CLIFFORD CORRESPONDENCE FOR ALGEBRAS
SARAH J. WITHERSPOON
Abstract. We give a Clifford correspondence for an algebra A over an alge-
braically closed field, that is an algorithm for constructing some finite-dimensional
simple A-modules from simple modules for a subalgebra and endomorphism alge-
bras. This applies to all finite-dimensional simple A-modules in the case that A is
finite-dimensional and semisimple with a given semisimple subalgebra. We discuss
connections between our work and earlier results, with a view towards applications
particularly to finite-dimensional semisimple Hopf algebras.
1. Introduction
Clifford theory is an important collection of results, due to A. H. Clifford [4],
relating representations of finite groups to representations of normal subgroups. One
of the results, the Clifford correspondence, is a one-to-one correspondence between
simple modules for a group and pairs (V, W ) where V is a simple module for a given
normal subgroup H of finite index, and W is a simple module for a twisted group
algebra of a subgroup of G/H. (These pairs are taken up to a standard equivalence.)
The correspondence is constructive, and may be used to build simple modules of G.
Many authors have generalized this result to other types of groups and rings, for
example to locally compact groups [10, 16], and to group-graded rings [6, 9, 21].
Others have extended at least part of the theory to arbitrary rings [15], non-normal
subgroups [1], and Hopf algebras [19, 20, 22].
The present work is an attempt to unify some of these generalizations, and to fill in
some gaps, with the goal of finding future applications. We extend some of the results
of Rieffel for arbitrary rings [15], at the expense of assuming the rings are algebras over
algebraically closed fields. Statements and proofs are presented module-theoretically,
providing an alternative view of Rieffel s theory. We connect our results to Alperin s
version of the Clifford correspondence for non-normal subgroups [1], which in fact
also holds for arbitrary rings.
This note applies in particular to Hopf algebras, extending some of the work in
Clifford theory for Hopf algebras. This may be the most promising potential applica-
tion of our work: In the case of finite-dimensional semisimple Hopf algebras, the full
theory is valid, and the constructions may be concrete enough to classify all simple
modules in some cases. There is still much to be learned about finite-dimensional
Research supported in part by National Security Agency Grant #MDA904-01-1-0067.
Date: July 20, 2001.
1
arXiv:math.RA/0108034 v1 6 Aug 2001
2 SARAH J. WITHERSPOON
semisimple Hopf algebras H. For example a conjecture of Kaplansky, that the di-
mension of a simple H-module should divide dim(H), has not been settled in this
full generality.
In [12] Clifford theory is used to prove Kaplansky s Conjecture when H is a finite-
dimensional semisimple Hopf algebra constructed by a sequence of crossed products
involving group algebras and their duals; in particular this includes all semisimple
Hopf algebras of prime power dimension in characteristic 0. (Other partial proofs of
this conjecture have been demonstrated by authors using completely different meth-
ods [8, 13].)
As further motivation, we mention some other recent applications of the Clifford
correspondence. In [14] Clifford theory is used to show that the simple modules for
many Hecke algebras and groups of interest may all be obtained from those of the
affine Hecke algebras of type A. Clifford theory arises in [2] in a classification of the
finite-dimensional simple modules for Hopf algebra extensions of some of the down-up
algebras (these Hopf algebras are essentially generalizations of the Borel subalgebra
of Uq(sl3)).
Let k be an algebraically closed field. All algebras will be algebras over k, and a
subalgebra will have the same identity element as the algebra. All modules will be
left modules unless otherwise specified.
2. Static modules and Alperin s correspondence
In this section we summarize some results from [1] that will be needed. Let A be
a ring with identity, M a fixed left A-module, and E = EndA(M)op. Our functions
will act on the left, so that M is a right module for E. (Alperin chooses to have his
functions act on the right, and accordingly takes E to be the endomorphism algebra
itself.) If N is another A-module, then the space HomA(M, N) is naturally a (left)
E-module under composition of functions (as E takes the opposite multiplication
from an endomorphism algebra). We say that N is static (with respect to M) if the
A-homomorphism
(2.1) Δ : M "E HomA(M, N) N
given by Δ(m " f) = f(m) is an isomorphism. A left E-module U is static (with
respect to M) if the E-homomorphism
(2.2) Θ : U HomA(M, M "E U)
given by Θ(u)(m) = m " u is an isomorphism. As M "E - and HomA(M, -) are
adjoint functors, there is a category equivalence between the full subcategories of
static A-modules and static E-modules:
Proposition 2.3 ([1, Lemma 1]). The category of static A-modules is equivalent
with the category of static E-modules. The equivalence is given by the functors
HomA(M, -) and M "E -.
CLIFFORD CORRESPONDENCE FOR ALGEBRAS 3
Alperin uses this category equivalence to give new proofs of Auslander equivalence
and Morita equivalence, as well as to give a version of the Dade-Cline equivalence
for non-normal subgroups of a finite group [1]. (The Dade-Cline equivalence [5, 7]
is a category equivalence which implies the Clifford correspondence for group-graded
rings.) In Section 3, we will use Proposition 2.3 to extend Rieffel s results further for
algebras over k.
Next we give Alperin s version of a Clifford correspondence for non-normal sub-
groups, valid for arbitrary rings, which will be used in Section 4. If M is a fixed
A-module, we say that another A-module N has an M-presentation if there is an
exact sequence
" M "M N 0
of A-modules. Letting B be a subring of A, we say this presentation is B-split if it
splits on restriction to B.
Now let V be a B-module and M the A-module induced from V , that is M = V Δ!A=
A "B V where A acts by multiplication on the first factor. Let E = EndA(V Δ!A)op
as above, and F = EndB(V Δ!A)op, so that E Δ
" F . If U is a left E-module, we define
the restriction of U to F to be the F -module F "E U (as in [1]), denoted U !F .
Proposition 2.4 ([1, Theorem]). The category of A-modules with B-split V Δ!A-presentations
is equivalent with the category of E-modules whose restrictions to F are projective.
The equivalence is given by the functors HomA(V Δ!A, -) and V Δ!A "E-.
Alperin s proof of the proposition is valid in this general setting, and it uses Propo-
sition 2.3. The proof involves verifying that:
(i) If N is an A-module with a B-split V Δ!A-presentation, then N is static (with
respect to V Δ!A) and HomA(V Δ!A, N)!F is F -projective.
(ii) If U is an E-module whose restriction to F is projective, then V Δ!A "EU has a
B-split V Δ!A-presentation, and U is static (with respect to V Δ!A).
These assertions can be verified exactly as in [1].
3. Stable Clifford correspondence
There are traditionally two parts to the Clifford correspondence. One part involves
the case where a module for a normal subgroup is stable in the whole group, and
the other part involves the connection between modules for a stabilizer subgroup and
modules for the whole group. Here we will deal with the stable part for algebras more
generally; the rest appears in the next section.
Let B " A be algebras over the algebraically closed field k, and V a B-module.
We say that V is A-stable if the restriction of V Δ!A= A "B V to B, denoted V Δ!A!B,
is isomorphic to a finite direct sum of copies of V .
Lemma 3.1. Let V be a finite-dimensional simple A-stable B-module, M = V Δ!A,
and E = EndA(M)op.
4 SARAH J. WITHERSPOON
(i) A finite-dimensional A-module N is static with respect to M if and only if N !B
is isomorphic to a direct sum of copies of V .
(ii) All finite-dimensional E-modules are static with respect to M.
"n
Proof. (i) First we claim that if N !B<" V , then the B-homomorphism
=
Ε : V "k HomA(M, N) N !B
given by Ε(v " f) = f(1 " v) is an isomorphism, where B acts only on the left
<"
factor of V "k HomA(M, N). To see this, note that HomA(M, N) HomB(V, N !B)
=
"n
as induction and restriction are adjoint functors. Since N !B<" V , the latter is
=
isomorphic to k"n by Schur s Lemma, from which it is clear that Ε is surjective. This
also shows that the dimensions of the modules are the same, so Ε is bijective.
In particular, we may let N = M as V is A-stable, and we obtain
<"
(3.2) V "k E M
=
as left B-modules and right E-modules (where E acts only on the right factor of
"n
V "k E). Therefore if N is any finite-dimensional A-module with N !B<" V , we
=
have B-module isomorphisms
<" <"
(3.3) M "E HomA(M, N) V "k HomA(M, N) N !B .
It is straightforward to check that the composite map is precisely the map Δ of (2.1),
an A-homomorphism.
Conversely, if N is static then by (3.2),
N !B<" (M "E HomA(M, N))!B<" V "k HomA(M, N)
= =
as B-modules. As B acts only on the left factor in V "k HomA(M, N), this is iso-
morphic to a direct sum of copies of V .
(ii) Let U be a finite-dimensional E-module. Then by (3.2), we have
<"
HomA(M, M "E U) HomB(V, (M "E U) !B)
=
<"
= HomB(V, (V "k U) !B)
" dim U
<" <"
= HomB(V, V ) k" dim U .
=
It follows that the map Θ : U HomA(M, M "E U) of (2.2) is surjective. As these
modules also have the same dimension, Θ must be an isomorphism. Therefore U is
static.
Combining the lemma with Proposition 2.3 yields:
Theorem 3.4. Let V be a finite-dimensional simple A-stable B-module and E =
EndA(V Δ!A)op. The category of finite-dimensional E-modules is equivalent with the
category of finite-dimensional A-modules whose restriction to B is a direct sum of
copies of V . The equivalence is given by the functors V Δ!A "E- and HomA(V Δ!A, -).
CLIFFORD CORRESPONDENCE FOR ALGEBRAS 5
We point out that if U is a finite-dimensional E-module, then
<"
(3.5) V Δ!A "EU V "k U
=
as B-modules (where B acts on the left factor only) by (3.2). The B-module V "k U
may then be given an A-module structure via this isomorphism. This may give a
simpler description of such A-modules, and it potentially allows an easy count of
their dimensions.
We next show that any finite-dimensional simple A-module containing a simple
A-stable B-module V on restriction to B may be described in this way:
Lemma 3.6. Let N be a finite-dimensional simple A-module such that N !B contains
a copy of a finite-dimensional simple A-stable B-module V . Then N !B is a direct
sum of copies of V .
Proof. Define Δ : V Δ!A N by Δ(a " v) = a · v (where we have identified V with a B-
submodule of N). Then Δ is a nonzero A-homomorphism, and so must be surjective,
as N is simple. Therefore N !B is a homomorphic image of V Δ!A!B, a direct sum of
copies of V , and so is itself such a direct sum.
In light of Lemma 3.6, we immediately have a corollary of Theorem 3.4:
Corollary 3.7 (Stable Clifford correspondence). Let V be a finite-dimensional sim-
ple A-stable B-module, and E = EndA(V Δ!A)op. There is a one-to-one correspon-
dence between finite-dimensional simple E-modules and finite-dimensional simple A-
modules containing V on restriction to B.
Remark 3.8. This generalizes the stable Clifford correspondence of finite group the-
ory: Let G be a finite group with normal subgroup H. Let A = kG and B = kH. A
B-module V is A-stable precisely when it is G-stable, that is all its conjugates under
g
G are isomorphic. (If g " G, the conjugate B-module V has underlying vector space
V , and module structure given by h ·g v = (g-1hg) · v for all h " H, v " V .) In this
case, E is isomorphic to a twisted group algebra of G/N. (In general, V Δ!A!B will
be isomorphic to a direct sum of copies of conjugates of V , one for each element of
G/N, and E will be isomorphic to a twisted group algebra of a subgroup of G/N.)
We now relate these results to those of Rieffel in [15]. We say that an ideal J of
B is A-invariant if AJ = JA. We say that B is normal in A if for every (two-sided)
ideal I of A, I )" B is A-invariant. (We caution that Rieffel s definition of normal is
somewhat different in case A and B are not both semisimple Artinian.)
Lemma 3.9. Let B " A be normal, V a finite-dimensional simple B-module, J =
annB V , and assume V Δ!A is finite-dimensional. Then V is A-stable if and only if J
is A-invariant.
Proof. Let M = V Δ!A, I = annA M, and assume V is A-stable. We claim that
J = I )" B. Note first that I )" B Δ
" J as V Δ
" M. Since M !B is a direct sum of copies
6 SARAH J. WITHERSPOON
of V , J annihilates M, so J Δ
" I )" B. Therefore J = I )" B. As B is normal in A, it
follows that J is A-invariant.
Assuming now that J is A-invariant, we again claim that J = I )" B. We have
I )" B Δ
" J as V Δ
" M. Now
J · M = JA "B V = AJ "B V = 0
as J is A-invariant and annihilates V . Therefore J Δ
" I )" B, and we have J = I )" B.
<"
It follows that B/(I )" B) = B/J Endk(V ) has a unique simple module up to
=
isomorphism, namely V . As I annihilates M, M !B is naturally a B/(I )" B)-module,
and so must be a direct sum of copies of V .
Remark 3.10. Rieffel gives a result analogous to Corollary 3.7, under the hypotheses
that A and B are semisimple Artinian rings and B is normal in A [15, Proposition
2.15]. He uses the hypothesis that J is A-invariant where we require V to be A-stable.
As applied to algebras over k, Corollary 3.7 is more general.
4. General Clifford correspondence
In this section we discuss the case in which a finite dimensional simple B-module
V is not necessarily A-stable. We will assume that A and B are finite-dimensional
semisimple algebras over k, as the results and proofs are more transparent in this
case.
We will give a module-theoretic definition of stabilizer S for V and two new proofs
of the full Clifford correspondence in this setting. (Rieffel has a more ring-theoretic
definition of stabilizer, and a full Clifford correspondence for semisimple Artinian
rings [15, ΕΌ2].) The first of our proofs is traditional, involving one step to get simple
S-modules, and another to get from simple S-modules to simple A-modules. The
second proof uses Alperin s correspondence (Proposition 2.4), and bypasses stabilizers
altogether.
Let B be a normal subring of A and V a finite-dimensional simple B-module. A
stabilizer (or stability subring) for V is a semisimple algebra S such that B Δ
" S Δ
" A,
B is a normal subring of S, V is S-stable, and V -soc(V Δ!A) = V -soc(V Δ!S). (Here,
the V -socle of a module is the sum of all its B-submodules that are isomorphic to
V .) This definition is equivalent to that of Rieffel [15, Definition 2.11] under the
hypotheses that A and B are semisimple, as the following lemma shows. The lemma
will be used in the first proof of the full Clifford correspondence.
Lemma 4.1. Assume A and B are finite-dimensional semisimple algebras with B "
A normal. Let V be a finite-dimensional simple B-module and J = annB V = pB
where p is a primitive central idempotent of B. Then a semisimple subalgebra S of A,
containing B as a normal subring, is a stabilizer for V if and only if J is S-invariant
and S + AJ + JA = A.
CLIFFORD CORRESPONDENCE FOR ALGEBRAS 7
Proof. Let S be a stabilizer for V . By Lemma 3.9, J is S-invariant. Note that
" dim V
<"
B/J V as B-modules, so that
=
<"
V -soc(A "B (B/J)) V -soc(S "B (B/J)) = S "B (B/J)
=
as S is a stabilizer for V . Also note that as a B-module, (1 - p)A "B (B/J) is a
direct sum of copies of V , since p annihilates it. This means that the B-submodule
(1 - p)A "B (B/J) of A "B (B/J) is contained in the image of the canonical inclusion
S "B (B/J) A "B (B/J).
<"
Applying the isomorphisms S "B (B/J) S/SJ (given by s " (b + J) sb + SJ)
=
<"
and A "B (B/J) A/AJ, we now have that (1 - p)A/AJ embeds in S/SJ under
=
the canonical map, that is (1 - p)A Δ
" S + AJ. Therefore
A = (1 - p)A + pA Δ
" S + AJ + JA,
so that S + AJ + JA = A as desired.
Conversely, suppose that S is a semisimple subalgebra of A containing B as a
normal subring, J is S-invariant, and S + AJ + JA = A. Then V is S-stable by
Lemma 3.9. As J annihilates V , we have
V Δ!A!B= (A "B V )!B = ((S + AJ + JA) "B V ) !B
= ((S + JA) "B V ) !B .
Note that (S "B V ) !BΔ
" V -soc(V Δ!A) as V is S-stable. Further, as J = pB, p acts as
the identity on JA"B V , which implies that JA"B V intersects V -soc(V Δ!A) trivially.
Therefore
V -soc(V Δ!A) = S "B V = V -soc(V Δ!S).
Remark 4.2. Rieffel shows that stabilizers exist (but are not unique) in case both A
and B are semisimple Artinian: S = B+(1-p)A(1-p) is a stabilizer for V contained
in every other stabilizer, and S = pAp + (1 - p)A(1 - p) (the centralizer of p) is a
stabilizer for V containing every other stabilizer.
Remark 4.3. In case H is a normal subgroup of a finite group G, A = kG and
B = kH, the stabilizer subgroup (or inertia subgroup) L for a simple B-module V
g
<"
consists of all g " G such that V V . The algebra S = kL is a stabilizer for V in
=
the above sense, and is the unique subalgebra of A arising from a subgroup of G that
is a stabilizer for V .
We now have two expressions for the endomorphism algebra E = EndA(V Δ!A)op.
For, as a B-homomorphism must map V into the V -socle of a module, we have vector
space isomorphisms
<" <"
(4.4) EndA(V Δ!A)op <" HomB(V, V Δ!A!B) HomB(V, V Δ!S!B) EndS(V Δ!S)op.
= = =
8 SARAH J. WITHERSPOON
It may be verified that the composite isomorphism preserves products in the endo-
morphism algebras.
Theorem 4.5. Assume A and B are finite-dimensional semisimple algebras, with B
normal in A. Let V be a finite-dimensional simple B-module and S a stabilizer for
V . There is a one-to-one correspondence between finite-dimensional simple S-modules
containing V on restriction to B, and finite-dimensional simple A-modules containing
V on restriction to B. The correspondence is given by induction of modules from S
to A.
Proof. We may assume S = A. First let N be a finite-dimensional simple S-module
with V contained in N !B. As V is S-stable, N !B must be a direct sum of copies of
V by Lemma 3.6. Let M be any simple A-submodule of N Δ!A. Then M is a direct
summand of N Δ!A, and so
<"
0 = HomA(N Δ!A, M) HomS(N, M !S).
=
As N is a simple S-module, this implies that N embeds in M !S. On the other hand,
if J = annB V = pB, then by Lemma 4.1,
N Δ!A!S = ((S + AJ + JA) "S N) !S
= ((S + JA) "S N) !S
= N " (JA "S N) !S
as J (and so p) annihilates N (N !B is a direct sum of copies of V ), but p acts as
the identity on JA "S N. This implies that there is a unique copy of N in N Δ!A!S.
Now by semisimplicity we have N Δ!A<" M " W for some A-module W , but restricting
=
to S, we see that W cannot contain N (as M !S contains the unique copy of N in
N Δ!A!S). Applying the above argument to any simple A-submodule of W leads to
<"
the contradiction that W must also contain N. Therefore M N Δ!A and N Δ!A is
=
simple.
Now let M be any simple A-module containing V on restriction to B. Let N be a
simple S-submodule of the image of S "B V under the map Δ : A "B V M given
by Δ(a " v) = a · v. By the previous arguments, N Δ!A is a simple A-module. By the
definition of N,
<"
HomA(N Δ!A, M) HomS(N, M !S) = 0.
=
<"
As both M and N Δ!A are simple A-modules, this implies M N Δ!A.
=
Now we are ready to give the full Clifford correspondence.
Theorem 4.6 (General Clifford correspondence). Assume that A and B are finite-
dimensional semisimple algebras with B normal in A. Let V be a finite-dimensional
simple B-module, and E = EndA(V Δ!A)op. There is a one-to-one correspondence be-
tween finite-dimensional simple E-modules and finite-dimensional simple A-modules
containing V on restriction to B.
CLIFFORD CORRESPONDENCE FOR ALGEBRAS 9
Proof. We give two proofs:
(i) Let S be a stabilizer for V . By Corollary 3.7 and (4.4), there is a one-to-one
correspondence between finite-dimensional simple E-modules and finite-dimensional
simple S-modules containing V on restriction to B. Theorem 4.5 provides the rest of
the correspondence.
(ii) Let F = EndB(V Δ!A)op. As V Δ!A!B is semisimple, F is a direct sum of matrix
algebras and so is semisimple itself. Thus every F -module is projective. Applying
Proposition 2.4, it remains to show that each simple A-module N containing V on
restriction to B has a V Δ!A-presentation (automatically B-split as B is semisimple).
Note that N is a quotient of V Δ!A via Δ : V Δ!A N, Δ(a " v) = a · v (where we
have identified V with a B-submodule of N). Let K be any simple component of
the kernel of Δ, so that K embeds in V Δ!A. By semisimplicity, K is also a quotient
of V Δ!A. Taking a direct sum of copies of V Δ!A, one for each simple summand of the
kernel of Δ, we obtain a V Δ!A-presentation
"V Δ!A V Δ!A M 0.
Note that the second proof only requires V Δ!A and V Δ!A!B to be semisimple A- and
B-modules, respectively. It does not require A and B to be semisimple algebras, nor
does it require B to be normal in A.
The correspondence of the theorem is given by the functors HomA(V Δ!A, -) and
V Δ!A "E- from Proposition 2.4. So the second proof and the construction of simple
A-modules (from V and simple E-modules) bypasses a stabilizer S altogether. This
may be useful in situations where stabilizers do not have desired properties. For
example, in the case of Hopf algebras (discussed in the next section), we do not know
whether there is always a stabilizer that is a Hopf subalgebra. On the other hand, in
specific applications, it may be useful to know a stabilizer S in order to obtain more
detailed information about the modules.
Remark 4.7. Rieffel gives a result analogous to Theorem 4.5, but for arbitrary rings
[15, Theorem 3.25]. It is quite possible that our methods could be extended to include
algebras more generally, and in any case Theorem 3.25 of [15] combined with Corollary
3.7 should yield a version of Theorem 4.6 valid under more general hypotheses. We
do not pursue this here as the ring-theoretic definitions required to state Rieffel s
Theorem 3.25 would lead us too far afield (see [15, ΕΌ3]). It would be interesting to
find a module-theoretic statement and proof of his theorem.
5. Clifford correspondence for Hopf algebras
Let B " A be a finite-dimensional H-Galois extension for a Hopf algebra H over
k (see [11] for definitions). Let V be a finite-dimensional B-module. By [17, Remark
3.2(4)], V is A-stable if and only if there is a left B-linear and right H-colinear
10 SARAH J. WITHERSPOON
isomorphism
<"
(5.1) Ε : A "B V V "k H.
This is Schneider s definition of A-stable (which does not require any finiteness as-
sumptions, see [17, ΕΌ3]). A stable Clifford correspondence in this context is given by
Schneider [19, Remark 5.8(2)] and van Oystaeyen and Zhang [20, Theorem 5.4]. Their
results are reformulated and extended in [22] to a general Clifford correspondence in
case H is cocommutative, using a stabilizer Hopf subalgebra defined by Schneider in
[17].
In this section, we summarize some of the results of [22] and show how they follow
from this note. At the same time, our results here or those of Rieffel in [15] give a
much more general version of the Clifford correspondence for Hopf algebras, which
may turn out to be useful.
Suppose that A = B#ΓH is a crossed product algebra of a k-algebra B with H (a
special case of an H-Galois extension). That is, Γ is a two-cocycle mapping H " H
to B and there is a measuring H " B B, denoted h " b h · b. These maps must
satisfy appropriate properties (see [11, Chapter 7]). We first show that B is a normal
subring of A, as defined in Section 3.
Proposition 5.2. The subalgebra B of A = B#ΓH is a normal subring of A.
Proof. Let I be any (two-sided) ideal of A. We must show that A(I )" B) = (I )" B)A.
Let a = b#h " A and i " I )" B. Then by definition of multiplication in A,
ai = (b#h)(i#1) = b(h1 · i)#h2
= (b(h1 · i)#1)(1#h2).
This is an element of (I )"B)A, as by [11, (7.2.4)], h1·i " I )"B. Therefore A(I )"B) Δ
"
(I )" B)A. The other containment is proved similarly.
If A is any finite-dimensional Hopf algebra with a normal Hopf subalgebra B (see
[11, Definition 3.4.1]), then A = B#ΓH for some two-cocycle Γ, where H = A/AB+
(B+ = B )" (Ker ΅)) [18], and so the lemma applies. Alternatively, it can be shown
directly that any normal Hopf subalgebra of a Hopf algebra is a normal subring.
Under these hypotheses, the stable Clifford correspondence for finite-dimensional
Hopf algebras [22, Theorem 3.1 or Corollary 3.3] follows immediately from our The-
orem 3.4 (or Corollary 3.7). The proof of the Clifford correspondence in [22] is based
on calculations involving the various structure maps, potentially allowing more de-
tailed information about the structure of A-modules (see [22, Theorem 2.2(i)]). The
proof we give of Theorem 3.4 here is essentially a generalization of those given for
Hopf algebras in [19, 20], and is based instead on facts about the relevant categories.
Under the hypothesis that H is cocommutative, we also gave in [22] a general
Clifford correspondence. This is the case in which a stabilizer coalgebra Hst of a
CLIFFORD CORRESPONDENCE FOR ALGEBRAS 11
B-module V , defined by Schneider, is a Hopf subalgebra of H. We will review this
definition.
Let V be a finite-dimensional B-module. If C Δ
" H is a subcoalgebra, let A(C) =
"-1(A " C) where "A : A A " H is the right H-comodule map arising from the
A
<"
crossed product A = B#ΓH. We say that C stabilizes V if V " C A(C) "B V as
=
left B-modules and right C-comodules [17, p. 216]. Let Hst = C, the sum over
all subcoalgebras C Δ
" H such that C stabilizes V . Assuming H is cocommutative,
Hst is a Hopf subalgebra of H by [17, Theorem 4.4]. (Note that H is pointed as k
is algebraically closed and H is cocommutative [11, p. 76]. In the more general case,
Hst is a subcoalgebra of H, but not necessarily a Hopf subalgebra.)
Let S = A(Hst) = B#ΓHst. By [17, Remark 3.2(4), and Theorems 4.4 and
5.4(1)(a)], S is a stabilizer for V (our definition). The cocommutative Clifford cor-
respondence [22, Corollary 3.5] in the semisimple case follows immediately from our
Theorem 4.6, and can be done in two steps (as in proof (i) of the theorem) involving
<"
the stabilizer S. The A-modules produced are of the form V Δ!A "EU A"S (V "k U)
=
for E-modules U by (3.5), and as S arises from a Hopf subalgebra of H, this may
allow an explicit description of their structure (see for example [22, Theorem 2.2(i)]).
On the other hand, Theorem 4.6 gives a Clifford correspondence more generally for
semisimple A and B (H need not be cocommutative), regardless of whether any of
the stabilizers S arise from Hopf subalgebras of H.
We make some final remarks that may be helpful in calculating examples. If the
measuring of B by H is trivial (that is h · b = ΅(h)b for all h " H and b " B), then
every B-module V is A-stable. Indeed, the map Ε of (5.1) is given by Ε((b#h)"Bv) =
Α(b)(v) " h where Α : B Endk(V ) is the structure map of the B-module V .
Let E = EndA(V Δ!A)op. By [17, Theorem 3.6] and Schur s Lemma, E is an H-
crossed product over the field k. That is, E is isomorphic to a twisted product
kΔ
[H] = k#Δ
H as any measuring of k by H is necessarily trivial. Therefore E-
modules are equivalent to projective representations of H [3, Proposition 2.4]. These
may be studied via the methods of Boca in [3].
References
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Department of Mathematics and Statistics, University of Massachusetts, Amherst,
MA 01003
(2001 02) Department of Mathematics and Computer Science, Amherst College,
Amherst, MA 01002
E-mail address:wither@math.umass.edu
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