, a_{m}) ↦ a_{h}. For details of this construction, see [Pr_{2}, Section 2] or [lB, Theorem 5].

Now observe that the GL_{m}-action on X = (M_{n})^{m} given by

(3.5)  g · (a_{1}, …, a_{m}) = (∑_{j=1}^{m} g_{1j}a_{j}, …, ∑_{j=1}^{m} g_{mj}a_{j})

commutes with the above PGL_{n}-action. Here g = (g_{ij}) ∈ GL_{m}, with g_{ij} ∈ k. Using formula (1.2), we see that this gives rise to a GL_{m}-action on UD(m, n) such that for g ∈ GL_{m},

(3.6)  g · X^{(h)} = ∑_{j=1}^{m} g′_{hj}X^{(j)} ,

where g^{−1} = (g′_{ij}). In other words, this GL_{m}-action on UD(m, n) is geometric, with associated G × PGL_{n}-variety X = (M_{n})^{m}. We will return to this important example later in this paper (in Example 5.5 and Sections 13 and 14), as well as in [RV_{5}].

3.7. Remark. The k-subalgebra of UD(m, n) generated by X^{(1)}, …, X^{(m)} is called the generic matrix ring generated by m generic n × n matrices; we denote it by G_{m,n}. Note that the action (3.6) of GL_{m} on UD(m, n) restricts to an action on G_{m,n}. Consequently, the GL_{m}-action on G_{m,n} is induced by the GL_{m}-action on (M_{n})^{m} in the sense of formula (1.2).

4. Brauer-Severi varieties

Let A/K be a central simple algebra of degree n. Throughout much of this paper, we associate to A a PGL_{n}-variety X/k such that A is the algebra of PGL_{n}-equivariant rational maps X −−→ M_{n}(k). Another variety that can be naturally associated to A is the Brauer-Severi variety BS(A), defined over K. Any algebra automorphism g: A ⟶ A, defined over the base field k, induces k-automorphisms of K and BS(A) such that the diagram

\xymatrix{BS(A) \ar[r]^{g_{∗}} \ar[d] & BS(A) \ar[d] \\ Spec(K) \ar[r]^{(g_{| K})_{∗}} & Spec(K) ,}

commutes; conversely, g can be uniquely recovered from this diagram. If a group G acts on A, it is natural to ask if BS(A) can be G-equivariantly represented by an algebraic variety over k. In this short section we will address this question, following a suggestion of the referee. Our main result, Proposition 4.1 below, will not be used in the sequel.

4.1. Proposition. Consider a geometric action ϕ of an algebraic group G on a central simple algebra A/K of degree n. Then there exists a morphism σ: S ⟶ Y of irreducible G-varieties (of finite type over k) such that

(a) S is a Brauer-Severi variety over Y;

(b) k(Y) = K and σ^{−1}(η) is the Brauer-Severi variety of A, where η is the generic point of Y;

(c) the G-actions on S and Y induce the action ϕ on A.

Proof. Let X be the G×PGL_{n}-variety associated to ϕ and H be the maximal parabolic subgroup of PGL_{n} consisting of matrices of the form

(\begin{matrix}∗ & 0 & … & 0 \\ ∗ & ∗ & … & ∗ \\ ⋮ & ⋮ & & ⋮ \\ ∗ & ∗ & … & ∗ \end{matrix}) .

Consider the natural dominant rational map σ: X/H −−→ X/PGL_{n} given by the inclusion k(X)^{PGL^{n}} ↪ k(X)^{H}. Recall that the rational quotient varieties X/H and X/PGL are a priori only defined up to birational isomorphism. However, we can choose models for these varieties such that the induced G-actions are regular; cf. [PV, Proposition 2.6 and Corollary 1.1]. For notational convenience, we will continue to denote these G-varieties by X/H and X/PGL_{n}. Note also that since the actions of G and PGL_{n} on X commute, the resulting map σ: X/H −−→ X/PGL_{n} is G-equivariant.

By [RV_{4}, Section 9], X/H is a Brauer-Severi variety over a dense open subset U of X/PGL_{n}, and is isomorphic to BS(A) over the generic point of X/PGL_{n}. Since σ is G-equivariant, X/H is a Brauer-Severi variety over g(U), for every g ∈ G. Setting Y to be the union of the g(U) inside X/PGL_{n}, as g ranges over G, and setting S to be the preimage of this set in X/H, we obtain a G-equivariant morphism σ: S ⟶ Y with desired properties. □

5. Algebraic actions

5.1. Definition. We shall say that the action of an algebraic group G on a (not necessarily commutative) k-algebra R is regular^{1}, if every finitedimensional k-subspace of R is contained in a G-invariant finite-dimensional k-subspace V, such that the G-action on V induces a homomorphism G ⟶ GL(V) of algebraic groups.

Every regular action of a connected algebraic group on a division algebra (or even a field) must be trivial (see, e.g., [V_{2}, A.1]), so this notion is too restrictive for our purposes. However, it naturally leads to the following definition, made in [V_{3}, §2]. (The special case where G is a torus had been considered earlier in [RV_{2}, §5].)

1Such actions are usually called rational; we prefer the term regular, since the term “rational action” has a different meaning in the context of birational invariant theory.

5.2. Definition. Let G be an algebraic group acting on a k-algebra A by k-algebra automorphisms. We call the action algebraic^{2} (over k) if there is a G-invariant subalgebra R of A and a G-invariant multiplicatively closed subset S of R consisting of central nonzerodivisors of R such that (1) G acts regularly on R, and (2) A = RS^{−1}.

Note that a regular action on A is algebraic (use S = {1}). We shall be primarily interested in the case where A is a central simple algebra; in this case R is an order in A (and in particular, R is prime). For basic properties of algebraic actions, see [V_{3}, §2].

The purpose of this section is to investigate the relationship between algebraic and geometric actions (cf. Definition 1.3).

5.3. Theorem. (a) Algebraic actions are geometric.

(b) Let G be an algebraic group acting geometrically on a central simple algebra A of degree n. Then the action of G on A is algebraic if and only if there is an associated G×PGL_{n}-variety X with the following two properties: X is affine, and the PGL_{n}-action on X is stable (cf. Definition 2.3(c)).

We begin with a result which is a G-equivariant version of [RV_{4}, Theorem 6.4].

5.4. Proposition. Let G be an algebraic group acting regularly on a finitely generated prime k-algebra R of PI-degree n. Then there is an n-variety Y with a regular G-action such that R is G-equivariantly isomorphic to k_{n}[Y].

See [RV_{4}, 3.1] for the definition of k_{n}[Y], the PI-coordinate ring of Y. The action of G on k_{n}[Y] is induced from the action of G on Y as in formula (1.2).

Proof. We may assume that G acts faithfully on R. There is a finitedimensional G-stable k-subspace W of R which generates R as a k-algebra. Set m = dim_{k}(W), and consider the generic matrix ring G_{m,n} with its GL_{m}- action as in Remark 3.7. Denote by V the k-subspace of G_{m,n} generated by the m generic n × n matrices. Let ψ_{0} : V → W be a k-vector space isomorphism. Define a regular action of G on V by making ψ_{0} G-equivariant. The action of G on V extends to a regular action on G_{m,n}. By the universal mapping property of G_{m,n}, ψ_{0} extends to a G-equivariant surjective k-algebra homomorphism ψ: G_{m,n} → R. Replacing G by an isomorphic subgroup of GL_{m}, we may assume that G acts on V as in (3.6). Then the action of G on G_{m,n} is induced (as in (1.2)) from the action of G on (M_{n})^{m} given by (3.5). Note that the actions of G and PGL_{n} on (M_{n})^{m} commute.

2In [V_{3}], S is not required to be central; it is, however, proved there that S can always be chosen to be central if A is a central simple algebra.

Let I be the kernel of ψ, and let Y = Z(I) ⊂ (M_{n})^{m} be the irreducible n-variety associated to I, see [RV_{4}, Corollary 4.3]. Note that Y is G-stable for the action of G on (M_{n})^{m}. By [RV_{4}, Proposition 5.3], I(Y) = I, so that R is G-equivariantly isomorphic to k_{n}[Y] = G_{m,n}/I(Y) = G_{m,n}/I. □

Proof of Theorem 5.3. (a) Let G be an algebraic group acting algebraically on a central simple algebra A of degree n. Let R be a G-stable finitely generated prime PI-algebra contained in A such that A is the total ring of fractions of R. By Proposition 5.4, there is an n-variety Y with a regular action of G such that R is G-equivariantly isomorphic to k_{n}[Y]. Then A is G-equivariantly isomorphic to the total ring of fractions of k_{n}[Y], i.e., to k_{n}(Y), see [RV_{4}, Proposition 7.3]. As the proof of Proposition 5.4 showed, (M_{n})^{m} is a G × PGL_{n}-variety (where G acts via some subgroup of GL_{m} as in (3.5)), and Y is a G-stable subset of (M_{n})^{m}. Hence the closure X of Y in (M_{n})^{m} is an affine G × PGL_{n}-variety. It is clear that k_{n}(Y) and k_{n}(X) are G-equivariantly isomorphic, and that the PGL_{n}-action on X is generically free and stable. So G acts geometrically on A, and the associated G × PGL_{n}-variety X has the two additional properties from part (b).

(b) If the action of G on A is algebraic then an associated G × PGL_{n}- variety X with desired properties was constructed in the proof of part (a).

Conversely, assume that there is an associated G×PGL_{n}-variety X which is affine and on which the PGL_{n}-action is stable. We may assume that A = k_{n}(X). So A is a central simple algebra with center K = k(X)^{PGL^{n}}; cf. Lemma 2.8. Since X is affine, and since PGL_{n}-orbits in X in general position are closed, k[X]^{PGL^{n}} separates PGL_{n}-orbits in general position, so that Q(k[X]^{PGL^{n}}) = k(X)^{PGL^{n}} = K; see [PV, Lemma 2.1]. (Here Q stands for the quotient field.) Denote by R the subalgebra of A consisting of the regular PGL_{n}-equivariant maps X → M_{n}. It is clearly G-invariant. Note that G acts regularly on k[X]. Consequently, G acts regularly on M_{n}(k[X]), the set of regular maps X → M_{n}. Hence, G also acts regularly on its subalgebra R. It remains to show that R is a prime subalgebra of A, and that its total ring of fractions is equal to A.

Let v ∈ (M_{n})^{2} be a pair of matrices generating M_{n} as k-algebra, and let x ∈ X be such that its stabilizer in PGL_{n} is trivial and such that its PGL_{n}- orbit is closed. Then by Lemma 2.6, there is a PGL_{n}-equivariant regular map X → (M_{n})^{2} such that f(x) = v. Write f = (f_{1}, f_{2}), where f_{1} and f_{2} are PGL_{n}-equivariant regular maps X ⟶ M_{n}, i.e., elements of R. Since f_{1}(x) and f_{2}(x) generate M_{n}, the central polynomial g_{n} ([Ro_{1}, p. 26]) does not vanish on R. Since g_{n} is t^{2}-normal, it vanishes on every proper K-subspace of A, see [Ro_{1}, 1.1.35]. Consequently RK = A, and R is prime and has PIdegree n. Clearly, R contains k[X]^{PGL^{n}}. Since R Q(k[X]^{PGL^{n}}) = RK = A, Q(R) = A. Hence, G acts algebraically on A. □

5.5. Example. It follows easily from Definition 5.2 that the action (3.6) of GL_{m} on UD(m, n) is algebraic. So by Theorem 5.3(b), there is an associated GL_{m} × PGL_{n}-variety X with the following two properties: X is affine, and the PGL_{n}-action on X is stable. Indeed, the natural associated variety X = (M_{n})^{m} has these properties.

6. Proof of Theorem 1.4

We begin with the following simple observation:

6.1. Remark. Consider a geometric action of an algebraic group G on a central simple algebra A, with associated G × PGL_{n}-variety X. Elements of A are thus PGL_{n}-equivariant rational maps a: X −−→ M_{n}. Such an element is G-fixed if and only if it factors through the rational quotient map X −−→ X/G. In other words, A^{G} is isomorphic to RMaps_{PGL_{n}}(X/G, M_{n}).

We are now ready to proceed with the proof of Theorem 1.4.

(a) We may assume that A = k_{n}(X). Combining Remark 6.1 with Lemma 2.8, we see that A^{G} is a central simple algebra of degree n if and only if Y = X/G is a generically free PGL_{n}-variety. The latter condition is equivalent to Stab_{G×PGL_{n}}(x) ⊆ G × {1} for x ∈ X in general position.

(b) First suppose that there is an a ∈ A^{G} with n distinct eigenvalues. Adding to a some constant in k, we may assume that the eigenvalues of a are distinct and nonzero. Hence for x ∈ X in general position, the eigenvalues of a(x) ∈ M_{n} are also distinct and nonzero. The stabilizer of a(x) in PGL_{n} is thus a maximal torus T_{x} of PGL_{n}. Let (g, p) ∈ Stab_{G×PGL_{n}}(x). Then a(x) = g(a)(x) = a(g^{−1}(x)) = a(p(x)) = p a(x)p^{−1}. Thus p ∈ T_{x}, so that Stab_{G×PGL_{n}}(x) ⊆ G × T_{x}.

We will now prove the converse. Assume Stab_{G×PGL_{n}}(x) is contained in G × T_{x} for some torus T_{x} of PGL_{n} (depending on x). Denote by Y the rational quotient PGL_{n}-variety X/G. To produce an a ∈ A^{G} with distinct eigenvalues, it suffices to construct a PGL_{n}-equivariant rational map a: Y −−→ M_{n} whose image contains a matrix with distinct eigenvalues. By our assumption, Stab_{PGL_{n}}(y) is contained in a torus T_{x} ⊂ PGL_{n} for y ∈ Y in general position. Hence, Stab_{PGL_{n}}(y) is diagonalizable (and, in particular, reductive). By [RV_{3}, Theorem 1.1], after replacing Y by a birationally equivalent PGL_{n}-variety, we may assume that Y is affine and the PGL_{n}- action on Y is stable.

We are now ready to construct a map a: Y −−→ M_{n} with the desired properties. Let y ∈ Y be a point whose orbit is closed and whose stabilizer S is diagonalizable, and let v ∈ M_{n} be a matrix with distinct eigenvalues. Then Stab(v) is a maximal torus in PGL_{n}; after replacing v by a suitable conjugate, we may assume S ⊆ Stab(v). Now Lemma 2.6 asserts that there exists a PGL_{n}-equivariant morphism a: Y ⟶ M_{n} such that a(y) = v. This completes the proof of Theorem 1.4. □

6.2. Example. Let G be a subgroup of PGL_{n}, acting by conjugation on A = M_{n}(k). The associated variety for this action is X = PGL_{n}, with G × PGL_{n} acting on it by (g, h) · x = hxg^{−1}; see Example 3.3. Since all of X is a single PGL_{n}-orbit, the stabilizer of any x ∈ X is conjugate to the stabilizer of 1_{PGL_{n}}, which is easily seen to be {(g, g) | g ∈ G}. So in this setting, Theorem 1.4(b) reduces to the following familiar facts:

(a) M_{n}(k)^{G} = M_{n}(k) if and only if G = {1}, and

(b) M_{n}(k)^{G} contains an element with n distinct eigenvalues if and only if G centralizes a maximal torus in GL_{n}, i.e., if and only if G is contained in maximal torus of PGL_{n}.

Using Lemma 2.10, we can rephrase Theorem 1.4(b) in a way that makes its relationship to Question 1.1(b) more transparent.

6.3. Corollary. Consider a geometric action of an algebraic group G on a central simple algebra A of degree n, with associated G × PGL_{n}-variety X. The following conditions are equivalent.

(a) A has a maximal étale subalgebra E of the form E = Z(A)[a] for some a ∈ A^{G}.

(b) A^{G} contains a separable element of degree n over Z(A).

(c) For x ∈ X in general position, Stab_{G×PGL_{n}}(x) is contained in G×T_{x}, where T_{x} is a torus in PGL_{n}. □

Here by a separable element of A we mean an element whose minimal polynomial over Z(A) is separable, i.e., has no repeated roots.

6.4. Remark. It is necessary in Corollary 6.3(b) to require that a is separable over Z(A). Indeed, in Example 6.2 set n = 2, A = M_{2}(k) and G = \{( \begin{matrix}1 & g \\ 0 & 1 \end{matrix} ) | g ∈ k\}. Then the fixed algebra A^{G} consists of all matrices of the form ( \begin{matrix}a & b \\ 0 & a \end{matrix} ) with a, b ∈ k. In particular, A^{G} contains elements of degree n = 2 over Z(A) = k, but the minimal polynomial of any such element has repeated roots.

7. The G-action on the center of A

Throughout this section, we consider a geometric action of an algebraic group G on a central simple algebra A of degree n with associated G×PGL_{n}- variety X. It is sometimes possible to deduce information about the G-action on A from properties of the G-action on the center Z(A). In this section, we find conditions on the G-action on Z(A) which allow us to answer question (a) in (1.1).

Recall that the field of rational functions on X/PGL_{n} is G-equivariantly isomorphic to the center Z(A) of A (see Lemma 2.8). Of course, a priori X/PGL_{n} is only defined up to birational isomorphism. From now on we will fix a particular model W equipped with a regular G-action and a Gequivariant rational quotient map for the PGL_{n}-action on X

π: X −−→ W .

It will not matter in the sequel which model W of X/PGL_{n} we use. Note that the G-variety W is just a birational model for the G-action on Z(A). In many (perhaps, most) cases, W is much easier to construct than X; for an example of this phenomenon, see Section 15.

We begin with a simple observation, relating stabilizers in X and W.

7.1. Lemma. Let X be a G × PGL_{n}-variety which is PGL_{n}-generically free. Denote by π: X −−→ X/PGL_{n} the rational quotient map for the PGL_{n}- action. Then for x ∈ X in general position, the projection G × PGL_{n} ⟶ G onto the first factor induces an isomorphism between Stab_{G×PGL_{n}}(x) and Stab_{G}(π(x)).

Proof. For x ∈ X in general position, π is defined at x, the fiber over π(x) is the orbit PGL_{n}x, and Stab_{PGL_{n}}(x) is trivial. For such x, the projection p restricts to a surjective map

Stab_{G×PGL_{n}}(x) ⟶ Stab_{G}(π(x))

whose kernel is Stab_{PGL_{n}}(x) = {1}, and the lemma follows. □

7.2. Proposition. (a) Suppose that for w ∈ W in general position, the stabilizer Stab_{G}(w) does not admit a non-trivial homomorphism to PGL_{n}. Then A^{G} is a central simple algebra of degree n = deg(A).

(b) Suppose that for w ∈ W in general position, Stab_{G}(w) is an abelian group consisting of semisimple elements and the n-torsion subgroup of Stab_{G}(w)/ Stab_{G}(w)^{0} is cyclic. Then there exists an a ∈ A^{G} with n distinct eigenvalues.

Note that the condition of part (a) is satisfied if the G-action on W is generically free.

Proof. (a) By Lemma 7.1, Stab_{G×PGL_{n}}(x) ⊆ G×{1} for x in general position in X. The desired conclusion follows from Theorem 1.4(a).

(b) Let H be the projection of Stab_{G×PGL_{n}}(x) to PGL_{n}. By Lemma 7.1, H is an abelian group consisting of semisimple elements, and H/H^{0} is a homomorphic image of Stab_{G}(w)/ Stab_{G}(w)^{0}. Using the fundamental theorem of finite abelian groups, one checks that surjective homomorphisms of finite abelian groups preserve the property that the n-torsion subgroup is cyclic. By [St, Corollary 2.25(a)], H is contained in a maximal torus of PGL_{n}. (Note that the torsion primes for PGL_{n} are the primes dividing n; see [St, Corollaries 1.13 and 2.7].) The desired conclusion now follows from Theorem 1.4(b). □

We will now use Proposition 7.2, to study inner actions. Recall that an automorphism ϕ of a central simple algebra A is called inner if there exists an invertible element a ∈ A such that ϕ(x) = axa^{−1} for every x ∈ A, and outer otherwise. By the Skolem-Noether theorem ϕ is inner if and only ϕ(x) = x for every x ∈ Z(A).

7.3. Corollary. Let G be a finite group or a torus acting geometrically on a central simple algebra A of degree n. The elements of G that act by inner automorphisms form a normal subgroup of G; denote this subgroup by N.

(a) If N = {1} (i.e., if G acts on A by outer automorphisms), then A^{G} is a central simple algebra of degree n.

(b) If N is a cyclic group, then there is an element a ∈ A^{G} with n distinct eigenvalues.

In the case where the group G is finite, part (a) is proved by algebraic means and under weaker hypotheses in [M, Theorem 2.7 and Corollary 2.10]. Note also that since every action of a finite group on a central simple algebra is algebraic (see Definition 5.2), our assumption that the action is geometric is only relevant if G is a torus. Moreover, if G is a torus then every geometric action is algebraic; see Corollary 8.4.

Proof. We may assume that the action is faithful. Indeed, if K ⊆ G is the kernel of this action, we can replace G by G/K and N by N/K.

Now let W be an irreducible G-variety whose function field k(W) is Gequivariantly isomorphic to Z(A) (over k); see the beginning of Section 7. Clearly an element of G acts trivially on W if and only if it acts on A by an inner automorphism. Now recall that if G is a finite group or a torus then the stabilizer in general position for the G-action on W is precisely the kernel N of this action; cf. Lemma 2.4.

The desired conclusions in parts (a) and (b) now follow from parts (a) and (b) of Proposition 7.2, respectively. □

8. Which geometric actions are algebraic?

Theorem 5.3(a) says that every algebraic action is geometric. It is easy to see that the converse is not true. For example, let Y be a generically free PGL_{n}-variety (e.g., we can take Y = PGL_{n} where PGL_{n} acts on Y by translations), and consider the G × PGL_{n}-variety X = (G/P) × Y, where G is a non-solvable connected algebraic group, and P is a proper parabolic subgroup. Here G acts by translations on the first factor, and PGL_{n} acts on the second factor. Since the PGL_{n}-action on X is generically free, A = k_{n}(X) is a central simple algebra of degree n. On the other hand, since G/P is complete, it is easy to see that X is not birationally isomorphic to an affine G × PGL_{n}-variety; hence by Theorem 5.3(b), this action is not algebraic.

Nevertheless, we will now show that under fairly mild assumptions, the converse of Theorem 5.3(a) holds, i.e., every geometric action is, indeed, algebraic.

8.1. Lemma. Let G be an algebraic group, and let X be an irreducible G × PGL_{n}-variety which is PGL_{n}-generically free. Assume that X has a stable affine model as a G×PGL_{n}-variety. Then the induced action of G on k_{n}(X) is algebraic.

Proof. We may assume without loss of generality that X itself is affine and stable as a G×PGL_{n}-variety. By Theorem 5.3(b) it suffices to show that X is stable as a PGL_{n}-variety, i.e., that PGL_{n}-orbits in general position in X are closed. Let x ∈ X be a point in general position. Then the G × PGL_{n}-orbit (G × PGL_{n})x is closed in X and can be naturally identified with the homogeneous space (G×PGL_{n})/H, where H = Stab_{G×PGL_{n}}(x). The PGL_{n}-orbit (PGL_{n}) · x is then identified with the image Z of PGL_{n} in (G × PGL_{n})/H. It thus remains to show that Z is closed in (G×PGL_{n})/H. Indeed, Z is also the image of the product PGL_{n}H, which is a closed subgroup of G × PGL_{n} (because PGL_{n} is normal; see [H, §7.4]). Since PGL_{n}H is a closed subgroup of G × PGL_{n} containing H, its image Z in (G × PGL_{n})/H is closed; see [H, §12.1]. □

8.2. Corollary. Let G be an algebraic group, and let X be an irreducible G × PGL_{n}-variety which is G × PGL_{n}-generically free. Then the induced action of G on k_{n}(X) is algebraic.

Proof. By [RV_{3}, Theorem 1.2(i)] X has a stable affine birational model as a G × PGL_{n}-variety. The desired conclusion is now immediate from Lemma 8.1. □

The criterion for a geometric action to be algebraic given by Lemma 8.1 can be further simplified by considering the G-action on the center of A, as in Section 7.

8.3. Proposition. Consider a geometric action of an algebraic group G on a central simple algebra A, and let W be a birational model for the G-action on Z(A). Then the G-action on A is algebraic, provided one of the following conditions holds:

(a) The G-action on W is generically free.

(b) The normalizer H = N_{G}(G_{w}) is reductive for w ∈ W in general position.

(c) G is reductive and the stabilizer G_{w} is reductive for w ∈ W in general position.

(d) G is reductive and W has a stable affine model as G-variety.

Proof. Let X be an associated G × PGL_{n}-variety for the G-action on A. Recall that the PGL_{n}-action on X is generically free and W is the rational quotient X/PGL_{n}. In view of Lemma 8.1, it suffices to show that X has a stable affine model as a G × PGL_{n}-variety.

(a) Immediate from Corollary 8.2 and Lemma 7.1.

(b) Choose x ∈ X in general position, and set w = π(x) ∈ W. Let S_{x} = Stab_{G×PGL_{n}}(x). We claim that N_{G×PGL_{n}}(S_{x}) is reductive for x ∈ X in general position. The desired conclusion follows from this claim by [RV_{3}, Theorem 1.2(ii)].

The proof of the claim is based on two simple observations. First of all, if H = N_{G}(G_{w}) is reductive, then so is S_{x} ≃ G_{w}. Indeed, the unipotent radical of R_{u}(G_{w}) is characteristic in G_{w}, hence, normal in H. Since H is reductive, this implies R_{u}(G_{w}) = {1}, i.e., G_{w} is reductive, as claimed.

Secondly, by Lemma 7.1, the normalizer N_{G×PGL_{n}}(S_{x}) is a priori contained in H × PGL_{n}, i.e.,

N_{G×PGL_{n}}(S_{x}) = N_{H×PGL_{n}}(S_{x}) .

Since both H × PGL_{n} and S_{x} are reductive, the normalizer N_{H×PGL_{n}}(S_{x}) is reductive as well; see [LR, Lemma 1.1]. This concludes the proof of the claim and thus of part (b).

(c) If G and G_{w} are both reductive then using [LR, Lemma 1.1] once again we see that N_{G}(G_{w}) is also reductive. Part (c) now follows from part (b).

(d) After replacing W by a stable affine model, we see that for w ∈ W in general position, the orbit Gw ≃ G/G_{w} is affine, so that G_{w} is reductive by Matsushima’s theorem, see [PV, Theorem 4.17]. Now use part (c). □

8.4. Corollary. Let G be an algebraic group whose connected component is a torus. Then every geometric action of G on a central simple algebra is algebraic.

Proof. In this case, every subgroup of G is reductive, so that part (c) of Proposition 8.3 applies. □

9. Proof of Theorem 1.5

9.1. The generic torus. Let T be a maximal torus in GL_{n}, and let N be the normalizer of the image of T in PGL_{n}. Since PGL_{n} permutes the maximal tori in GL_{n} transitively, one can think of PGL_{n}/N as the variety of maximal tori of GL_{n} (or equivalently, of PGL_{n}). We briefly recall how one can construct a PGL_{n}-equivariant rational map

π: M_{n} −−→ PGL_{n}/N

which sends a non-singular matrix α ∈ M_{n} with distinct eigenvalues to the unique maximal torus in GL_{n} containing α. The map π is sometimes called the generic torus of GL_{n}; cf. [Vos, 4.1].

Denote by Gr(n, n^{2}) the Grassmannian of n-dimensional subspaces of M_{n}. The action of PGL_{n} on M_{n} induces a regular action of PGL_{n} on Gr(n, n^{2}). Define a rational, PGL_{n}-equivariant map π_{1} : M_{n} −−→ Gr(n, n^{2}) by sending a non-singular matrix α with distinct eigenvalues to Span(1, α, …, α^{n−1}). The unique maximal torus T_{(α)} of GL_{n} containing α is characterized by Span(T_{(α)}) = π_{1}(α). The image of π_{1} consists thus of a single PGL_{n}-orbit O. Since the stabilizer of both T and Span(T) is N, gN ↦ g Span(T)g^{−1} defines an isomorphism π_{2} : PGL_{n}/N → O. Here T is the maximal torus in GL_{n} which we chose (and fixed) at the beginning of this section and N is the normalizer of the image of T in PGL_{n}. Now π = π_{2}^{−1} ∘ π_{1} is a PGL_{n}- equivariant rational map M_{n} −−→ PGL_{n}/N such that for any α as above, π(α) = gN if and only if gT g^{−1} is the unique torus of GL_{n} containing α.

9.2. Proof of Theorem 1.5. (a) Suppose A = k_{n}(X) has a G-invariant maximal étale subalgebra E. It follows easily from the primitive element theorem that there is an a ∈ E so that E = Z(A)[a]. Choose one such a. By Lemma 2.10, a has distinct eigenvalues. Adding some constant in k to a, we may assume that the eigenvalues of a are distinct and nonzero. Then for x ∈ X in general position a(x) is a matrix whose eigenvalues are distinct and nonzero. We now define a rational map φ: X −−→ PGL_{n}/N by φ(x) = π(a(x)). This map is PGL_{n}-equivariant by construction. Moreover, for every g ∈ G, g(a) ∈ E commutes with a. Thus, for x ∈ X in general position, a(x) and g^{−1}(a)(x) = a(g(x)) lie in the same maximal torus, and consequently, φ(x) = φ(g(x)).

Conversely, suppose there exists a G × PGL_{n}-equivariant rational map X −−→ PGL_{n}/N. After removing the indeterminacy locus from X, we may assume this map is regular. We may also assume that PGL_{n} acts freely on X. Let X_{0} be the preimage of the coset N ∈ PGL_{n}/N in X. Note that X_{0} is G × N-invariant, that X = PGL_{n} · X_{0}, and that the N-action on X_{0} is generically free. Moreover, X is birationally isomorphic as PGL_{n}-variety to PGL_{n} ∗_{N} X_{0}, see [P, Theorem 1.7.5].

Let Δ ≃ 𝔸^{n} be the variety of diagonal n×n-matrices. By [Re, Proposition 7.1] there exists an N-equivariant rational map a: X_{0} −−→ Δ whose image contains a matrix with distinct eigenvalues. (Note that here we use the fact that Δ is a vector space and N acts on it linearly.) This rational map then naturally extends to a PGL_{n}-equivariant rational map

X ≃ PGL_{n} ∗_{N} X_{0} −−→ PGL_{n} ∗_{N} Δ ≃ M_{n}

induced by (g, x_{0}) ↦ (g, a(x_{0})). By abuse of notation, we denote this extended rational map by a as well.

We now view a as an element of A = k_{n}(X). Since the image of a contains a matrix with distinct eigenvalues, Lemma 2.10 tells us that E = Z(A)[a] is a maximal étale subalgebra of A. It remains to show that E is G-invariant. To do this it suffices to prove that g(a) ∈ E for every g ∈ G. Since E = C_{A}(E), we only need to establish that g(a) commutes with a, i.e., that the commutator b = [a, g(a)] equals 0. Indeed, for any x ∈ X_{0},

b(x) = [a(x), a(g^{−1}(x))] = [a(x), a(y)] ,

where y = g^{−1}(x) ∈ X_{0}. By our construction a maps every element of X_{0} to a diagonal matrix. In particular, a(x) and a(y) commute, and thus b(x) = 0 for every x ∈ X_{0}. Since b is a PGL_{n}-equivariant rational map X −−→ M_{n} and since PGL_{n} · X_{0} = X, we conclude that b = [a, g(a)] is identically zero on X, as claimed. This completes the proof of part (a).

(b) The action of G×PGL_{n} on PGL_{n}/N has stabilizer of the form G×N(S) at every point, where S is a maximal torus of PGL_{n}. Part (b) is now an immediate consequence of part (a).

(c) Assume that A has a G-invariant maximal étale subalgebra. Let x ∈ X be a point in general position. We claim that

(9.3)  dim(Gx ∩ PGL_{n}x) ≤ n − 1 .

Indeed, Gx ∩ PGL_{n}x is easily seen to be the image of the morphism from Stab_{G×PGL_{n}}(x) to X given by (g, p) ↦ px. Since Stab_{G×PGL_{n}}(x) ⊆ G × N(T_{x}) by part (b), we conclude that

dim(Gx ∩ PGL_{n}x) ≤ dim N(T_{x}) = n − 1

as claimed.

Consider the rational quotient map π: X −−→ X/PGL_{n}. We may assume without loss of generality that π is defined at x. Now restrict π to the (welldefined) rational map π_{G^{0}x}: G^{0}x −−→ X/PGL_{n}, where G^{0} is the connected component of G. For y ∈ G^{0}x in general position, the fiber over π_{G^{0}x}(y) is G^{0}x ∩ PGL_{n}y = G^{0}y ∩ PGL_{n}y. By (9.3),

dim(Gx) = dim(G^{0}x) ≤ dim(X/PGL_{n}) + n − 1
= dim(X) − dim(PGL_{n}) + n − 1
= dim(X) − n^{2} + n .

So dim(X) − dim(Gx) ≥ n^{2} − n. This proves part (c). □

10. Proof of Theorem 1.7

We begin by spelling out what it means for an algebraic group action on a central simple algebra to be split in terms of the associated variety.

10.1. Lemma. A geometric action of an algebraic group G on a central simple algebra A of degree n is G-split in the sense of Definition 1.6 if and only if its associated G × PGL_{n}-variety is birationally isomorphic to X_{0} × PGL_{n}, for some G-variety X_{0}.

Here G acts on the first factor and PGL_{n} acts on the second factor by translations.

Proof. Suppose X = X_{0} ×PGL_{n}. Then we have the following G-equivariant isomorphisms,

RMaps_{PGL_{n}}(X, M_{n}) ≃ RMaps(X_{0}, M_{n}) ≃ M_{n}(k) ⊗_{k} k(X_{0}) ,

where the first isomorphism is given by f ↦ f|_{X_{0}×1_{PGL_{n}}} for every PGL_{n}- equivariant rational map f: X −−→ M_{n}. In other words, the induced Gaction on A = k_{n}(X) is G-split in the sense of Definition 1.6.

Conversely, suppose a geometric G-action on A is G-split. Denote the associated G × PGL_{n}-variety by X. Let X_{0} = X/PGL_{n} be the rational quotient of X by the PGL_{n}-action. Note that k(X_{0}) = Z(A). Then, as we saw above, RMaps_{PGL_{n}}(X_{0} × PGL_{n}, M_{n}) is G-equivariantly isomorphic to M_{n}(k) ⊗_{k} k(X_{0}), which is G-equivariantly isomorphic to A (because A is G-split). By Corollary 3.2, we conclude that X is birationally isomorphic to X_{0} × PGL_{n}. □

10.2. Corollary. Consider a geometric action of an algebraic group G on a central simple algebra A, with associated G × PGL_{n}-variety X. Then for any G-variety X_{0} the following are equivalent:

(a) L = k(X_{0}) is a G-splitting field for A.

(b) There exists a dominant rational map f: X_{0} × PGL_{n} −−→ X which is G × PGL_{n}-equivariant.

Here G acts on the first factor of X_{0}×PGL_{n} and PGL_{n} acts on the second factor by translations, as in Lemma 10.1.

Proof. (a) ⟹ (b). The G-action on A′ = A ⊗_{Z(A)} L ≃ M_{n}(k) ⊗_{k} L is geometric, with associated variety X′ = X_{0} × PGL_{n}; see Lemma 10.1. The embedding j: A ↪ A′ induces a G × PGL_{n}-equivariant dominant rational map j_{∗}: X′ −−→ X; see Lemma 3.1.

(b) ⟹ (a): Let X′ = X_{0} × PGL_{n}. By Lemma 3.1, f induces a Gequivariant embedding f^{∗}: A ↪ A′ of central simple algebras, where A′ = k_{n}(X′) ≃ M_{n}(k) ⊗_{k} k(X_{0}); see Lemma 10.1. In other words, A′ is Gequivariantly isomorphic to A ⊗_{Z(A)} k(X_{0}). □

10.3. Proof of Theorem 1.7. Let X be the associated G × PGL_{n}-variety for the G-action on A. Consider the dominant morphism f: X × PGL_{n} ⟶ X given by (x, h) ↦ hx. If we let (g, h) ∈ G × PGL_{n} act on X × PGL_{n} by (g, h) · (x, h′) = (gx, hh′), as in Lemma 10.1 and Corollary 10.2, then we can easily check that f is G×PGL_{n}-equivariant. By Corollary 10.2, we conclude that L = k(X) is a G-splitting field for A. Moreover,

trdeg_{Z(A)} L = trdeg_{k}(L) − trdeg_{k} Z(A)
= dim(X) − dim(X/PGL_{n}) = n^{2} − 1,

as claimed. Note that if G acts algebraically on A, we may assume that X is affine by Theorem 5.3(b). □

11. More on G-splitting fields

In this section we discuss G-splitting fields in the case where G is a connected group. Our main result is the following:

11.1. Proposition. Consider a geometric action of a connected algebraic group G on a central simple algebra A of degree n. Then there exists an affine G-variety X_{0} such that L = k(X_{0}) is a G-splitting field of A and

(11.2)  trdeg_{Z(A)} L = dim Stab_{G×PGL_{n}}(x) = dim Stab_{G}(w) ,

where x and w are points in general position in the associated G × PGL_{n}- variety X and in the rational quotient W = X/PGL_{n}, respectively. In particular,

trdeg_{Z(A)} L ≤ dim(G) .

Note that for w ∈ W in general position we have

dim Stab_{G}(w) = dim(G) − dim(Gw)
= dim(G) − (dim(W) − dim(W/G)) .

(11.3) so that the integer dim Stab_{G}(w) for w ∈ W in general position, which appears in the statement of Proposition 11.1 is well defined. Similarly, the integer dim Stab_{G×PGL_{n}}(x) for x ∈ X in general position is also well-defined. Since trdeg_{Z(A)^{G}} Z(A) = dim(W) − dim(W/G), (11.2) can be restated in algebraic terms as

(11.2′)  trdeg_{Z(A)} L = dim(G) − trdeg_{Z(A)^{G}} Z(A) .

In general, the value for trdeg_{Z(A)} L given in (11.2) and (11.2′) is the smallest possible, see Remark 11.8. Our proof of Proposition 11.1 will rely on the following lemma.

11.4. Lemma. Let H be a connected algebraic group and let V be an irreducible H-variety. Then there exists an irreducible variety Y and an H-equivariant dominant morphism Y × H ⟶ V such that dim(Y) = dim(V/H).

The action of H on Y × H is induced by the trivial action on Y and by the translation action on H.

Proof. See [P, (1.2.2)] or [PV, Proposition 2.7], where the term quasi-section is used to describe Y. □

11.5. Proof of Proposition 11.1. By Lemma 11.4 (with H = G × PGL_{n}) there is a G×PGL_{n}-equivariant dominant morphism f: Y ×(G×PGL_{n}) ⟶ X, where

(11.6)  dim(Y) = dim(X/(G × PGL_{n})) = dim(W/G) .

Note that since G×PGL_{n} acts trivially on Y, we can take Y to be affine. Setting X_{0} = Y × G (as a G-variety) and applying Corollary 10.2, we conclude that L = k(X_{0}) is a G-splitting field for A. By our construction, X_{0} = Y ×G is affine. Since the second equality in (11.2) is an immediate consequence of Lemma 7.1, we only need to check that trdeg_{Z(A)} L = dim Stab_{G}(w) for w ∈ W in general position. Indeed,

trdeg_{Z(A)} L = trdeg_{k}(L) − trdeg_{k} Z(A) = dim(X_{0}) − dim(X/PGL_{n})
= dim(Y) + dim(G) − dim(X/PGL_{n})
= dim(G) − (dim(W) − dim(W/G)) = dim Stab_{G}(w) ,

where the two last equalities follow from (11.6) and (11.3), respectively. □

Specializing Proposition 11.1 to the case of torus actions, we recover a result which was proved in [V_{3}] for algebraic actions in arbitrary characteristic.

11.7. Corollary. Suppose a torus T acts geometrically (or equivalently, algebraically; cf. Corollary 8.4) on a central simple algebra A. Let H be the kernel of the T -action on Z(A). Then there exists a T -variety X_{0} such that L = k(X_{0}) is a T -splitting field for A and trdeg_{Z(A)} L = dim(H).

Proof. Let X be the associated G×PGL_{n}-variety and W = X/PGL_{n}, as before. By Lemma 2.4, applied to the T -action on W, we have H = Stab_{T} (w) for w ∈ W in general position. The corollary now follows from Proposition 11.1. □

11.8. Remark. If the T -action on A is faithful then the value of trdeg_{Z(A)} L given by Corollary 11.7 is the smallest possible. Indeed, since the T -action on both A and L = k(X_{0}) is algebraic (cf. Corollary 8.4), [V_{3}, Theorem 2(b)] tells us that trdeg_{Z(A)} L ≥ dim(H) for every T -splitting field of the form L = k(X_{0}), where X_{0} is a T -variety. □

11.9. Remark. Suppose a torus T acts geometrically (or equivalently, algebraically; cf. Corollary 8.4) on a division algebra D. Then [V_{3}, Theorem 2(c)] asserts that D has a T -splitting field L of the form k(X_{0}) such that [L : Z(D)] < ∞.

We now give an alternative proof of this result based on Corollary 11.7. Let T_{0} ⊂ T be the kernel of the T -action on D. After replacing T by T/T_{0}, we may assume the action is faithful. Let H be the kernel of the T -action on Z(A), i.e., the subgroup of T acting by inner automorphisms. By Corollary A.2, H is a finite group. By Corollary 11.7, there exists a T -splitting field L = k(X_{0}) such that trdeg_{Z(D)} L = dim(H) = 0. Since L is finitely generated over k (and hence, over Z(D)), we conclude that [L : Z(D)] < ∞. □

12. An example: algebraic actions of unipotent groups

In this and the subsequent three sections we will present examples, illustrating Theorems 1.4, 1.5, and 1.7. We begin by applying Theorems 1.4 and 1.5 in the context of unipotent group actions on division algebras.

12.1. Proposition. Let U be a unipotent group acting algebraically on a finite-dimensional division algebra D. Then D^{U} is a division algebra of the same degree as D.

Proof. Say D has degree n, and let X be the associated U × PGL_{n}-variety. By Lemma 7.1, for x ∈ X in general position, Stab_{U×PGL_{n}}(x) is a unipotent group (it is isomorphic to a subgroup of U). Consequently, the projection H_{x} of this group to PGL_{n} is unipotent.

On the other hand, by [V_{3}, Proposition 7], D has a U-invariant maximal subfield. In view of Theorem 1.5(b), this implies that H_{x} is a subgroup of the normalizer of a maximal torus in PGL_{n}; in particular, H_{x} has no non-trivial unipotent elements. This is only possible if H_{x} = {1}, i.e., if

Stab_{U×PGL_{n}}(x) ⊆ U × {1} .

The desired conclusion now follows from Theorem 1.4(a). □

12.2. Remark. The condition that D is a division algebra is essential here. Suppose G = U is a non-trivial unipotent subgroup of PGL_{n} acting on A = M_{n}(k) by conjugation, as in Example 3.3. Since A is a finite-dimensional k-vector space, this action is easily seen to be algebraic. On the other hand, the fixed algebra A^{U} is not a central simple algebra of degree n; cf. Example 6.2(a) (see also Remark 6.4).

13. An example: the GL_{m}-action on UD(m, n), m ≥ n^{2}

We now return to the GL_{m}-action on the universal division algebra A = UD(m, n), described in Example 3.4. In this section we will assume that m ≥ n^{2}; in the next section we will set m = n = 2. The case where m ≤ n^{2} − 1 will be considered in [RV_{5}].

13.1. Proposition. Let A = UD(m, n), where m ≥ n^{2}. Then (a) A^{GL^{m}} = k

(b) trdeg_{Z(A)} L ≥ n^{2} −1 for every GL_{m}-splitting field L of A of the form L = k(X_{0}), where X_{0} is a GL_{m}-variety.

Part (b) shows that the value of trdeg_{Z(A)} L given by Theorem 1.7 is optimal for this action.

Proof. The variety (M_{n})^{m} is an associated GL_{m}×PGL_{n}-variety for the GL_{m}- action on A; see Example 3.4. The key fact underlying the proof of both parts is that for m ≥ n^{2}, (M_{n})^{m} has a dense GL_{m}-orbit; denote this orbit by X. Since the actions of GL_{m} and PGL_{n} commute, X is PGL_{n}-stable, and therefore is also an associated GL_{m} × PGL_{n}-variety for the GL_{m}-action on A.

(a) By Remark 6.1, A^{GL^{m}} = RMaps_{PGL_{n}}(X/GL_{m}, M_{n}). Since X is a single GL_{m}-orbit, the rational quotient X/GL_{m} is a point (with trivial PGL_{n}- action). Clearly, every PGL_{n}-equivariant rational map f: {pt} −−→ M_{n} is regular and has its image in the center of M_{n}. In other words,

A^{GL^{m}} = RMaps_{PGL_{n}}(X/GL_{m}, M_{n}) = RMaps({pt}, k) = k ,

as claimed.

(b) By Corollary 10.2 there exists a dominant rational map f: X_{0} × PGL_{n} = X′ −−→ X. Choose x′ ∈ X′, so that f is defined at x′ and set x = f(x′). Denote by S and S′ the stabilizers in GL_{m} × PGL_{n} of x and x′, respectively. Note that S′ ⊆ S ⊆ GL_{m} × PGL_{n}. Since GL_{m} acts transitively on X, the projection of S to PGL_{n} is all of PGL_{n}. On the other hand, we clearly have S′ ⊆ G × {1}. Consequently, dim(S) − dim(S′) ≥ dim(PGL_{n}) = n^{2} − 1, and if O′ is a GL_{m} × PGL_{n}-orbit in general position in X′, then dim(O′) − dim(X) ≥ n^{2} − 1. We thus conclude that

trdeg_{Z(A)} L = trdeg_{k} L − trdeg_{k} Z(A)
= dim(X′/PGL_{n}) − dim(X/PGL_{n})
= dim(X′) − dim(X) ≥ dim(O′) − dim(X) ≥ n^{2} − 1 ,

as claimed. □

13.2. Remark. One can show directly that the GL_{m}-splitting field L for A = UD(m, n) given by Proposition 11.1 satisfies the inequality of Proposition 13.1(b) (assuming, of course, that m ≥ n^{2}). Indeed, since G has a dense orbit in W = X/PGL_{n}, for w ∈ W in general position,

dim Stab_{GL_{m}}(w) = dim(GL_{m}) − dim(W) = m^{2} − dim(X/PGL_{n}) .

Since the associated variety X = (M_{n})^{m} has dimension mn^{2}, this yields

trdeg_{Z(A)} L = dim Stab_{GL_{m}}(w) = m(m − n^{2}) + (n^{2} − 1) ≥ n^{2} − 1 ,

as claimed.

14. An example: the GL_{2}-action on UD(2, 2)

In this section we will use Theorem 1.4 to study the natural GL_{m}-action on the universal division algebra UD(m, n), described in Example 3.4, for m = n = 2. Note that this case exhibits some special features that do not recur for other values of m and n ≥ 2; see Proposition 13.1(a) (for m ≥ n^{2}) and [RV_{5}] (for m ≤ n^{2} − 1).

14.1. Proposition. The fixed algebra UD(2, 2)^{GL_{2}} is a non-central subfield of UD(2, 2) of transcendence degree 1 over k.

Recall from Example 3.4 that the GL_{2}-action on UD(2, 2) is defined as follows. Denote by X and Y the two generic 2 × 2 matrices generating UD(2, 2). Then for g ∈ GL_{2}, we have g(X) = αX + βY, and g(Y) = γX + δY, where g^{−1} = ( \begin{matrix}α & β \\ γ & δ \end{matrix} ). Recall also that the associated variety for the GL_{2}-action on UD(2, 2) is X = (M_{2})^{2}. In order to use Theorem 1.4 to prove Proposition 14.1, we first need to determine the stabilizer in general position for the GL_{2} × PGL_{2}-action on (M_{2})^{2}.

14.2. Lemma. For x ∈ (M_{2})^{2} in general position, Stab_{GL_{2}×PGL_{2}}(x) is isomorphic to ℤ/2ℤ.

Proof. By Lemma 7.1, Stab_{GL_{2}×PGL_{2}}(x) is isomorphic to Stab_{GL_{2}}(y) for the GL_{2}-action on W = X/PGL_{2}, which is a birational model for the GL_{2}- action on the center Z of UD(2, 2). In this case there is a particularly simple birational model, which we now describe.

It is well known that Z is freely generated (as a field extension of k) by the five elements tr(X), tr(Y), tr(X^{2}), tr(Y^{2}) and tr(XY); see [Pr_{1}, Theorem 2.2]. In other words, the categorical (and, hence, the rational) quotient for the PGL_{2}-action is 𝔸^{5}. The group GL_{2} acts on 𝔸^{5} linearly. In fact, the representation of GL_{2} on 𝔸^{5} = X/PGL_{2} can be decomposed as V_{2} ⊕ V_{3}, where V_{2} is the natural 2-dimensional representation (we can think of it as Span_{k}(tr(X), tr(Y))) and V_{3} is its symmetric square. (We can think of V_{3} as Span_{k}(tr(X^{2}), tr(Y^{2}), tr(XY)).)

The question we are asking now reduces to the following: What is the stabilizer, in GL_{2}, of a pair (v, q), in general position, where v is a vector in k^{2} and q is a quadratic form in 2 variables? Indeed, since GL_{2} acts transitively on non-degenerate quadratic forms in two variables, we may assume that q is a fixed form of rank 2, e.g., q = x^{2} + y^{2}. The stabilizer of q is thus the orthogonal group O_{2}, and our question further reduces to the following: what is the stabilizer in general position for the natural linear action of O_{2} on k^{2}? The answer is easily seen to be ℤ/2ℤ, where the non-trivial element of Stab_{O_{2}}(v) is the orthogonal reflection in v; see Example 2.5. □

Proof of Proposition 14.1. Note that the GL_{2}-action on X = (M_{2})^{2} is generically free (it is isomorphic to the direct sum of 4 copies of the natural 2-dimensional representation of GL_{2}). Thus the image of the stabilizer Stab_{GL_{2}×PGL_{2}}(x) under the natural projection to the second factor is ℤ/2ℤ. Since this image is non-trivial, Theorem 1.4(a) tells us that UD(2, 2)^{GL^{2}} is not a division subalgebra of UD(2, 2) of degree 2. In other words, it is a subfield of UD(2, 2). On the other hand, Theorem 1.4(b) tells us that UD(2, 2)^{GL^{2}} is not contained in the center Z of UD(2, 2). Indeed, every subgroup of PGL_{2} of order 2 is contained in a torus. Hence, Stab_{GL_{2}×PGL_{2}}(x) is contained in GL_{2}×T_{x}, where T_{x} is a maximal torus of PGL_{2}. It follows from Theorem 1.4(b) that the subfield UD(2, 2)^{GL^{2}} is not central in UD(2, 2).

Finally, note that UD(2, 2)^{GL^{2}} is algebraic over Z^{GL^{2}}, since the minimal polynomial of any element of UD(2, 2)^{GL^{2}} over Z is unique, so must have coefficients in Z^{GL^{2}}. It follows from Lemmas 7.1 and 14.2 that the GL_{2}- action on W = X/PGL_{2} has a finite stabilizer in general position. Hence the transcendence degree of Z^{GL^{2}} = k(X/PGL_{2})^{GL^{2}} (over k) is dim(X/PGL_{2})− dim(GL_{2}) = 1. □

14.3. Remark. This argument also shows that UD(2, 2)^{SL^{2}} is a division algebra of degree 2.

14.4. Remark. One can exhibit an explicit non-central GL_{2}-fixed element of UD(2, 2) as follows. Let

S_{3}(A_{1}, A_{2}, A_{3}) = ∑_{σ∈S_{3}}(−1)^{σ}A_{σ(1)}A_{σ(2)}A_{σ(3)}

be the standard polynomial in three variables; cf., [Ro_{1}, p. 8]. Set a = [X, Y] = XY − Y X and b = S_{3}(X, Y, a). Using the fact that [A_{1}, A_{2}] and S_{3}(A_{1}, A_{2}, A_{3}) are multilinear and alternating, it is easy to see that for g ∈ GL_{2}, g(a) = a/det(g) and g(b) = b/det^{2}(g). Specializing X to ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) and Y to ( \begin{matrix}1 & 1 \\ 1 & 0 \end{matrix} ), an elementary computation shows that a and b specialize to ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) and ( \begin{matrix}−4 & 0 \\ 0 & −2 \end{matrix} ), respectively. This shows that det(a) ≠ 0 and that b is non-central. Now, b/det(a) is a non-central GL_{2}-fixed element of UD(2, 2).

Note also that a and b are non-commuting SL_{2}-invariant elements of UD(2, 2). This gives an explicit proof of Remark 14.3.

15. An example: a finite group action on a cyclic algebra

In this section we present an example of a finite group action on a cyclic algebra. This example illustrates Lemma 7.1 and Theorem 1.5 and, in particular, shows that the converse to Theorem 1.5(b) is false.

Let p be a prime integer, and ζ a primitive p-th root of unity in k. Let P = k{x, y} be the skew-polynomial ring with generators x and y, subject to the relation

xy = ζyx .

Let A be the division algebra of fractions of P; it is a central simple algebra of degree n = p. Note that A is the symbol algebra (u, v)_{p} whose center is Z(A) = k(u, v), where u = x^{p} and v = y^{p} are algebraically independent over k.

For (a, b) ∈ (ℤ/pℤ)^{2}, define an automorphism σ_{(a,b)} of A by

(15.1)  σ_{(a,b)}(x) = ζ^{a}x and σ_{(a,b)}(y) = ζ^{b}y .

These automorphisms of A form a group K which is isomorphic to (ℤ/pℤ)^{2}. Next, we define an automorphism τ of A by

(15.2)  τ(x) = y and τ(y) = x^{−1}y^{−1} .

Note that τ is well-defined since

τ(x)τ(y) − ζτ(y)τ(x) = y(yx)^{−1} − ζx^{−1} = y(ζ^{−1}xy)^{−1} − ζx^{−1} = 0 .

Elementary calculations show that τ has order three, and that τ^{−1}σ_{(a,b)}τ = σ_{(b,−a−b)}. Consequently, the subgroup G of automorphisms of A generated by K and τ is a semidirect product G = K ⋊ H, where K ≃ (ℤ/pℤ)^{2} and H = ⟨τ⟩ ≃ ℤ/3ℤ.

One easily checks that sending τ to the matrix ( \begin{matrix}0 & −1 \\ 1 & −1 \end{matrix} ) defines a representation

ϕ_{p}: H → SL_{2}(ℤ/pℤ) ,

and thus an action of H on (ℤ/pℤ)^{2}.

Let X be the G × PGL_{n}-variety associated to the action of G on the central simple algebra A of degree n = p. That is, X is an irreducible G × PGL_{n}-variety which is PGL_{n}-generically free, and A is G-equivariantly isomorphic to k_{n}(X).

15.3. Proposition. (a) For x ∈ X in general position, there exists a

maximal torus T_{x} of PGL_{n} such that Stab_{G×PGL_{n}}(x) ⊆ G × N(T_{x}).

(b) A has a G-invariant maximal subfield if and only if the 2-dimensional representation ϕ_{p}: H → SL_{2}(ℤ/pℤ) is reducible over ℤ/pℤ.

(c) The converse to Theorem 1.5(b) is false.

Before we proceed with the proof, two remarks are in order. First of all, every finite group action on a central simple algebra is automatically geometric (and algebraic).

Secondly, an explicit model for X is not immediately transparent (a description of X as a PGL_{n}-variety can be found in [RY, Lemma 5.2]). On the other hand, the G-variety W associated to the G-action on the center of A (see the beginning of Section 7) is easy to describe: We can take W to be the two-dimensional torus W = (k^{∗})^{2} = Spec(k[u, v, u^{−1}, v^{−1}]), where as before, u = x^{p} and v = y^{p}. It follows from (15.1) and (15.2) that the K-action on W is trivial, and that the action of τ is induced from τ(u) = v, τ(v) = (x^{−1}y^{−1})^{p} = ϵ · u^{−1}v^{−1}, where ϵ = 1 if p > 2 and ϵ = −1 if p = 2.

We now proceed with the proof of Proposition 15.3.

Proof. (a) Since G is a finite group, Stab_{G}(w), for w ∈ W in general position, is precisely the kernel of the G-action on W. We claim that the kernel is equal to K. That it contains K is immediate from (15.1), since every element of K preserves both u = x^{p} and v = y^{p}. On the other hand, the H-action on W is faithful, because H is a simple group acting nontrivially on Z(A) = k(W). We have thus shown that Stab_{G}(w) = K for w ∈ W in general position.

By Lemma 7.1, Stab_{G×PGL_{n}}(x) ≃ K ≃ (ℤ/pℤ)^{2} for x in general position in X. In particular, the projection of this group to PGL_{n} is a finite abelian subgroup of PGL_{n}. By [SS, II.5.17], every finite abelian subgroup of PGL_{n} lies in the normalizer of a maximal torus T_{x}. Thus

Stab_{G×PGL_{n}}(x) ⊆ G × N(T_{x}) ,

as claimed.

(b) First we will describe the K-invariant maximal subfields of A, then determine which ones of them are also invariant under H. Note that since A is a division algebra of prime degree p, every nontrivial field extension L of the center Z(A) is a maximal subfield of A.

The group K ≃ (ℤ/pℤ)^{2} acts trivially on Z(A); its action on A decomposes as a direct sum of p^{2} one-dimensional character spaces Span_{Z(A)}(x^{i}y^{j}), where 0 ≤ i, j ≤ p−1. These spaces are associated to the p^{2} distinct characters of (ℤ/pℤ)^{2}; hence, every K-invariant Z(A)-vector subspace L contains x^{i}y^{j} for some 0 ≤ i, j ≤ p − 1. Moreover, if L is a K-invariant maximal subfield of A then Z(A)(x^{i}y^{j}) ⊆ L, where 0 ≤ i, j ≤ p − 1 and (i, j) ≠ (0, 0). Since [L : Z(A)] = p and x^{i}y^{j} ∉ Z(A), we conclude that L = Z(A)(x^{i}y^{j}). We will denote Z(A)(x^{i}y^{j}) by L_{(i,j)}.

Now suppose (i, j) and (r, s) are non-zero elements of (ℤ/pℤ)^{2}. We claim that L_{(i,j)} = L_{(r,s)} if and only if (i, j) and (r, s) are proportional, i.e., if and only if they lie in the same 1-dimensional ℤ/pℤ-subspace of (ℤ/pℤ)^{2}. Indeed, if (i, j) and (r, s) are proportional then up to a multiple from Z(A), x^{i}y^{j} and x^{r}y^{s} are powers of one another. Since neither one is central, they generate the same maximal subfield. Conversely, since a maximal subfield has dimension p over Z(A), it can contain only p − 1 distinct x^{i}y^{j} with (0, 0) ≠ (i, j) ∈ (ℤ/pℤ)^{2}. Since there are p − 1 nonzero ℤ/pℤ-multiples of (i, j), this proves the claim.

We have thus shown that the K-invariant maximal subfields of A are in bijective correspondence with 1-dimensional ℤ/pℤ-subspaces of (ℤ/pℤ)^{2}: a 1-dimensional subspace V corresponds to the maximal subfield L_{V} = Z(A)(x^{i}y^{j}), where (i, j) is a non-zero element of V.

It is clear from (15.2) that τ(L_{V}) = L_{τ(V)}, where τ acts on (ℤ/pℤ)^{2} via the representation ϕ_{p}. To sum up: A has a maximal G-invariant subfield ⟺ τ preserves one of the K-invariant maximal subfields L_{V} ⟺ (ℤ/pℤ)^{2} has a τinvariant 1-dimensional ℤ/pℤ-subspace V ⊂ (ℤ/pℤ)^{2} ⟺ the representation ϕ_{p} of H is reducible.

(c) In view of parts (a) and (b) it suffices to show that the representation ϕ_{p} of H is irreducible if and only if p ≡ 2 (mod 3). If p = 3, ϕ_{p} is reducible, since in this case ( \begin{matrix}1 \\ −1 \end{matrix} ) is an eigenvector for the matrix ( \begin{matrix}0 & −1 \\ 1 & −1 \end{matrix} ). Now assume that p ≠ 3. Then Maschke’s theorem implies that ϕ_{p}(τ) is diagonalizable over the algebraic closure of ℤ/pℤ. The eigenvalues of ϕ_{p}(τ) are then necessarily third roots of unity, including at least one primitive third root of unity.

Thus the action of H on (ℤ/pℤ)^{2} is reducible ⟺ ϕ_{p}(τ) is diagonalizable over ℤ/pℤ ⟺ the eigenvalues of ϕ_{p}(τ) belong to ℤ/pℤ ⟺ ℤ/pℤ contains a primitive third root of unity ⟺ 3 | p − 1.

Consequently, the representation ϕ_{p} irreducible if and only if p ≡ 2 (mod 3). □

Appendix A. Inner actions on division algebras

In this appendix we continue to assume that k is an algebraically closed base field of characteristic zero, and that every division algebra is finitedimensional over its center, which in turn is a finitely generated field extension of k. (Some of the lemmas below hold in greater generality; see Remark A.5.) Our main result is the following theorem.

A.1. Theorem. Let G be an algebraic group acting on a division algebra D of degree n by inner automorphisms. Then the kernel N of this action contains the connected component G^{0} of G, and G/N is a finite abelian n-torsion group.

Here the algebraic group G is treated as an abstract group; in particular, the (inner) action of G on D is not assumed to be algebraic or geometric. Consequently, our proof has a rather different flavor from the other arguments in this paper. Instead of using algebraic geometry, we exploit, in the spirit of [RV_{1}], the fact that connected algebraic groups are generated, as abstract groups, by their divisible subgroups. Note that the special case of Theorem A.1, where G is a torus is proved in [RV_{1}, Corollary 5.6].

Before we prove Theorem A.1, we deduce an easy consequence.

A.2. Corollary. Let G be an algebraic group acting faithfully and geometrically on a division algebra D of degree n. Then the normal subgroup of G acting by inner automorphisms is a finite abelian n-torsion group.

Proof. Since G acts geometrically, the normal subgroup H of G consisting of the elements acting by inner automorphisms (i.e., acting trivially on Z(D)) is closed, so itself an algebraic group. Now apply Theorem A.1 to the faithful action of H on D. □

We now turn to the proof of Theorem A.1, beginning with two lemmas.

A.3. Lemma. The group of inner automorphisms of a division algebra contains no divisible subgroups.

Proof. Assume to the contrary that there is a nontrivial divisible group H acting faithfully on a division algebra D by inner automorphisms. By [RV_{1}, Corollary 3.2], the torsion subgroup of H acts trivially on D, so it must be trivial. Hence H is a torsion-free divisible group, i.e., a direct sum of copies of (ℚ, +); cf. [Sc, 5.2.7]. By [RV_{1}, Lemma 3.3(a)], there is a subfield L of D containing the center K of D such that H embeds into L^{∗}/K^{∗}. Thus (ℚ, +) embeds into L^{∗}/K^{∗}. By [RV_{1}, Lemma 5.5]^{3}, this implies that K is not finitely generated over the algebraically closed field k, a contradiction. □

A.4. Lemma. Let D be a division algebra of degree n whose center K contains all roots of unity.

(a) Suppose x ∈ D has the following properties: det(x) = 1, and x^{m} ∈ K for some integer m ≥ 1. Then x ∈ K.

(b) If G is a finite group acting faithfully on D by conjugation, then G is an abelian n-torsion group.

As the statement of the lemma implies, here K is not assumed to contain an algebraically closed base field.

Proof. (a) Suppose x^{m} = a for some integer a ∈ K. Taking the determinant (i.e., reduced norm) on both sides, we obtain a^{n} = 1. Thus, after replacing m by mn, we may assume x^{m} = 1. Since the polynomial f(t) = t^{m} −1 splits over K, we conclude that x ∈ K.

(b) Suppose g ∈ G acts by conjugation by d_{g}. Then for every g, h ∈ G, the commutator x = d_{g}d_{h}d_{g}^{−1}d_{h}^{−1} satisfies the conditions of part (a), where m can be taken to be the order of ghg^{−1}h^{−1} in G. Thus x ∈ K and consequently, g and h commute in G. This shows that G is abelian.

To prove that G is n-torsion, choose g ∈ G and consider the element x = (d_{g})^{n}/ det(d_{g}). Once again, x satisfies the conditions of part (a), with m the order of g^{n} in G. Thus x ∈ K, and consequently, g^{n} = 1 in G, as claimed. □

3We take the opportunity to correct an error in the proof of [RV_{1}, Lemma 5.5]. The third paragraph of that proof should read: “If π_{i} ∘ ϕ is not injective, its image is a torsion group. Since π ∘ ϕ is injective, π_{i}(ϕ(ℚ)) is not torsion for some i. Hence, for this i, ψ = π_{i} ∘ ϕ is injective, so that ψ(ℚ) is nontrivial. Thus by the argument in the previous paragraph, ψ(ℚ) is not contained in K^{∗}.”

Proof of Theorem A.1. Let S be a torus of G, or a closed subgroup which is isomorphic to (k, +). We claim that S ⊆ N. Since S is a divisible group, so is S/N ∩ S; cf. [Sc, 5.2.19]. Since S/N ∩ S acts faithfully on D, Lemma A.3 tells us that S/N ∩ S = {1}, i.e., S ⊆ N, as claimed.

Now recall that every element g ∈ G^{0} has a Jordan decomposition product g = g_{s}g_{u}, where g_{s} is semisimple and g_{u} is unipotent; cf., e.g., [H, Theorem 15.3]. Since g_{s} lies in a torus of G, g_{s} ∈ N. Similarly, g_{u} ∈ N; cf., e.g., [H, Lemma 15.1C]. Thus G^{0} ⊆ N, as claimed. The desired conclusion now follows from Lemma A.4. □

A.5. Remark. Lemmas A.3 and A.4 also hold in prime characteristic, and so does Theorem A.1, provided G is reductive (since then G^{0} is generated as abstract group by the tori it contains).

Appendix B. Regular actions on prime PI-algebras

It is a consequence of Posner’s theorem that every prime PI-algebra R of PI-degree n can be realized as a subalgebra of n × n-matrices over some commutative domain C. Given an action of a group G on R, it is natural to ask whether one can always find such an embedding R ↪ M_{n}(C) which is G-equivariant for some action of G on M_{n}(C). We now deduce from Theorem 1.7 a rather strong affirmative answer in the case of regular actions of algebraic groups (see Definition 5.1) on prime PI-algebras. Such actions were extensively studied in [V_{1}] and [V_{2}].

B.1. Proposition. Let R be a prime PI-algebra of PI-degree n, which is finitely generated as k-algebra. Let G be an algebraic group acting regularly on R. Then there is a finitely generated commutative k-algebra C which is a domain, and a regular action of G on C such that R embeds G-equivariantly into M_{n} ⊗_{k} C. Here G acts trivially on M_{n}.

In the case where G is a torus, this assertion was proved in [V_{3}, Corollary 9].

Proof. Let A be the total ring of fractions of R; it is a central simple algebra of degree n, and G acts algebraically on A. Note that since R is finitely generated as k-algebra, the center of A is a finitely generated field extension of k. By Theorem 1.7, there is a G-splitting field L = k(X_{0}) for A, where X_{0} is an affine G-variety, i.e., the G-action on L is algebraic; cf. Definition 5.2. This gives rise to a G-equivariant embedding φ: R → M_{n} ⊗_{k} L = A′. Hence G also acts algebraically on A′, so that A′ contains a unique largest subalgebra S_{A′} on which G acts regularly, and which contains every subalgebra of A′ on which G acts regularly. Denote by S_{L} the corresponding subalgebra of L. Since S_{A′} contains M_{n} ⊗_{k} k, it follows that S_{A′} = M_{n} ⊗_{k} S_{L}. Since G acts regularly on φ(R), φ(R) ⊆ M_{n} ⊗_{k} S_{L}. Since R is finitely generated, and since G acts regularly on S_{L}, there is a finitely generated G-invariant subalgebra C of S_{L} such that φ(R) ⊆ M_{n} ⊗_{k} C. □

References

[B] N. Bourbaki, Algebra II, Chapters 4–7, Translated from the French by P. M. Cohn and J. Howie, Springer, Berlin, 1990.

[H] J. Humphreys, Linear Algebraic Groups, corrected second printing, Springer, New York, 1981.

[J] N. Jacobson, Basic Algebra II, Freeman, 1980.

[lB] L. Le Bruyn, The Artin-Schofield theorem and some applications, Comm. Algebra 14 (1986), no. 8, 1439–1455.

[LR] D. Luna, R. W. Richardson, A generalization of the Chevalley restriction theorem, Duke Math. J. 46 (1979), no. 3, 487–496.

[M] S. Montgomery, Fixed rings of finite automorphism groups of associative rings, Lecture Notes in Mathematics, 818. Springer, Berlin, 1980.

[P] V. L. Popov, Sections in invariant theory, The Sophus Lie Memorial Conference (Oslo, 1992), Scand. Univ. Press, Oslo, 1994, pp. 315–361.

[PV] V. L. Popov and E. B. Vinberg, Invariant Theory, Algebraic Geometry IV, Encyclopedia of Mathematical Sciences 55, Springer, 1994, 123–284.

[Pr_{1}] C. Procesi, Non-commutative affine rings, Atti Accad. Naz. Lincei Mem. Cl. Sci.

Fis. Mat. Natur. Sez. I (8) 8 (1967), 237–255. [Pr_{2}] C. Procesi, The invariant theory of n × n matrices, Advances in Math. 19 (1976),

no. 3, 306–381.

[Re] Z. Reichstein, On the notion of essential dimension for algebraic groups, Transform. Groups 5 (2000), no. 3, 265–304.

[RV_{1}] Z. Reichstein and N. Vonessen, Torus actions on rings, J. Algebra 170 (1994),

781–804. [RV_{2}] Z. Reichstein and N. Vonessen, Rational central simple algebras, Israel J. Math. 95

(1996), 253–280. [RV_{3}] Z. Reichstein and N. Vonessen, Stable affine models for algebraic group actions, J.

Lie Theory, 14, no. 2 (2004), 563–568. [RV_{4}] Z. Reichstein and N. Vonessen, Polynomial identity rings as rings of functions, J.

Algebra, to appear. Preprint available at http://arxiv.org/math.RA/0507152 [RV_{5}] Z. Reichstein and N. Vonessen, Group actions and invariants in algebras of generic

matrices, Adv. in Appl. Math., to appear. Preprint available at

      http://arxiv.org/math.RA/0507548

[RY] Z. Reichstein and B. Youssin, Conditions satisfied by characteristic polynomials in fields and division algebras, J. Pure Appl. Algebra 166 (2002), no. 1-2, 165–189.

[Ri] R. W., Richardson, Jr., Deformations of Lie subgroups and the variation of isotropy subgroups, Acta Math. 129 (1972), 35–73.

[Ro_{1}] L. H. Rowen, Polynomial Identities in Ring Theory, Academic Press, 1980. [Ro_{2}] L. H. Rowen, Ring Theory, Volume II, Academic Press, 1988.

[Sa] D. S. Sage, Group actions on central simple algebras, J. Algebra 250 (2002), 18–43.

[Sc] W. R. Scott, Group theory, second edition, Dover Publications, Inc., New York, 1987.

[SS] T. A. Springer, R. Steinberg, Conjugacy classes, Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69), pp. 167–266, Lecture Notes in Math., Vol. 131, Springer, Berlin, 1970.

[St] R. Steinberg, Torsion in reductive groups, Advances in Math. 15 (1975), 63–92. [V_{1}] N. Vonessen, Actions of linearly reductive groups on affine PI-algebras, Mem. Amer.

Math. Soc. 414, 1989. [V_{2}] N. Vonessen, Actions of linearly reductive groups on PI-algebras, Trans. Amer.

Math. Soc. 335 (1993), 425–442.

[V_{3}] N. Vonessen, Actions of solvable algebraic groups on central simple algebras, Algebras and Representation Theory, to appear. Preprint available at

   http://www.math.umt.edu/vonessen/publications.html

[Vos] V. E. Voskresenskiĭ, Algebraic groups and their birational invariants, American Mathematical Society, Providence, RI, 1998.

Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada

E-mail address: reichste@math.ubc.ca
URL: www.math.ubc.ca/~reichst

Department of Mathematical Sciences, University of Montana, Missoula, MT 59812-0864, USA

E-mail address: Nikolaus.Vonessen@umontana.edu
URL: www.math.umt.edu/vonessen