GROUP ACTIONS ON CENTRAL SIMPLE ALGEBRAS

DANIEL S. SAGE SCHOOL OF MATHEMATICS INSTITUTE FOR ADVANCED STUDY PRINCETON, NJ 08540

Abstract. Let G be a group, F a field, and A a finite-dimensional central simple algebra over F on which G acts by F -algebra automorphisms. We study the subalgebras and ideals of A which are preserved by the group action. We prove a structure theorem and two classification theorems for invariant subalgebras under suitable hypotheses on A. We illustrate these results in the case of compact connected Lie groups and give some other applications. We also classify invariant ideals.

1. Introduction

Let G be a group, F a field, and V a finite-dimensional F -vector space on which G acts by F -linear automorphisms. A fundamental problem in representation theory is to classify the G-invariant subspaces of V, in other words, to determine those subspaces of V which inherit a G-action from V. For the case when G is a compact group and F = C, this question has been answered completely. The representation can be decomposed canonically into a direct sum of subrepresentations V = U_{1} ⊕ ⋯⊕U_{m}, where each U_{i} is the direct sum of n_{i} copies of an irreducible representation V_{i} and the V_{i}’s are pairwise nonisomorphic. The G-invariant subspaces of U_{i} are parametrized by subspaces of C^{n_{i}} while the subrepresentations of V are direct sums of subrepresentations of the U_{i}’s which may be chosen independently. As long as a decomposition of V into irreducible components is given explicitly (which may be very difficult in practice), this classification is also entirely explicit.

Let us now replace the vector space V with a finite-dimensional F -algebra A. We suppose further that A is a G-algebra, i.e G acts on A by F -algebra automorphisms, so that the G-action is well-behaved with respect to ring multiplication. The natural analogue of the problem considered above is to determine those Ginvariant subspaces of A which have significance in terms of the multiplicative structure of A. In particular, we would like to classify the G-invariant ideals (left, right, and two-sided) and subalgebras. These are just special cases of the general problem of understanding the multiplication of subrepresentations of A. If M and N are two subrepresentations of A, then MN, the F -linear span of the set {mn | m ∈ M, n ∈ N} ⊆ A, is also G-invariant. We thus obtain a multiplication on the set of subrepresentations of A. Invariant ideals and algebras are now easily expressed in terms of this multiplication; an invariant left ideal is a subrepresentation I such that AI ⊂ I, an invariant subalgebra is a subrepresentation B such that BB ⊂ B, and so on.

Research partially supported by NSF grants DMS 96-233533 and DMS 97-29992.

These problems are much more difficult than the classification of G-invariant subspaces. It is unreasonable to expect to find a way of determining G-invariant ideals and subalgebras that works for all A, even for G compact and F = C. Indeed, if we let G act trivially on A, then this result would give a uniform way of classifying ideals and subalgebras. It is thus necessary to limit the class of algebras under consideration.

In this paper, we restrict attention to central simple algebras over F. Our major goal is to prove a structure theorem (Theorem 3.8) and two classification theorem (Theorems 3.15 and 3.23) for invariant subalgebras under suitable conditions. We illustrate these results in the case of compact connected Lie groups (Theorem 4.3). We also classify invariant ideals (Theorem 5.2).

We now give a more detailed outline of the paper. Let A be a central simple algebra over F, and suppose G acts on A by F -algebra automorphisms. In section two, we show that the unique simple module V of A is a projective representation of G. Moreover, A is isomorphic to the algebra of D-endomorphisms of this projective representation, where D is a certain central division algebra.

In the third section, we impose the hypothesis that V is irreducible and show that invariant subalgebras are semisimple with a very special structure. Indeed, we prove that an invariant subalgebra B must be simply embedded; this means that both B and its centralizer in A are direct products of isomorphic simple algebras. We then introduce two complementary constructions of invariant subalgebras. In the first construction, we take an appropriate simple H-algebra for a finite index subgroup H and obtain an invariant subalgebra by induction. In fact, all invariant subalgebras arise in this way; the first classification theorem parameterizes invariant subalgebras in terms of induction data. The second construction produces central simple invariant subalgebras from factorizations of V into the tensor product of projective representations. All invariant central simple subalgebras are obtained from this procedure.

Combining these results gives the second classification theorem, an entirely explicit parameterization of the invariant subalgebras for F algebraically closed. This classification shows that the set of invariant subalgebras of A encodes complicated information about G and V, involving both how V can be expressed as an induced representation Ind_{H}^{G}(W) and how W can be factored into the tensor product of projective representations. It should be observed that for F = C and G finite, knowing the character table of G does not suffice to determine all invariant subalgebras. In fact, even in the simplest case where V is a primitive representation, the character table of a covering group of G is needed to find all invariant subalgebras.

We conclude the section with two applications of the classification theorem. In the first, we prove that for G finite and F algebraically closed, the set of invariant subalgebras is finite, and we describe how finiteness fails in the general case. In the second, we show that when V is primitive, there are no nonunital invariant subalgebras besides {0}.

In section four of the paper, we illustrate the structure and classification theorems for a topological or Lie group G. Here, all invariant subalgebras are simple as long as the connected component of G acts irreducibly on V. We apply our results to obtain a theorem of Etingof giving a particularly elegant solution to the classification problem when G is a compact connected Lie group and F = C. In fact, suppose G is semisimple and simply connected, say G = G_{1} × ⋯ × G_{n} with each G_{i} simple. The representation V is then isomorphic to V_{1} ⊗ ⋯ ⊗ V_{n}, for some irreducible representations V_{i} of G_{i}. We show that the G-invariant subalgebras of A are parametrized by the subsets J of {i | V_{i} ≠ C} via J ↦ ⨂_{j∈J} End_{C}(V_{j}) and that the only nonunital invariant subalgebra is {0}. In particular, if G is simple, the invariant subalgebras are C and A.

In the final section, we briefly consider the much simpler problem of understanding invariant ideals. Dropping the hypothesis that V is irreducible, we prove that there is a natural one-to-one correspondence between G-invariant D-submodules of V and invariant left (and right) ideals of A, where D = End_{A}(V). Indeed, we show that if G is compact and A is the endomorphism algebra of a complex representation, then the parametrization of invariant left and right ideals of A is the same as the classical parametrization of invariant subspaces of V. In particular, V is irreducible if and only if there are no invariant proper left (right) invariant ideals, and V is multiplicity free if and only if there are a finite number of left (right) invariant ideals.

We have also obtained results on the general problem of multiplication of subrepresentations in central simple algebras when G is a compact, simply reducible group. (A group is simply reducible if the conjugacy classes are stable under inversion and the tensor product of irreducible representations is multiplicity-free. The most familiar examples of simply reducible groups are S_{3}, S_{4}, the quaternion group, SU(2), and SO(3).) However, since the proofs use quite different techniques, these results will appear in another paper [S].

Our initial motivation for studying group actions on central simple algebras came from a problem in solid state physics. The analysis of G-actions on real and complex central simple algebras is important in understanding how physical properties such as conductivity, elasticity, and piezoelectricity of a composite material depend on the properties of its constituents. These physical characteristics are described by elements of a symmetric tensor space Sym^{2}(T), where T is a certain real representation of the rotation group SO(n). In general, a property of a composite depends heavily on the microstructure, i.e. the arrangement of the component materials. Let M ⊂ Sym^{2}(T) be the set of all possible values of a fixed property for composites made with their constituents taken in prescribed volume fractions. Typically, M is the closure of an open set in Sym^{2}(T) and may be described by a system of inequalities, so that away from the boundary of M, it is possible to make any desired small change in the property by varying the microgeometry. However, in certain unusual situations, some of the inequalities become equations, determining a proper closed submanifold E in which M is locally closed. The submanifold E and also the equations defining E are called exact relations for the property. The variability of the property with microstructure is thus drastically reduced when an exact relation is present. Recent work of Grabovsky, Milton, and Sage has shown how to classify exact relations in terms of the multiplication of SO(n)-subrepresentations in the endomorphism algebra End_{R}(T); in particular, invariant algebras and ideals of this central simple algebra have great physical significance [G, GS, GMS].

It is a great pleasure to thank Yury Grabovsky for first bringing these problems to my attention and for explaining their importance in physics. I would also like to thank Daniel Allcock for several helpful comments and Pavel Etingof for letting me use his unpublished result on invariant subalgebras of compact connected Lie groups.

2. Preliminaries

Let A be a finite-dimensional central simple algebra over the field F, and let V be a simple (left) A-module. The module V is unique up to isomorphism and is a finite-dimensional vector space over F. By Schur’s Lemma, the ring D = End_{A}(V) is a central division algebra, and V is naturally a left D-module. It is well-known that A is isomorphic to End_{D}(V), and from now on, we assume without loss of generality that A = End_{D}(V).

It is easy to construct examples of central simple algebras on which the group G acts by F -algebra automorphisms. Recall that a mapping ρ : G → GL(V) is called a projective representation of G over F if ρ(1) = 1_{V} and if there exists α : G × G → F^{∗} such that ρ(xy) = α(x, y)ρ(x)ρ(y) for all x, y ∈ G. (Equivalently, we can view a projective representation as a homomorphism G → P GL(V).) The map α is a 2-cocycle. Let ḡ be the basis vector corresponding to g ∈ G in the twisted group algebra F^{α}G. A projective α-representation is just an F^{α}G-module via ḡv = ρ(g)(v), and we also use this notation. (For linear representations, we just write gv.) The map π : G → GL(A) then makes A into a (linear) representation of G with (π(g)f)(v) = ρ(g)(f(ρ(g)^{−1}(v)) for all g ∈ G, f ∈ A, and v ∈ V. Moreover, the linear map π(g) is in fact an algebra automorphism. It turns out that all central simple algebras on which G acts via algebra automorphisms are of this type.

Proposition 2.1. Suppose that G acts on A = End_{D}(V) by F -algebra automorphisms, i.e. A is a representation of G via a homomorphism G →^{π} Aut(A). Then V is a projective representation of G determined up to projective equivalence, and the G-action on A is the natural action induced by the projective G-action on V.

Proof. Any automorphism of A is inner by the Skolem-Noether theorem. Hence, we obtain a function ρ̂ : G → A^{×} ⊂ GL(V) such that π(g)(a) = ρ̂(g)aρ̂(g)^{−1} for all g ∈ G and a ∈ A. Since π(1) = 1_{A}, we have ρ̂(1) ∈ Z(A)^{×} = F^{∗}. Setting ρ(g) = ρ̂(g)/ρ̂(1) gives ρ(1) = 1_{V}. Also, the equation π(gh) = π(g)π(h) implies that ρ(gh)ρ(h)^{−1}ρ(g)^{−1} is central and therefore a nonzero multiple of the identity. It follows that (V, ρ) is a projective representation of G giving rise to π.

In our study of invariant subalgebras, we will concentrate on the case of irreducible representations V. Accordingly, we introduce the definition:

Definition. A central simple G-algebra over F is called G-simple if the associated projective representation V is irreducible.

3. Invariant Subalgebras

We now consider subalgebras of the algebra A = End_{D}(V) which are preserved by the group action. (All subalgebras will be assumed to contain 1 unless otherwise specified.) In general, invariant subalgebras can be very badly behaved. For example, if we let G act trivially on End_{F} (V), then every subalgebra is invariant. This means that if V has dimension n, then End_{F} (V) contains every n-dimensional F -algebra as an invariant subalgebra. Moreover, it is not even true that the ring of invariants A^{G} need be semisimple, if G is infinite or G is finite with the characteristic of F dividing |G| [M]. We will therefore need to place additional restrictions on the G-algebra A.

We assume for the remainder of this section that A is G-simple. Note that under this hypothesis, the possible pathologies involving A^{G} are avoided, since by Schur’s lemma, A^{G} is a division algebra. We will show that all G-invariant subalgebras of A are semisimple with a very special structure. We will then prove two classification theorems. The first parameterizes invariant subalgebras in terms of induction data while the second provides a complete and explicit classification when F is algebraically closed.

3.1. Semisimplicity. As a first step, we show that invariant subalgebras are semisimple with isomorphic simple components.

Proposition 3.1. Let B be an invariant subalgebra of A. Then B is semisimple, and the Wedderburn components of B are all isomorphic as F -algebras. Moreover, if U is any simple B-submodule of V, then for each g ∈ G, ḡU is also a simple B-submodule, and any simple B-module is isomorphic to some ḡU.

Proof. The inclusion of B in A makes the A-module V into a B-module. Let U be a simple B-submodule of V; for example, take U to be a B-submodule of minimal dimension as an F -vector space. Consider the translate ḡU for g ∈ G. Note that the G-invariance of B implies that

bḡ(u) = ḡḡ^{−1}bḡ(u) = ḡ(g^{−1} · b)(u) ∈ ḡU (1)

for all b ∈ B and u ∈ U. Here, we have used the fact that \overline{g^{−1}} = α(g, g^{−1})ḡ^{−1}, where α is the cocycle defined by (V, ρ). Thus, ḡU is a B-submodule of V. Moreover, ḡU is simple, since the same argument shows that if W is a submodule of ḡU, then \overline{g^{−1}}W is a submodule of U. The sum ∑_{g∈G} ḡU is evidently a nonzero G-invariant subspace of V, and by irreducibility, V = ∑_{g∈G} ḡU. Thus, V is a semisimple B-module, and we can choose g_{1}, …, g_{l} ∈ G such that V = ⊕_{i=1}^{l}ḡ_{i}U.

Let u_{1}, …, u_{k} be an F -basis for U. The map B → ⊕_{i=1}^{l}k(\overline{g_{i}}U) given by b ↦ (b\overline{g_{1}}u_{1}, …, b\overline{g_{1}}u_{k}, …, b\overline{g_{l}}u_{1}, …, b\overline{g_{l}}u_{k}) is a B-homomorphism. If b is in the kernel, then b kills an F -basis of V, and since b ∈ A ⊆ End_{F} (V), we have b = 0; hence, the map is injective. This shows that B is a semisimple F -algebra, and any simple B-module is isomorphic to ḡU for some g ∈ G. The simple components of B are of the form End_{D_{g}}(ḡU), where D_{g} = End_{B}(ḡU). To complete the proof, it suffices to verify that End_{D_{g}}(ḡU) is isomorphic to End_{D′}(U), where D′ = D_{1}.

We first show that the division algebras D′ and D_{g} are isomorphic via the map d → ḡdḡ^{−1}. Using the formula for the B-action on ḡU given in (1), we have ḡdḡ^{−1}(bḡu) = ḡdḡ^{−1}(ḡ(g^{−1} · b)u)) = ḡd((g^{−1} · b)u) = ḡ(g^{−1} · b)d(u) = bḡd(u) = bḡdḡ^{−1}(ḡu) for all d ∈ D′ and u ∈ U, so that ḡdḡ^{−1} ∈ D_{g}. It is clear that this is an F -algebra homomorphism. In fact, it is an isomorphism with inverse map D_{g} → D′ given by d̂ ↦ \overline{g^{−1}}d̂\overline{g^{−1}}^{−1}. This follows since ḡ\overline{g^{−1}} ∈ F^{∗} and elements of D′ and D_{g} are F -linear.

Now suppose f ∈ End_{D′}(U). The F -map ḡfḡ^{−1} : ḡU → ḡU is D_{g} linear asḡfḡ^{−1}(ḡdḡ^{−1}(ḡu)) = ḡf(du) = ḡdf(u) = ḡdḡ^{−1}(ḡfḡ^{−1}(ḡu)). Thus, we have an F -algebra homomorphism End_{D′}(U) → End_{D_{g}}(ḡU), f ↦ ḡfḡ^{−1}, which is in

fact an isomorphism with inverse f̂ ↦ \overline{g^{−1}}f̂\overline{g^{−1}}^{−1}.

Corollary 3.2. The invariant subalgebra B is simple if and only if any for any simple B-submodule U of V, the B-modules U and ḡU are isomorphic for all g ∈ G.

3.2. Symmetrically embedded subalgebras. Although the proposition places significant restrictions on the structure of a G-invariant subalgebra, it turns out that the subalgebra must satisfy a much more stringent condition which depends on the ambient algebra A. For the time being, let B = B_{1} ⊕ … ⊕ B_{l} be an arbitrary semisimple subalgebra of A = End_{D}(V) where the B_{i}’s are simple F - algebras with corresponding simple modules W′_{i}. Note that V = ⊕_{i=1}^{l}m′_{i}W′_{i} with positive multiplicities m′_{i} (or else 1_{B_{i}}(V) = 0 for some i, contradicting the fact that the central primitive idempotent 1_{B_{i}} is a nonzero element of A). The subalgebra B consists of D-linear maps, so V can also be viewed as a (B, D^{op})-bimodule or equivalently as a B ⊗_{F} D-module. Since D is central simple, B ⊗ D is semisimple with Wedderburn components B_{1}⊗D, …, B_{l}⊗D, and we can write V = ⊕_{i=1}^{l}m_{i}W_{i}, where the W_{i}’s are the simple B ⊗ D-modules, with W_{i} = W′_{i} ⊗ D isomorphic to a minimal left ideal of B_{i} ⊗_{F} D. Again, each m_{i} is nonzero. In fact, we can say more.

Lemma 3.3. Let V′_{i} and V_{i} be the isotypic B and B ⊗_{F} D-submodules of V for W′_{i} and W_{i} respectively. Then V′_{i} = V_{i} and m_{i} = m′_{i}/ dim_{F} D. In particular, V′_{i} is a D-submodule of V.

Proof. Recall that V′_{i} and V_{i} are the one-eigenspaces of the central primitive idempotents 1_{B_{i}} and 1_{B_{i}} ⊗ 1_{D}. Since these are the same maps on V, we have V′_{i} = V_{i} for all i. Also dim W_{i} = (dim D)(dim W′_{i}), so m′_{i} = m_{i} dim D.

Definition. A semisimple subalgebra B of A = End_{D}(V) is called symmetrically embedded if the Wedderburn components of B are all isomorphic as F -algebras and if the simple B modules appearing in V have the same multiplicity m′, i.e. if m′ = m′_{1} = ⋯ = m′_{l}. (It is equivalent to replace either condition with the analogous statement involving B ⊗ D.)

More explicitly, an element b ∈ B acts on each copy of W_{i} in the same way, so is represented by a block diagonal matrix (in M_{dim_{D} V} (D^{op})) with m_{1} + ⋯ + m_{l} blocks, m_{i} of which consist of the dim_{D} W_{i} × dim_{D} W_{i} matrix corresponding to b|_{W_{i}}. If B is symmetrically embedded, then the blocks are all the same size and for each i, the matrix for b|_{W_{i}} appears m times. Note that this implies that dim_{D} V = ml dim_{D} W_{i} = ml dim_{F} W′_{i} for any i.

It is clear from the above lemma that whether a subalgebra satisfies the above property does not depend on the central division algebra D. Indeed, we have:

Proposition 3.4. Suppose that V is a module for two central division algebras D and D′. If B is a semisimple subalgebra of both A = End_{D}(V) and A′ = End_{D′}(V), then B is symmetrically embedded in A if and only if it is symmetically embedded in A′.

Proof. Since both A and A′ are subalgebras of End_{F} (V), we can assume without loss of generality that D′ = F, and this case follows immediately from the lemma.

In order to see the importance of symmetrically embedded subalgebras, we need to recall some information about centralizers of semisimple subalgebras of central simple algebras. Let Z_{A}(B) denote the centralizer in A of the subalgebra B. We call B a Howe subalgebra if it equals its double centralizer Z_{A}Z_{A}(B) and say that the pair (B, Z_{A}(B)) is a dual pair. A strong version of the Double Centralizer Theorem states that if B is semisimple, then Z_{A}(B) is also semisimple and B is a Howe subalgebra [J, Theorem 4.10]. In other words, the mapping B ↦ Z_{A}(B) provides a duality operator on the set of semisimple subalgebras of A. It is possible to calculate the Wedderburn structure of Z_{A}(B) by an argument due to Moeglin, Vignéras, and Waldspurger [MVW, p.12]. Note that f ∈ End_{B⊗_{F} D}(V) if and only if f is a Dlinear map which commutes with the action of B, i.e. if and only if f ∈ Z_{A}(B). Using the decomposition V = ⊕_{i=1}^{l}m_{i}W_{i} of V into simple B ⊗_{F} D-submodules, it is immediate that End_{B⊗D}(V) ≅ ⊕_{i=1}^{l}End_{B⊗D}(m_{i}W_{i}) ≅ ⊕_{i=1}^{l}M_{m_{i}}(D_{i}), where D_{i} = End_{B⊗_{F} D}(W_{i}) is a division algebra over F. Since W_{i} = W′_{i} ⊗ D, D_{i} is canonically isomorphic to End_{B}(W′_{i}). Summing up, we have:

Theorem 3.5. There is a duality on the set of semisimple subalgebras of A given by B ↦ Z_{A}(B) which preserves the number of Wedderburn components of the subalgebras. Moreover, if V ≅ ⊕_{i=1}^{l}m_{i}W_{i} is the decomposition of V into simple B ⊗_{F} D-modules and D_{i} is the division algebra End_{B}(W′_{i}) = End_{B⊗_{F} D}(W_{i}), then Z_{A}(B) ≅ ⊕_{i=1}^{l}M_{m_{i}}(D_{i}).

There are also two maps from a semisimple subalgebra to the set of self-dual (i.e. commutative) semisimple subalgebras, given by B ↦ Z(B), the center of B, and B ↦ Z_{0}(B), the F -linear span of the central primitive idempotents of B. These are respectively the largest and smallest self-dual subalgebras with the same central primitive idempotents as B. It is clear that both maps are constant on dual pairs. With the notation of the theorem, Z(B) ≅ ⊕_{i=1}^{l}Z(D_{i}) and Z_{0}(B) ≅ F^{l}.

We can now reformulate the concept of a symmetrically embedded subalgebra in terms of centralizers.

Proposition 3.6. A semisimple subalgebra B is symmetrically embedded in A if and only if both B and Z_{A}(B) are direct sums of isomorphic simple F -algebras.

Proof. Suppose B is symmetrically embedded. By definition, the Wedderburn components of B are all isomorphic. The Jacobson density theorem implies that B ⊗_{F} D ≅ End_{D_{i}}(W_{i}) for all i, and by the structure theorem for simple Artinian algebras, the D_{i}’s are all isomorphic as F -algebras. Since the multiplicities of the W_{i}’s in V are the same, it follows from Theorem 3.5 that Z_{A}(B) is a direct sum of isomorphic simple F -algebras.

Conversely, suppose that both B and Z_{A}(B) have isomorphic Wedderburn components. Then M_{m_{i}}(D_{i}) ≅ M_{m_{j}}(D_{j}) for all i and j, and so the m_{i}’s are equal by the structure theorem for simple Artinian algebras. Thus, B is symmetrically embedded in A.

Next, we need an easy, but important lemma on centralizers of invariant subalgebras.

Lemma 3.7. Let R be a G-algebra, and S a G-invariant subalgebra. Then the centralizer Z_{R}(S) is also an invariant subalgebra. In particular, the center of S Z(S) = Z_{S}(S) is an invariant subalgebra.

Proof. This follows immediately from the fact that (g · z)s = g · (z(g^{−1} · s)) = g · ((g^{−1} · s)z) = s(g · z) for all g ∈ G, s ∈ S, and z ∈ Z_{R}(S).

Combining the lemma with Propositions 3.1 and 3.6, we obtain the structure theorem:

Theorem 3.8. Let B be an invariant subalgebra of A. Then B is symmetrically embedded in A.

3.3. Induction. We now describe a fundamental construction of invariant subalgebras. We will then show that all invariant subalgebras are of this type and obtain a classification of them.

We first need to introduce induction of G-algebras. Let H be a subgroup of finite index in G, and suppose that C is an H-algebra. We show how to define a natural G-algebra structure on Ind_{H}^{G}(C) making Ind_{H}^{G} into a functor from the category of H-algebras into the category of G-algebras.

Proposition 3.9. There is a unique G-algebra structure on Ind_{H}^{G}(C) = F G⊗_{F H} C extending the H-algebra 1⊗C such that distinct G-translates of 1⊗C annihilate each other. If {g_{1}, …, g_{n}} is a left transversal for H in G, then the algebra multiplication is given by (g_{i} ⊗ b)(g_{j} ⊗ b′) = δ_{ij}(g_{i} ⊗ bb′) for b, b′ ∈ C. As F -algebras, Ind_{H}^{G}(C) is isomorphic to C^{n}. Furthermore, this definition makes Ind_{H}^{G} into a functor from the category of H-algebras into the category of G-algebras.

Proof. Uniqueness is clear. To show existence, recall that the coinduced representation Hom_{F H}(F G, C) (with G acting by (g·f)(x) = f(xg) for x, g ∈ G) is isomorphic to Ind_{H}^{G}(C) via the map ϕ ↦ ∑_{i=1}^{n} g_{i} ⊗ ϕ(g_{i}^{−1}). If ϕ and ψ are F H-linear, then ϕψ is as well, since (ϕψ)(hy) = (hϕ(y))(hψ(y)) = h · (ϕψ(y)) for h ∈ H and y ∈ F G. Thus, pointwise multiplication makes Hom_{F H}(F G, C) into a G-algebra; translating the multiplication back to Ind_{H}^{G}(C) gives the desired formula. The elements g_{i} ⊗ 1 are pairwise orthogonal central idempotents summing to the identity element in Ind_{H}^{G}(C), which is thereby isomorphic to ⨁_{i=1}^{n}(g_{i} ⊗ C) ≅ C^{n} as F -algebras.

Now let C′ be another H-algebra, and let ψ : C → C′ be an H-algebra map. It is immediate that the G-module map Ind_{H}^{G}(ψ) is also an algebra homomorphism. (Under the above identifications, it is just ψ ⊕ … ⊕ ψ : C^{n} → (C′)^{n}.) Thus, Ind_{H}^{G} is a functor.

Remarks. 1. If H does not have finite index in G, then Ind_{H}^{G}(B) is a nonunital G-algebra. Indeed, the coinduced representation is still a G-algebra, and Ind_{H}^{G}(B) is isomorphic to the nonunital subalgebra of F H-maps which are finitely supported modulo H.

2. If B is an interior H-algebra, i.e. H acts on B by inner automorphisms, then there is another way of defining an induced G-algebra originally introduced by Puig. These two concepts are quite different. Indeed, the underlying G-module in Puig’s construction is not Ind_{H}^{G}(B), but instead Ind_{H}^{G}(B) ⊗_{F H} F G. The resulting F -algebra structure is isomorphic to M_{n}(B) instead of B^{n} [T, §16].

It is easy to check that this functor satisfies the usual properties of induction.

Proposition 3.10. Let H be a subgroup of G of finite index, and suppose that C and C′ are H-algebras.

1. Ind_{H}^{G}(C⊕C′) ≅ Ind_{H}^{G}(C)⊕Ind_{H}^{G}(C′) and Ind_{H}^{G}(C∩C′) = Ind_{H}^{G}(C)∩Ind_{H}^{G}(C′) as G-algebras.

2. If C is an H-subalgebra of C′, then Ind_{H}^{G}(C) is a G-subalgebra of Ind_{H}^{G}(C′), and C = C′ if and only if Ind_{H}^{G}(C) = Ind_{H}^{G}(C′).

3. If H ≤ K ≤ G, then Ind_{K}^{G}(Ind_{H}^{K}(C)) = Ind_{H}^{G}(C).

We now return to our construction of invariant subalgebras. Suppose that V = Ind_{H}^{G}(W), where W is a D-module which is a projective representation of H. The cocycle defining ρ_{W} is just the restriction of α to H × H. It is automatic that W is irreducible. Since W is a direct summand of V_{H} =^{def} Res_{H}^{G}(V), H must act on W by D-linear automorphisms; this means that W is an F^{α}H ⊗ D-module. Note that the induced representation V comes equipped with a distinguished choice of F^{α}H-submodule isomorphic to W (namely ¯1 ⊗ W), and the invariant subalgebra we construct below depends on this choice. For ease of notation, we view W as this fixed H-submodule of V. Let T = {g_{1} = 1, g_{2}, …, g_{l}} be a left transversal of H, and set W_{i} = \overline{g_{i}} ⊗ W, a D-subspace of V = F^{α}G ⊗_{F^{α}H} W. Define a map Ψ_{(H,W,T)} : Ind_{H}^{G}(End_{D}(W)) → A = End_{D}(V) via the formula Ψ_{(H,W,T)}((g_{i} ⊗ f))(\overline{g_{j}} ⊗ w) = δ_{ij}\overline{g_{i}} ⊗ f(w) for f ∈ End_{D}(W), w ∈ W and extending by linearity.

Lemma 3.11. The map Ψ_{(H,W)} = Ψ_{(H,W,T)} is independent of the choice of transversal. It is an injective G-algebra homomorphism whose image is the block-diagonal subalgebra ⊕_{i=1}^{l}End_{D}(W_{i}). In particular, this subalgebra is G-invariant.

Proof. Let Ψ = Ψ_{(H,W,T)}. It is easy to see that Ψ is an embedding of algebras with the specified image, so we need only check that Ψ is an intertwining map. Fix g ∈ G. There exists a permutation σ = σ_{g} ∈ S_{l} and elements h_{i} ∈ H such that gg_{i} = g_{σ(i)}h_{σ(i)} for all i. First note that

\begin{aligned}Ψ(g · (g_{i} ⊗ f))(\overline{g_{j}} ⊗ w) = Ψ(gg_{i} ⊗ f)(\overline{g_{j}} ⊗ w) & = Ψ(g_{σ(i)} ⊗ h_{σ(i)} · f)(\overline{g_{j}} ⊗ w) \\ & = δ_{σ(i)j}\overline{g_{j}} ⊗ (h_{j} · f)(w). \end{aligned}

On the other hand, a similar calculation using the definition of multiplication in F^{α}G gives

(g · Ψ(g_{i} ⊗ f))(\overline{g_{j}} ⊗ w) = δ_{σ(i)j}β\overline{g_{j}} ⊗ \overline{h_{j}}f(\overline{h_{j}}^{−1}w),

where

β = α(g_{j}, h_{j})^{−1}α(h_{j}, h_{j}^{−1})α(g, g^{−1}g_{j}h_{j})α(g^{−1}g_{j}h_{j}, h_{j}^{−1})^{−1}α(g^{−1}, g_{j})α(g, g^{−1})^{−1}.

Applying the cocycle condition and the fact that α(x, 1) = 1 = α(1, x) for all x ∈ G, we get

\begin{aligned}β & = α(g_{j}, h_{j})^{−1}α(h_{j}, h_{j}^{−1})α(g_{j}h_{j}, h_{j}^{−1})^{−1}α(g, g^{−1}g_{j})α(g^{−1}, g_{j})α(g, g^{−1})^{−1} \\ & = α(g_{j}, h_{j})^{−1}α(h_{j}, h_{j}^{−1})α(g_{j}h_{j}, h_{j}^{−1})^{−1} = 1, \end{aligned}

as desired.

The verification that Ψ does not depend on the transversal is similar, but easier.

Let C be an invariant subalgebra of the H-algebra End_{D}(W). It now follows from Proposition 3.10 and the lemma that Ψ(Ind_{H}^{G}(C)) is a G-invariant subalgebra of A = End_{D}(V). More precisely,

Proposition 3.12. The map C ↦ Θ_{(H,W,C)} =^{def} Ψ_{(H,W)}(Ind_{H}^{G}(C)) defines an injective lattice homomorphism from the H-invariant subalgebras of End_{D}(W) to the G-invariant subalgebras of ⊕_{i=1}^{l}End_{D}(W_{i}) ⊂ A.

It is not true that an invariant subalgebra of A can be expressed uniquely in terms of this construction if the initial data (namely H, W, and C) are allowed to vary. Indeed, conjugate data (i.e. gHg^{−1}, ḡW, and g · C ⊆ End_{D}(ḡW) for some g ∈ G) produces the same invariant subalgebra. However, we will see below that uniqueness does hold if we restrict ourselves to conjugacy classes of initial data with C simple.

3.4. The first classification theorem. We now show that all invariant subalgebras are obtained from this induction procedure. First, we associate a transitive permutation representation of G to any invariant subalgebra B. By 3.1, we can write B = B_{1} ⊕ ⋯ ⊕ B_{l}, where the B_{i} are simple. The restriction of the G-action π to B gives rise to a permutation representation of G on the set of B_{i}’s because the algebra automorphism π(g) must permute the minimal two-sided ideals of B. More explicitly, let X = {e_{1}, …, e_{l}} with e_{i} = 1_{B_{i}} be the set of central primitive idempotents of B. Since e_{i} is the unique nonzero idempotent in the center of B_{i}, it is clear that if π(g)(B_{i}) = B_{j}, then π(g)(e_{i}) = e_{j}. We thus obtain a homomorphism π̄_{B} : G → S_{l}, where we have identified S(X) with S_{l} in the obvious way. Note that X is also the set of central primitive idempotents in Z_{A}(B) and Z(B). Accordingly, dual pairs give rise to the same permutation representation, as do any invariant subalgebras with the same center.

The permutation representation π̄_{B} can also be defined in terms of the B-isotypic components of V. Recall that V = ⊕_{i=1}^{l}V_{i} where V_{i} = V′_{i} is the isotypic Bsubmodule of V corresponding to B_{i}. Fix g ∈ G and v ∈ V_{j}, and, write ḡ^{−1}(v) = ∑_{i=1}^{l} v_{i}^{g} with v_{i}^{g} ∈ V_{i}. Note that ḡ^{−1}(g ·e_{i})(v) = e_{i}(ḡ^{−1}(v) = v_{i}^{g}. But by definition, g · e_{i} = e_{π̄_{B}(g)(i)}, giving v_{i}^{g} = ḡ^{−1}(g · e_{i})(v) = δ_{j,π̄_{B}(g)(i)}ḡ^{−1}(v) = δ_{i,π̄_{B}(g^{−1})(j)}ḡ^{−1}(v). This implies that ḡ^{−1}(V_{j}) ⊆ V_{π̄_{B}(g^{−1})(j)} for all j. Applying this to g^{−1} (or using the fact that ρ(g)^{−1} is surjective) gives the reverse inclusion. Thus, G permutes the V_{i}’s, and this permutation is just π̄_{B}.

Proposition 3.13. The permutation representation π̄_{B} is transitive.

Proof. Let U be a simple B-submodule of V isomorphic to a minimal left ideal of B_{1}. By definition, e_{1} is the identity map on U. Let e_{i} be any central primitive idempotent, and choose g ∈ G such that ḡU is a simple B_{i} module. For all u ∈ U, we have (g · e_{1})(ḡu) = ḡ(e_{1}(ḡ^{−1}(ḡu))) = ḡ(e_{1}(u)) = ḡ(u). Since e_{i} is the unique central primitive idempotent acting as the identity on ḡU, this implies that g · e_{1} = e_{i}.

If G acts on B by inner automorphisms, then the G-action preserves the simple components of B. We thus obtain the useful corollary:

Corollary 3.14. If G acts on the invariant subalgebra B by inner automorphisms, then B is simple.

Let H_{i} = {g ∈ G | g · e_{i} = e_{i}} be the inertia subgroup of e_{i}. Note that it has finite index l in G. It is immediate that V_{i} is an F^{α}H_{i} ⊗ D submodule of V, and the transitivity of π̄_{B} implies that V = Ind_{H_{i}}^{G}(V_{i}), i.e. V is isomorphic to the induced representation and has distinguished H_{i}-submodule V_{i}. Moreover, V_{i} is an (F -)irreducible projective representation of H_{i} because if M were a proper subrepresentation, then Ind_{H_{i}}^{G}(M) would be a proper G-submodule of V, contradicting the irreducibility of V. The algebra B_{i} is a simple H_{i}-subalgebra of End_{D}(V_{i}), and we are precisely in the situation of the fundamental construction. The uniqueness part of Proposition 3.9 shows that B = Θ_{(H_{i},V_{i},B_{i})}. We have thus realized B in l different ways, all of which have conjugate initial data.

Now suppose that B = Θ_{(H,W,C)}. By definition, W is the isotypic B-submodule corresponding to the simple component C (i.e. 1 ⊗ C) of B = Ind_{H}^{G}(C), implying that W = V_{j} and C = B_{j} for some j. Also, H is the stabilizer of B_{j}, so in fact H = H_{j}.

We are now ready to state the first classification theorem. Let 𝒟 be the set of equivalence classes of triples (H, W, C) where H is a subgroup of finite index in G, W is an F^{α}H ⊗ D submodule of V such that V ≅ Ind_{H}^{G}(W), and C is a simple invariant subalgebra of the H-algebra End_{D}(W). Also, let 𝒟_{(H,W)} ⊂ 𝒟 be the subset of classes with a representative of the form (H, W, C).

Theorem 3.15 (First Classification Theorem). Let A = End_{D}(V) be a G-simple central simple algebra. The map (H, W, C) ↦ Θ_{(H,W,C)} gives a bijective correspondence between 𝒟 and the set of unital G-invariant subalgebras of A. This bijection preserves dual pairs and centers; if B = Θ_{(H,W,C)}, then Z_{A}(B) = Θ_{(H,W,Z_{End_{D} (W)}(C))} and Z(B) = Θ_{(H,W,Z(C))}. Similarly, Z_{0}(B) (the F -linear span of the Wedderburn components of B) is just Θ_{(H,W,Z_{0}(C))} = Θ_{(H,W,F 1_{End_{D} (W)})}. Furthermore, the image of 𝒟_{(H,W)} under the correspondence is precisely the set of invariant subalgebras B with Z_{0}(B) = Θ_{(H,W,F 1_{End_{D} (W)})}.

Proof. We have already shown that there is a bijection between invariant subalgebras and triples (H, W, C) where W is an F^{α}H-submodule of V such that the obvious map W → ¯1⊗W extends to an isomorphism V ≅ Ind_{H}^{G}(W). This amounts to saying that V is the internal direct sum of the translates \overline{g_{i}}W (and so V can be viewed as equal and not just isomorphic to Ind_{H}^{G}(W)). The following lemma shows that any subrepresentation of V_{H} isomorphic to W satisfies this condition.

Lemma 3.16. Suppose that V = Ind_{H}^{G}(W) with V irreducible. Then if W′ is any subrepresentation of V_{H} isomorphic to W, V is the internal direct sum of the \overline{g_{i}}W′’s.

Proof. By Frobenius reciprocity, there is a linear isomorphism Hom_{F^{α}H}(W, V_{H}) ≅ Hom_{F^{α}G}(V, V) given by f ↦ f̂, with f̂(\overline{g_{i}} ⊗ w) = \overline{g_{i}}f(w). Let f : W → V_{H} be an H-map with image W′. Since V is irreducible, f̂ is an isomorphism. Accordingly, V is the direct sum of the distinct G-translates of f(W) = W′.

It only remains to prove the last three statements. We have shown that as an F -algebra, Θ_{(H,W,C)} is just C^{[G:H]} embedded in the block diagonal subalgebra ⊕_{i=1}^{l}End_{D}(W_{i}) ⊂ A. Since taking finite direct sums commutes with taking dual pairs, centers, and Z_{0}, the result follows.

Remarks. 1. Since an invariant subalgebra B can always be expressed trivially as Θ_{(G,V,B)}, it is clear that a nonsimple B can arise from nonconjugate initial data. The class in 𝒟 corresponding to B consists of the triples with minimal H (or W or C).

2. Let F be an infinite field. If V ≅ Ind_{H}^{G}(W) and V_{H} does not have a unique subrepresentation isomorphic to W, then A has an infinite number of invariant subalgebras. Indeed, in this case, the W -isotypic submodule of V_{H} is a direct sum of t ≥ 2 submodules isomorphic to W, so there are an infinite number of submodules W′ isomorphic to W. At most [G : H] of these submodules can be conjugate, and each class gives rise to a distinct invariant subalgebra Θ_{(H,W′,F)}.

Before proceeding, we give two examples in the case A = End_{F} (V).

Examples. 1. Let V be primitive, i.e. suppose that V is not induced from any proper subgroup. Then all invariant subalgebras of A are simple.

2. The theorem shows that V is a monomial representation, i.e. it is induced from a linear character, if and only if End_{F} (V) has a G-invariant split Cartan subalgebra 𝔥. Indeed, this can be shown directly. By choosing an appropriate basis for V, we can view 𝔥 as the subalgebra of diagonal matrices in M_{n}(F). Note that for 𝔥 to be G-invariant means precisely that its normalizer N(𝔥) contains ρ(G). But N(𝔥) is the set of monomial matrices, and it is well known that V is monomial if and only if ρ(G) consists of monomial matrices with respect to some basis for V. [I, p.67].

The correspondence in this theorem becomes much simpler when V has nice rationality properties. Recall that a projective F -representation V is called absolutely irreducible if V_{E} = V ⊗E is an irreducible projective E-representation for every algebraic extension E of F. Equivalently, the division algebra End_{G}(V) =^{def} End_{F^{α}G}(V) is just the ground field F. Note that if F is algebraically closed, then all irreducible representations are absolutely irreducible.

Lemma 3.17. Let A be G-simple. If K = End_{G}(V), then D = End_{A}(V) ⊆ K. In particular, if V is absolutely irreducible, then D = F and A = End_{F} (V).

Proof. Choose d ∈ D. Then we have d(ρ(g)v) = ρ(g)(dv) for g ∈ G, v ∈ V, since ρ(g) ∈ A. Hence, d ∈ K.

If V is absolutely irreducible, we call such A = End_{F} (V) absolutely G-simple.

Now suppose that H is a subgroup of finite index and W is an (irreducible) F^{α}H-module such that V ≅ Ind_{H}^{G}(W). Here, we are not viewing W as a specific subspace of V. If V is absolutely irreducible, then Hom_{F^{α}G}(Ind_{H}^{G}(W), V) is onedimensional. By Frobenius reciprocity, the same is true for Hom_{F^{α}H}(W, V_{H}). This implies that there is a unique subrepresentation of V_{H} isomorphic to W, since otherwise there would be linearly independent H-maps W → V_{H}. Similarly, we must have End_{H}(W) = F. Summing up:

Proposition 3.18. Let V be absolutely irreducible, and suppose that V ≅ Ind_{H}^{G}(W) where H is a subgroup of finite index and W is an irreducible F^{α}H-module. Then there is a unique subrepresentation of V_{H} isomorphic to W. Moreover, W is absolutely irreducible.

Let 𝒟̃ be the set of conjugacy classes of triples where W is only defined up to isomorphism, i.e. W is no longer viewed as a specific subspace of V. In other words, 𝒟̃ consists of the classes of 𝒟 modulo H-isomorphism of the second variable. It is clear that triples in 𝒟 representing the same class in 𝒟̃ give rise to invariant subalgebras that are isomorphic as G-algebras. If V is absolutely irreducible, the previous proposition shows that the projection 𝒟 → 𝒟̃ is a bijection. Accordingly, we get the first statement of the corollary:

Corollary 3.19. Let A = End_{F} (V) be absolutely G-simple. The map (H, W, C) ↦ Θ_{(H,W,C)} gives a bijective correspondence between 𝒟̃ and the set of unital G-invariant subalgebras of A. In addition, Θ_{(H,W,C)} is separable; equivalently, Z(C) is a separable field extension of F.

Proof. Write B = Θ_{(H,W,C)}. Extending scalars to the algebraic extension E gives the invariant subalgebra B_{E} of the central simple E-algebra A_{E} ≅ End_{D_{E}}(V_{E}). Since V_{E} is irreducible, Proposition 3.1 applies, showing that B_{E} is semisimple. Thus, B is separable.

3.5. Invariant central simple algebras. Theorem 3.15 shows that in order to understand the invariant subalgebras of A, it suffices to understand simple invariant subalgebras of certain associated G-simple algebras. We now classify the invariant central simple subalgebras of any G-simple A.

Let B be a simple subalgebra of A = End_{D}(V) with simple B-module W′ and simple B ⊗ D-module W = W′ ⊗ D. The B ⊗ D-module V is isotypic, say V ≅ mW. Let L = End_{B}(W′) = End_{B⊗D}(W) and set U = (L^{op})^{m}. We obtain the factorization V ≅ W ⊗_{L^{op}} U ≅ (W′ ⊗_{L^{op}} U) ⊗_{F} D. As shown in the proof of Theorem 3.5, Z_{A}(B) = End_{L^{op}}(U); also, B ≅ End_{L}(W′) ≅ End_{L⊗D}(W). In addition, any dual pair of simple subalgebras arises in this way.

Proposition 3.20. Let A = End_{D}(V) be a central simple algebra. If V ≅ W ⊗_{L^{op}} U ≅ (W′⊗_{L^{op}}U)⊗_{F} D with W′ an L-module, U an L^{op}-module, and W = W′⊗D an L ⊗ D-module, then End_{L}(W′) and End_{L^{op}}(U) is a dual pair of simple subalgebras. Conversely, any dual pair of simple subalgebras comes from such a factorization. In addition, the subalgebras are central simple if and only if L is a central division algebra.

Using this result, we can classify invariant central simple subalgebras. Let L be a central division algebra, and let W′ and U be L and L^{op} modules respectively which are projective representations given by G →^{ρ_{W′}} End_{L}(W′)^{×} and G →^{ρ_{U}} End_{L^{op}}(U)^{×}. Set V = (W′ ⊗_{L^{op}} U) ⊗_{F} D, and let τ denote the canonical isomorphism End_{L}(W′) ⊗_{F} End_{L^{op}}(U) →^{τ} End_{D}(V) given by τ(f_{1} ⊗ f_{2})(w′ ⊗ u ⊗ d) = f_{1}(w)⊗f_{2}(u)⊗d. Then ρ_{V} : G → End_{D}(V)^{×} defined by ρ_{V} (g) = τ(ρ_{W′}(g)⊗ρ_{U}(g)) makes V into a projective representation. It is easy to check that τ becomes a Galgebra isomorphism. If ρ_{W′} and ρ_{U} are twisted by (one-dimensional) projective characters, then the new G-action on V is projectively equivalent to the old one.

Conversely, suppose that V is a projective representation, and End_{L}(W′) and End_{L^{op}}(U) are invariant. The map τ is thus a G-algebra isomorphism. By Proposition 2.1, the G-actions on these subalgebras come from projective representations (W′, ρ_{W′}) and (U, ρ_{U}). Hence, τ^{−1}(ρ_{W′} ⊗ ρ_{U}) and ρ_{V} define the same G-algebra structure on End_{D}(V), implying that they are projectively equivalent, i.e. differ by a projective character. Modifying ρ_{U} by this twist, we get ρ_{V} = τ(ρ_{W′} ⊗ ρ_{U}). It is obvious that if V is irreducible, then both W′ and U must be as well. This proves the following theorem:

Theorem 3.21. Let A = End_{D}(V) be G-simple. Suppose that V ≅ (W′⊗_{L^{op}}U)⊗_{F} D is a factorization such that L is a central division algebra and W′ and U are (irreducible) projective representations of G (via L and L^{op} linear automorphisms respectively). Then A ≅ End_{L}(W′)⊗End_{L^{op}}(U) as G-algebras and the images of the two factors in A are a dual pair of invariant central simple subalgebras. Conversely, any such dual pair arises in this way.

Remark. If D = F, invariant central simple subalgebras come from expressing V as the tensor product of projective representations. In general, finding all (or even some) factorizations for a given V is a difficult problem. See for example [St].

We can say more when V is absolutely irreducible. Recall that in this case, D = F and W = W′ ⊗_{F} D = W′. Since End_{F^{α}G}(V) = F, any two G-maps W ⊗_{L^{op}} U →^{∼} V are scalar multiples of each other and thus give the same dual pair of invariant central simple subalgebras. Thus, the specific factorization does not matter.

Corollary 3.22. If A is absolutely G-simple, then there is a one-to-one correspondence between pairs of irreducible projective representations (W, U) modulo projective equivalence such that V ≅ (W ⊗_{L^{op}} U) and dual pairs of invariant central simple subalgebras.

3.6. The second classification theorem. We are ready to make the correspondence in Theorem 3.15 entirely explicit when F is algebraically closed. Let ℰ′ be the set of quadruples (H, W, W_{1}, W_{2}) where H is a subgroup of G of finite index, W is an irreducible projective representation of H such that V ≅ Ind_{H}^{G}(W), and W_{1}, and W_{2} are irreducible projective representations of H such that W ≅ W_{1} ⊗_{F} W_{2}. We then let ℰ be the set of equivalence classes of ℰ′ where two quadruples (H, W, W_{1}, W_{2}) and (H′, W′, W′_{1}, W′_{2}) are equivalent if there exists g ∈ G such that H′ = H^{g}, W′ = W^{g}, and W′_{i} is projectively equivalent to W_{i}^{g}. We let ℰ_{(H,W)} ⊂ ℰ be the subset of classes with a representative of the form (H, W, W_{1}, W_{2}). In addition, we denote by C(W_{1}, W_{2}) the image of End_{F} (W_{1}) ⊗ 1 under the isomorphism End_{F} (W_{1}) ⊗ End_{F} (W_{2}) → End_{F} (W). The trivial factorizations give C(F, W) = F and C(W, F) = End_{F} (W). We can now state the second classification theorem.

Theorem 3.23 (Second Classification Theorem). Let F be algebraically closed and A = End_{F} (V) a G-simple algebra. Then the map (H, W, W_{1}, W_{2}) ↦ Θ_{(H,W,C(W_{1},W_{2}))} gives a bijective correspondence between ℰ and the set of invariant subalgebras of A. Moreover, the duality on invariant subalgebras is given by interchanging the W_{i}’s, i.e. Z_{A}(Θ_{(H,W,C(W_{1},W_{2}))}) = Θ_{(H,W,C(W_{2},W_{1}))}. The image of ℰ_{(H,W)} under the correspondence is precisely the set of invariant subalgebras B with center Θ_{(H,W,C(F,W))}.

Proof. Recall that 𝒟̃ is the set of classes of triples (H, W, C) where H and W are defined as in ℰ and C is a (central) simple subalgebra of End_{F} (W) (using the fact that F is algebraically closed). Since V is absolutely irreducible, Corollary 3.19 shows that invariant subalgebras are parameterized by this set. Applying Corollary 3.22, we see that the map (H, W, W_{1}, W_{2}) ↦ (H, W, C(W_{1}, W_{2})) induces a bijection ℰ → 𝒟̃, and we obtain the desired correspondence. Since Z_{End_{F} (W)}(C(W_{1}, W_{2})) = C(W_{2}, W_{1}) and Z(C(W_{1}, W_{2})) = C(F, W) , the last statements follow from Theorem 3.15.

Remark. Note that the cocycle α does not determine the cocycles defined by ρ_{W_{1}} and ρ_{W_{2}}. In particular, even if V is a linear representation, it is not possible to avoid considering projective representations when studying invariant subalgebras of End_{F} (V).

It is convenient to reformulate this correspondence in terms of covering groups. Recall that G̃ is an F^{∗}-generalized covering (or representation) group for G if it is a central extension of G satisfying the projective lifting property for projective representations over F. It is known that F^{∗}-generalized covering groups always exist. If F is algebraically closed and G is finite, then we can choose G̃ finite of order |G||H^{2}(G, F^{∗})|; such a group is called an F^{∗}-covering group for G [BT].

We now assume that F is algebraically closed (so D and L are just F and W = W′). Suppose that the projective representation V factors as V ≅ W ⊗_{F} U. Choose a linear representation (V, ρ̃_{V}) of G̃ lifting ρ_{V} and similarly for W and U. A priori, V is only projectively equivalent to W ⊗ U over G̃. However, if V_{1} and V_{2} are linear representations which are projectively equivalent, then V_{1} ≅ V_{2} ⊗ λ, where λ is a linear character. Thus, by choosing a different lift for ρ_{W}, we obtain linear representations of G̃ such that V ≅ W ⊗_{F} U as G̃-modules. On the other hand, it is obvious that any such factorization gives an isomorphism of projective representations for G.

This allows us to redefine ℰ_{(H,W)}. Let H̃ be a generalized covering group for H, and fix a lift of W to a linear representation of H̃. If (W_{1}, W_{2}) and (W′_{1}, W′_{2}) are two pairs of linear representations of H̃ satisfying W ≅ W_{1} ⊗ W_{2} ≅ W′_{1} ⊗ W′_{2}, we say they are equivalent if for some linear character λ of H̃, W′_{1} ≅ W_{1} ⊗ λ and W′_{1} ≅ W_{1} ⊗ λ^{−1}. Denote the set of such classes by ℱ_{(H,W)}. The previous observations give the following result.

Lemma 3.24. There is a natural bijection between ℰ_{(H,W)} and ℱ_{(H,W)}.

Let Y be a complete set of representatives of the conjugacy classes of pairs (H, W). Then the ℰ_{y}’s partition ℰ. Set ℱ = ∐_{y∈Y} ℱ_{y}. We obtain a modified second classification theorem:

Theorem 3.25. Let F be algebraically closed and A = End_{F} (V) a G-simple algebra. Then the map (H, W, W_{1}, W_{2}) ↦ Θ_{(H,W,C(W_{1},W_{2}))} gives a bijective correspondence between ℱ and the set of invariant subalgebras of A. Duals and centers of invariant subalgebras are given by the same formulas as before.

It is possible to avoid all explicit mention of projective representations in classifying invariant subalgebras. In order to do this, choose a generalized covering group G̃ of G and fix a lift of V to a representation of G̃. Since the G and G̃ invariant subspaces of A are the same, we can apply the above procedure to the G̃-simple algebra A. Note that this will require choosing a generalized covering group ˜ G̃ of G̃!

3.7. Finiteness results. If F is not algebraically closed, it is not true in general that a simple G-algebra A will have a finite number of invariant subalgebras, even when G is finite. We have already seen a way that finiteness can fail if F is infinite and V is not absolutely irreducible. Namely, if V ≅ Ind_{H}^{G}(W) and V_{H} does not have a unique subrepresentation isomorphic to W, then for any simple H-invariant C ⊂ End_{D}(W), the set {Θ_{(H,W′,C)} | W′ ⊂ V, W′ ≅ W } will be infinite. Note that these subalgebras are all nonsimple.

Furthermore, the set of invariant subalgebras can be infinite even when V is primitive. Indeed, we have the proposition:

Proposition 3.26. Let A = End_{F} (V) where V is an irreducible projective representation of G, and suppose that the division algebra End_{G}(V) is not a field. Then 𝒟_{(G,V)} is infinite, i.e End_{F} (V) has an infinite number of simple invariant subalgebras.

Proof. Note that any subalgebra of End_{G}(V) = (End_{F} (V))^{G} is G-invariant, so the following lemma gives the result.

Lemma 3.27. Let D be a noncommutative central F -division algebra. Then D contains an infinite number of distinct subfields.

Proof. Choose noncommuting elements u, v ∈ D, and consider the subfields F_{a} = F (u + av) for a ∈ F. Wedderburn’s theorem on finite division rings shows that the field F is infinite, so it suffices to show that F_{a} = F_{b} if and only if a = b. If F_{a} = F_{b}, then u + av and u + bv commute, implying that auv + bvu = buv + avu. Therefore, (a − b)(uv − vu) = 0, and since uv ≠ vu, a = b follows.

However, these pathologies cannot occur when F is algebraically closed.

Theorem 3.28. Let F be algebraically closed, G a finite group, and A = End_{F} (V) a G-simple algebra. Then A has a finite number of invariant subalgebras.

Proof. Replacing G by a covering group (which is also finite), we can assume without loss of generality that V is a linear representation of G. Since the set of invariant subalgebras and ℱ = ∐_{y∈Y} ℱ_{y} have the same cardinality (using the notation of Theorem 3.25), it suffices to show that Y and the ℱ_{y}’s are finite.

Recall that a finite group H has at most |H| non-isomorphic irreducible representations over any field K. (This follows from the Jordan-Hölder Theorem, since any irreducible KH-module can be realized as a composition factor of KH.) The set Y is finite because it is contained in the set of all pairs (H, W) where H is a subgroup of G and W is an isomorphism class of irreducible F H-modules. Also, ℱ_{(H_{y},W_{y})} is finite, since it is smaller than the set of arbitrary pairs of isomorphism classes of irreducible F H̃-modules, where H̃ is a covering group for H.

3.8. Nonunital invariant subalgebras. We conclude this section with an application to nonunital invariant subalgebras.

Proposition 3.29. Let F be an algebraically closed field and V an irreducible primitive projective representation of G. Then {0} is the only nonunital invariant subalgebra of A = End_{F} (V). Equivalently, any nonzero subrepresentation of A closed under multiplication must contain the identity.

Proof. We begin with a lemma.

Lemma 3.30. Let F be an algebraically closed field. For t ≥ 2, the matrix algebra M_{t}(F) has no nonunital subalgebras of codimension one.

Proof. Suppose that Q is a nonunital subalgebra of codimension one. First note that any element of Q must be singular. To see this, take q ∈ Q invertible, so that det q ≠ 0. It is a well-known corollary of the Cayley-Hamilton theorem that q^{−1} can be expressed as a polynomial in q, so q^{−1} ∈ Q. This implies that Q contains the identity, a contradiction. Thus, Q ⊆ V (det), the hypersurface of M_{t}(F) cut out by the determinant. But Q is also a codimension one linear subvariety, so Q = V (f) for some homogeneous degree one polynomial f. As a result, f divides det, and this cannot be true, since the determinant is an irreducible polynomial of degree t.

Now, let Q be an nonunital invariant subalgebra. Then Q′ = Q+F 1_{A} is a unital invariant subalgebra. We know from the first example after Theorem 3.15 that Q′ is simple, hence isomorphic to M_{t}(F) for some t ≥ 1. If t = 1, then Q = {0}. Applying the lemma finishes the proof.

4. Invariant subalgebras for topological and Lie groups

In this section, we illustrate our results on invariant subalgebras in the case where V is a continuous irreducible complex projective representation of a compact connected Lie group. For the moment, we consider a more general situation. Suppose that G is a topological group, A = End_{D}(V) is a G-simple algebra endowed with a T_{1} topology, and G acts continuously on A. For example, the topology on A could come from F having the structure of a T_{1} topological field or End_{F} (V) could be given the Zariski topology. So far, this setting includes every abstract group G and G-algebra considered in the previous section by giving G and A the discrete topology. In order to avoid this type of triviality, we further assume that the connected component of the identity G^{o} (a closed normal subgroup) acts irreducibly on V. We call such an algebra topologically G-simple.

Proposition 4.1. Every invariant subalgebra of a topologically G-simple algebra A is simple.

Proof. A G-invariant algebra is also G^{o}-invariant, so it suffices to assume that G is connected. Let X be the set of central primitive idempotents of an invariant subalgebra B. The transitivity of π_{b} shows that X is connected. However, since A is T_{1}, X is discrete. This implies that X is a singleton, i.e. B is simple.

If we further assume that F is algebraically closed, Theorem 3.22 now applies to give a classification of the invariant subalgebras of A = End_{F} (V) in terms of factorizations V ≅ W_{1} ⊗ W_{2} modulo projective equivalence.

We now assume that F = C and G is a compact Lie group. Note that a continuous homomorphism G → Aut_{F −alg}(A) ⊂ GL(A) is a continuous homomorphism G → P GL(V). Thus, if A is a continuous G-algebra, then V is a continuous projective representation.

Lemma 4.2. Suppose that G is a simple compact connected Lie group, and let V (λ) and V (μ) be irreducible representations with highest weights λ and μ. Then V (λ) ⊗ V (μ) is irreducible if and only if λ or μ is 0.

Proof. Since V (λ + μ) is a component of V (λ) ⊗ V (μ), it suffices to compare the dimension of these representations. The Weyl dimension formula states that

dim V (λ) = ∏_{α∈R^{+}} \frac{⟨α, λ + ρ⟩}{⟨α, ρ⟩},

where R^{+} is the set of positive roots, ρ is half the sum of the positive roots, and ⟨ , ⟩ is the Killing form. The equation ⟨α, λ + μ + ρ⟩⟨α, ρ⟩ + ⟨α, λ⟩⟨α, μ⟩ = ⟨α, λ + ρ⟩⟨α, μ + ρ⟩ shows that

\frac{⟨α, λ + μ + ρ⟩}{⟨α, ρ⟩} ≤ \frac{⟨α, λ + ρ⟩}{⟨α, ρ⟩} \frac{⟨α, μ + ρ⟩}{⟨α, ρ⟩},

with equality if and only if ⟨α, λ⟩⟨α, μ⟩ = 0. Here we have used the fact that ⟨α, λ⟩ ≥ 0 and ⟨α, ρ⟩ > 0 for every positive root α and dominant weight λ. If β is the highest root, then ⟨β, ν⟩ > 0 for any nonzero dominant weight ν. Multiplying over all positive roots, it follows easily that dim V (λ + μ) < dim V (λ) dim V (μ) if and only if both λ and μ are nonzero.

Let G be a compact connected Lie group. It is well known that the universal covering group of G is of the form G̃ = G_{1} × ⋯ × G_{s} × R^{n}, where each G_{i} is a simple, simply connected, compact Lie group. Let V be an irreducible projective representation of G. Then V can be lifted to an irreducible representation of G̃, which can be expressed as V_{1} ⊗ … ⊗ V_{s} ⊗ L, where V_{i} is a complex irreducible representation of G_{i} and L is a character of R^{n}. This means that V is projectively equivalent to Ṽ = V_{1} ⊗ … ⊗ V_{s}. Moreover, simple Lie groups have no nontrivial characters, so projective and linear equivalence are the same for representations of G_{1} × ⋯ × G_{s}. The lemma shows that any factorization of Ṽ = W ⊗ W′ into the tensor product of two representations of G̃ must have W and W′ as complementary partial products of V_{1} ⊗ … ⊗ V_{s}. More precisely, let I = {i | V_{i} ≠ C} and take J ⊂ I. Set W_{J} = ⨂_{i=1}^{s} W_{Ji} and W′_{J} = ⨂_{i=1}^{s} W′_{Ji}, where W_{Ji} is V_{i} if i ∈ J and C otherwise and W′_{Ji} is V_{i} if i ≠ J and C otherwise. We get a factorization Ṽ = W_{J} ⊗W′_{J}, and J ↦ W_{J} gives a one-to-one correspondence between the subsets of I and the factors of Ṽ. This observation combined with Theorem 3.22 proves the following theorem due to Etingof:

Theorem 4.3. Let G be a compact connected Lie group, and let A = End_{C}(V) where V is an irreducible projective representation of G projectively equivalent to V_{1} ⊗ … ⊗ V_{s}. Then there is a bijective correspondence between 𝒫(I), the power set of I = {i | V_{i} ≠ C}, and the set of invariant subalgebras of A, given by J ↦ End_{C}(W_{J}). Moreover, the duality operator corresponds to taking complements in 𝒫(I), i.e. it is given by End_{C}(W_{J}) ↦ End_{C}(W_{I−J}).

By Theorem 3.22, any nontrivial invariant subalgebras contains 1_{A}, so we obtain the corollary:

Corollary 4.4. There are exactly 2^{|I|} + 1 subrepresentations of End_{C}(V) which are closed under matrix multiplication: 2^{|I|} unital subalgebras and {0}.

In particular, if G is a simple compact connected Lie group, then no topologically G-simple algebra has any nontrivial invariant subalgebras. It would be interesting to find classes of finite group satisfying this property and to find a group-theoretic characterization of such groups. It is not true that finite simple groups have this property. In the notation of the Atlas of Finite Groups, U_{4}(2) has irreducible representations χ_{3} and χ_{4} of dimensions five and six respectively such that χ_{3}⊗χ_{4} ≅ χ_{12} is also irreducible [C].

5. Invariant ideals

In this section, we briefly describe the G-invariant ideals of A. We no longer assume that A is G-simple, so A ≅ End(V) where V is an arbitrary finite-dimensional projective representation of G.

We now recall the ideal structure of A. Let 𝒮(V) denote the set of D-subspaces of V partially ordered by inclusion. This poset is in fact a complete lattice, with the greatest lower bound and least upper bound of a collection of subspaces given by their intersection and sum respectively. Similarly, the sets ℒ(A) and ℛ(A) of left and right ideals of A are complete lattices. It will be convenient to work with the dual lattice ℒ(A)^{∗} of left ideals under reverse inclusion (and with the supremum and infimum reversed). If L is a D-submodule of V, we define the annihilator and coannihilator of L by Ann(L) = {f ∈ A | f(L) = 0} and Coann(L) = {f ∈ A | f(V) ⊂ L}; these are respectively left and right ideals of A. We then have the well-known fact that all left and right ideals of A are of this form.

Proposition 5.1. The maps 𝒮(V) →^{Ann} ℒ(A)^{∗} and 𝒮(V) ⟶^{Coann} ℛ(A) are isomorphisms of complete lattices. The inverses are given by I ↦ ⋂_{f∈I} Ker(f) and J ↦ ∑_{f∈J} f(V), where I ∈ ℒ(A) and J ∈ ℛ(A).

Remark. In matrix language, this simply says that a left ideal consists of all matrices (with respect to some basis depending on the ideal) with zeroes in given columns while a right ideal consists of all matrices with zeros in given rows.

Let 𝒮_{G}(V) ⊂ 𝒮(V) be the complete sublattice of all D-subspaces of V preserved by the G-action on V. Similarly, we define the complete sublattices ℒ_{G}(A) ⊂ ℒ(A) and ℛ_{G}(A) ⊂ ℛ(A) of G-invariant left and right ideals of A. It is natural to conjecture that the sublattices ℒ_{G}(A) and ℛ_{G}(A) are just the images of 𝒮_{G}(V) under the above isomorphisms, i.e. invariant left and right ideals are annihilators and coannihilators respectively of subrepresentations of V. This is indeed the case.

Theorem 5.2. The restrictions of the maps Ann and Coann define isomorphisms of complete lattices 𝒮_{G}(V) →^{Ann} ℒ_{G}(A)^{∗} and 𝒮_{G}(V) ⟶^{Coann} ℛ_{G}(A).

Proof. In order to prove the first isomorphism, it suffices to show that Ann(𝒮_{G}(V)) ⊂ ℒ_{G}(A)^{∗} and Ann^{−1}(ℒ_{G}(A)^{∗}) ⊂ 𝒮_{G}(V). If L is a subrepresentation of V and f ∈ Ann(L), then (g · f)(v) = ḡ(f(ḡ^{−1}(v))) = ḡ(0) = 0 for all g ∈ G and v ∈ L. Thus, Ann(L) is G-invariant. Conversely, if I is an invariant left ideal and v ∈ Ann^{−1}(I) = ⋂_{f∈I} Ker(f), then we also have v ∈ ⋂_{f∈I} Ker(g · f). Since ρ(g) is bijective, this gives f(ḡ^{−1}v) = 0 for all g ∈ G and f ∈ I. It follows that Ann^{−1}(I) is G-invariant.

The proof for invariant right ideals is similar.

Remarks. 1. Since A is simple, the only two-sided ideals are {0} and A which are of course G-invariant. However, it is a general fact that if B is an arbitrary G-algebra on which G acts by inner automorphisms, then all two-sided ideals are G-invariant. Indeed, if I is a two-sided ideal and the action of g on B is given by conjugation by b_{g} ∈ B^{×}, then gI = b_{g}Ib_{g}^{−1} ⊂ I.

2. Suppose that F is algebraically closed and V is a completely reducible linear representation of G, say V ≅ n_{1}V_{1} ⊕ ⋯ ⊕ n_{m}V_{m} where the V_{i}’s are pairwise nonisomorphic irreducible representations. Then the G-invariant left (and right) ideals of End_{F} (V) are parametrized by ∏_{i=1}^{m}{subspaces of F^{n_{i}}}.

3. Analogous results hold for certain spaces of homomorphisms between two linear representations of G. If V and W are two representations of G defined over D, then Hom_{D}(V, W) is a representation whose G-action is compatible with the (End(W), End(V))-bimodule structure. A similar proof shows that the lattice of subrepresentations of V is isomorphic to the lattice of invariant left End(W)- submodules of Hom_{D}(V, W) while the lattice of subrepresentations of W is isomorphic to the lattice of invariant right End(V)-submodules of Hom_{D}(V, W) under reverse inclusion.

This theorem allows us to characterize certain properties of representations in terms of the associated endomorphism algebras.

Corollary 5.3. 1. The projective representation V is irreducible if and only if

End_{F} (V) has no proper invariant one-sided ideals.

2. Let D be a central division algebra, and suppose V is a D-module on which G acts (projectively) by D-linear automorphisms. Then V is D-irreducible (i.e. has no G-invariant D-submodules) if and only if End_{D}(V) has no proper invariant one-sided ideals.

3. Suppose that F is an infinite field and V is completely reducible. Then V is multiplicity free if and only if End_{F} (V) has a finite number of invariant one-sided ideals.

Proof. The first two statements are clear from the theorem. The last follows from the second remark and the fact that for an infinite field, a vector space has an infinite number of subspaces if and only if it has dimension larger than one.

It is worth noting that in spite of the strong connection between subrepresentations and invariant ideals, the group action on a subrepresentation does not determine the action on the corresponding left and right invariant ideals or vice versa.

If B is a semisimple (finite-dimensional) algebra on which G acts by inner automorphisms, this theorem can be used to determine the invariant ideals of B. Let B = B_{1}⊕⋯⊕B_{s} where the simple component B_{i} can be viewed as End_{D_{i}}(V_{i}) where D_{i} is a finite-dimensional division algebra over F and V_{i} is a finite-dimensional D_{i}- module. By the first remark, the two-sided ideal B_{i} is invariant and is thus a simple algebra on which G acts by inner automorphisms. Since left and right ideals of B are just direct sums of left and right ideals of B_{i}, we obtain the following corollary:

Corollary 5.4. The maps ∏_{i=1}^{s} 𝒮_{G}(V_{i}) → ℒ_{G}(B)^{∗} and ∏_{i=1}^{s} 𝒮_{G}(V_{i}) → ℛ_{G}(B) given by (L_{1}, …, L_{s}) ↦ ⨁_{i=1}^{s} Ann(L_{i}) and (L_{1}, …, L_{s}) ↦ ⨁_{i=1}^{s} Coann(L_{i}) respectively are isomorphisms of complete lattices.

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School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540 Current address: Department of Mathematics, Louisiana State University, Baton Rouge, LA

70803

   E-mail address: sage@math.lsu.edu