Equivariant map superalgebras
Suppose a group acts on a scheme and a Lie superalgebra . The corresponding equivariant map superalgebra is the Lie superalgebra of equivariant regular maps from to . We classify the irreducible finite dimensional modules for these superalgebras under the assumptions that the coordinate ring of is finitely generated, is finite abelian and acts freely on the rational points of , and is a basic classical Lie superalgebra (or , , if is trivial). We show that they are all (tensor products of) generalized evaluation modules and are parameterized by a certain set of equivariant finitely supported maps defined on . Furthermore, in the case that the even part of is semisimple, we show that all such modules are in fact (tensor products of) evaluation modules. On the other hand, if the even part of is not semisimple (more generally, if is of type I), we introduce a natural generalization of Kac modules and show that all irreducible finite dimensional modules are quotients of these. As a special case, our results give the first classification of the irreducible finite dimensional modules for twisted loop superalgebras.
Key words and phrases:Lie superalgebra, basic classical Lie superalgebra, loop superalgebra, equivariant map superalgebra, finite dimensional representation, finite dimensional module
2010 Mathematics Subject Classification:17B65, 17B10
Lie superalgebras, which are generalizations of Lie algebras to include a -grading, are of considerable interest to both mathematicians and physicists. Significant mathematical progress has been made in the theory of these algebras. For example, Kac has classified the simple finite dimensional Lie superalgebras over an algebraically closed field of characteristic zero in [Kac77] and the irreducible finite dimensional representations of the so-called basic classical Lie superalgebras in [Kac77c, Kac77, Kac78]. Nevertheless, the general theory of Lie superalgebras and their representations still remains much less developed than the corresponding theory of Lie algebras. For example, while tremendous progress has been made in the classification of irreducible finite dimensional representations of (twisted) loop algebras and their generalizations (equivariant map algebras), the analogous theory in the super case is in its infancy.
In the current paper, we consider a certain important class of Lie superalgebras. Let be a scheme and let be a “target” Lie superalgebra, both defined over an algebraically closed field of characteristic zero. Furthermore, let be a finite group acting on and by automorphisms. Then the Lie superalgebra of -equivariant regular maps from to is called an equivariant map superalgebra. In the case that is, in fact, a Lie algebra, we call it an equivariant map algebra.
A special important case of the above construction is when is the one-dimensional torus. In this case, is called a twisted loop (super)algebra when acts nontrivially and an untwisted loop (super)algebra in the case that acts trivially. Loop (super)algebras and their twisted analogues play an vital role in the theory of affine Lie (super)algebras and quantum affine algebras. They are also an important ingredient in string theory. In the non-super case, their representation theory is fairly well developed. In particular, the irreducible finite dimensional representations were classified by Chari and Pressley ([Cha86, CP86, CP98]). Subsequently, many generalizations of this work (also in the non-super case) have appeared in the literature, for example, in [Bat04, CFK, CFS08, CM04, FL04, Lau10, Li04, Rao93, Rao01]. In [NSS09], the irreducible finite dimensional representations were classified in the completely general setting of equivariant map algebras. There it was shown that all such representations are tensor products of evaluation representations and one-dimensional representations. In [NS11], the extensions between these representations were computed and the block decompositions of the categories of finite dimensional representations were described.
Despite the above-mentioned progress in our knowledge of the representation theory of equivariant map algebras, relatively little is known if the target is in fact a Lie superalgebra (with nonzero odd part). Perhaps the best starting point is when is one of the basic classical Lie superalgebras (see Section 2.2), since these have many properties in common with semisimple Lie algebras. In the special case of untwisted multiloop superalgebras with basic classical target, the irreducible finite dimensional representations have been classified in [RZ04, Rao11]. However, beyond this, almost nothing is known. For example, even the irreducible finite dimensional representations of twisted loop superalgebras have not been classified. This is in sharp contrast to the non-super case.
In the current paper we give a complete classification of the irreducible finite dimensional modules for an arbitrary equivariant map superalgebra when the target is a basic classical Lie superalgebra and the group is abelian and acts freely on the set of rational points of a scheme with finitely generated coordinate algebra. In particular, our classification covers the case of the twisted loop superalgebras (with basic classical target). Our main result (Theorem LABEL:thm:classification-equivariant) is that all irreducible finite dimensional modules are generalized evaluation modules and that they can be naturally parameterized by a certain set of equivariant finitely supported maps defined on . In fact, even more can be said. In the case that the even part of is semisimple, all irreducible finite dimensional modules are evaluation modules, just as for equivariant map algebras. However, this is not the case if the even part of is not semisimple. In this situation (more generally, when is of type I), we introduce a natural generalization of Kac modules, which are certain modules induced from modules for the equivariant map algebra , where is the even part of (and hence a Lie algebra). We then show that all irreducible finite dimensional modules can be described as irreducible quotients of these Kac modules. In the untwisted setting (i.e. when is trivial), this classification also applies when , .
A natural generalization of the category of finite dimensional modules is the category of quasifinite modules (that is, modules with finite dimensional weight spaces). In Theorem LABEL:thm:hw-quasifinite-classification, we give a characterization of the quasifinite irreducible highest weight )-modules when is any basic classical Lie superalgebra or , . We do not assume that is finitely generated there. Our characterization is similar to [Sav11, Prop. 5.1], which describes the quasifinite modules for the map Virasoro algebras.
This paper is organized as follows. In Section 2 we review some results on commutative algebras and Lie superalgebras that we will need. We introduce the equivariant map superalgebras in Section 3. In Section 4 we discuss their quasifinite and highest weight modules and give a characterization of the quasifinite modules. We define generalized evaluation modules and prove some important facts about them in Section LABEL:sec:eval-rep. In Section LABEL:sec:kac-modules, we discuss a generalization of Kac modules to the setting of equivariant map superalgebras. Finally, we classify the irreducible finite dimensional modules in the untwisted setting in Section LABEL:sec:classification-untwisted and in the twisted setting in Section LABEL:sec:classification-equivariant.
We let be the set of nonnegative integers and be the set of positive integers. Throughout, is an algebraically closed field of characteristic zero and all Lie superalgebras, associative algebras, tensor products, etc. are defined over unless otherwise specified. We let denote the coordinate ring of a scheme . Thus is a commutative associative unital -algebra. We let denote the set of -rational points of . Recall that is a -rational point of if its residue field is . Thus and we have equality if is finitely generated. We assume that is finitely generated in Sections LABEL:sec:eval-rep–LABEL:sec:classification-equivariant. When we refer to the dimension of , we are speaking of its dimension as a vector space over (as opposed to referring to a geometric dimension). Similarly, when we say that an ideal of is of finite codimension, we mean that is finite dimensional as a vector space over . We use the term reductive Lie algebra only for finite dimensional Lie algebras.
The author would like to thank Shun-Jen Cheng, Daniel Daigle, Dimitar Grantcharov, Erhard Neher and Hadi Salmasian for helpful discussions.
This document contains some small corrections to typographical errors that appear in the published version. It was decided that these errors are not significant enough to warrant publishing an erratum, but that it would helpful to correct the arXiv version. The changes are listed below.
In this section, we review some mostly well-known results on commutative associative algebras and Lie superalgebras that will be used in the sequel. For the results on Lie superalgebras (Section 2.2), we refer the reader to [FSS00, Kac77, Kac77b] for further details.
2.1. Commutative algebras
The support of an ideal of an algebra is
Note that the support of an ideal is often defined to be the set of prime (rather than maximal) ideals containing it. So our definition is more restrictive.
If is an ideal of finite codimension in an algebra , then has finite support.
Let be an ideal of finite codimension in . It suffices to show that has a finite number of maximal ideals. Let be a sequence of pairwise distinct maximal ideals of . Define . Then and thus the length of the sequence is bounded by the dimension of . ∎
If is an ideal of finite support in a finitely generated algebra , then is of finite codimension in .
If is of finite support, then has finitely many maximal ideals. Since is finitely generated, this implies that is finite dimensional (see, for example, [Fu11, Lem. 1.9.2]). ∎
If and are ideals of an algebra and and have disjoint supports, then .
Suppose and are ideals of an algebra and and have disjoint supports. Then since . Thus we can write with and . Let . Then , where . Therefore . The reverse inclusion is obvious. ∎
Lemma 2.4 ([Am69, Prop. 7.14]).
In a Noetherian ring, every ideal contains a power of its radical.
2.2. Lie superalgebras
Let be a finite dimensional simple Lie superalgebra over . Then is called classical if the representation of on is completely reducible. A simple Lie superalgebra is classical if and only if its even part is a reductive Lie algebra. If is a classical Lie superalgebra, then the representation of on is either
irreducible, in which case we say is of type II, or
the direct sum of two irreducible representations, in which case we say is of type I.
A classical Lie superalgebra is called basic if there exists a non-degenerate invariant bilinear form on . (The classical Lie superalgebras that are not basic are called strange.)
In Table 1, we list all of the basic classical Lie superalgebras (up to isomorphism) that are not Lie algebras, together with their even part and their type. We also include the Lie superalgebra , , which is a 1-dimensional central extension of .
For any basic classical Lie superalgebra , there exists a distinguished -grading of that is compatible with the -grading (i.e. each graded piece is a -graded subspace of ) and such that
if is of type I, then for , , , and
if is of type II, then for , , .
If , then we have a distinguished -grading as in (a) above. We simply let be the preimage of the zero graded piece of the type I Lie superalgebra under the canonical projection from to .
By definition, a Cartan subalgebra of is just a Cartan subalgebra of the even part . Fix such a Cartan subalgebra . If is a basic classical Lie superalgebra or , then we can choose a Borel subalgebra containing and . (Here we refer to the distinguished -grading on mentioned above.) Since the adjoint action of on is diagonalizable, we obtain a decomposition
where are subalgebras, , and . A root is called positive (resp. negative) if (resp. ). A root is called even (resp. odd) if (resp. ). Let and denote the set of even and odd roots respectively. Then is the set of all roots. We let , , denote the corresponding subsets of positive/negative roots. For a basic classical Lie superalgebra, the zero weight space of is equal to (see [Kac77b, Prop. 5.3]). It follows that this property also holds for . Let and be the weight lattice and positive root lattice of respectively.
Lemma 2.5 ([Kac77, Prop. 5.2.4]).
All the irreducible finite dimensional representations of a solvable Lie superalgebra are one-dimensional if and only if .
Suppose is a Lie superalgebra and is an irreducible -module such that for some ideal of and nonzero vector . Then .
By assumption and so is nonzero. Furthermore, since is an ideal of , it is easy to see that is a submodule of . Since is irreducible, this implies that , completing the proof of the lemma. ∎
In the Lie superalgebra case, the Poincaré-Birkhoff-Witt Theorem (or PBW Theorem) has the following form (see, for example, [Var04, Thm. 7.2.1]).
Let be totally ordered bases for (over ) respectively. Then the standard monomials
form a basis (over ) for the universal enveloping superalgebra . In particular, if (i.e. is odd), then is finite dimensional.
3. Equivariant map superalgebras
In this section we introduce our main object of study: the equivariant map superalgebras. Recall that is a Lie superalgebra and is a scheme with coordinate ring .
Definition 3.1 (Map superalgebra).
We call the Lie superalgebra of regular functions on with values in a map (Lie) superalgebra. The -grading on is given by for , and the multiplication on is pointwise. That is, multiplication is given by extending the bracket
by linearity. It is easily verified that satisfies the axioms of a Lie superalgebra.
An action of a group on a Lie superalgebra and on a scheme will always be assumed to be by Lie superalgebra automorphisms of and scheme automorphisms of . Recall that Lie superalgebra automorphisms respect the -grading. A -action on induces a -action on .
Definition 3.2 (Equivariant map superalgebra).
Let be a group acting on a scheme (hence on ) and a Lie superalgebra by automorphisms. Then acts naturally on by extending the map , , , , by linearity. We define
to be the subsuperalgebra of points fixed under this action. In other words, is the subsuperalgebra of consisting of -equivariant maps from to . We call an equivariant map (Lie) superalgebra. Note that if is the trivial group, this definition reduces to Definition 3.1.
Example 3.3 (Multiloop superalgebras).
Fix positive integers . Let
and suppose that acts on . Note that this is equivalent to specifying commuting automorphisms , , of such that . For , let be a primitive -th root of unity. Let , where is the -algebra of Laurent polynomials in variables (in other words, is the -dimensional torus ), and define an action of on by
is the (twisted) multiloop superalgebra of relative to and . In the case that is trivial (i.e. for all ), we call often call it an untwisted multiloop superalgebra. If , is simply called a (twisted or untwisted) loop superalgebra.
If acts on , hence on , then acts on by [EH00, I-40]. Since the coordinate rings of and are both , the equivariant map superalgebras corresponding to and are the same. Therefore, we lose no generality in assuming that is an affine scheme and we will often do so in the sequel.
Definition 3.5 (, ).
For a -module , we define to be the largest -invariant ideal of satisfying . In other words, is the sum of all -invariant ideals such that . If is the representation corresponding to , we set .
If is a perfect Lie superalgebra (i.e. ) and is a )-module, then
Let . It suffices to show that is an ideal of . It is easy to see that is a linear subspace of . Now let and . Then, since is perfect,
and so . Thus is an ideal of . ∎
Definition 3.7 (Support).
Let be a -module. We define the support of to be
If is the representation corresponding to , we also set . We say that has reduced support if is a radical ideal (equivalently, is reduced).
4. Quasifinite and highest weight modules
In this section we introduce certain classes of modules that will play an important role in our exposition. We assume that is either a reductive Lie algebra (considered as a Lie superalgebra with zero odd part), a basic classical Lie superalgebra, or , . Thus, in particular, is a reductive Lie algebra. If is a basic Lie superalgebra or , , we choose a triangular decomposition of as in Section 2.2. If is a reductive Lie algebra, we choose any triangular decomposition. We set . We also identify with the subsuperalgebra .
Definition 4.1 (Weight module).
A -module is called a weight module if its restriction to is a weight module, that is, if
The such that are called weights of . A nonzero element of for some is called a weight vector of weight .
Definition 4.2 (Quasifinite module).
A -module is called quasifinite if it is a weight module and all weight spaces are finite dimensional.
Definition 4.3 (Highest weight module).
A -module is called a highest weight module if there exists a nonzero vector such that , , and . Such a vector is called a highest weight vector.
Note that every highest weight module is a weight module. This follows from the fact that the highest weight vector is a weight vector by definition and generates the entire module.
We fix the usual partial order on given by
Every irreducible finite dimensional -module is a highest weight module.
Let be an irreducible finite dimensional -module. Since all irreducible finite dimensional representations of the abelian Lie algebra are one-dimensional, there exists a nonzero vector fixed by . Thus is a weight vector. Since is irreducible as a -module, it follows that generates and hence that is a weight module. Now let be a maximal weight of and let be a nonzero vector in . Then it follows from the irreducibility of that is a highest weight vector. ∎
Definition 4.6 (Irreducible highest weight modules ).
Fix . We define an action of on (considered to be in degree zero) by declaring to act via and to act by zero. We denote the resulting module by and consider the induced module
It is clear that this is a highest weight module. It follows that it possesses a unique maximal submodule and we define
Every irreducible highest weight )-module (hence, by Lemma 4.5, every irreducible finite dimensional )-module) is isomorphic to for some .
Let be a highest weight -module. Then .
This follows easily from the fact that any -module endomorphism of must send a highest weight vector to a highest weight vector, together with the fact that the highest weight space of is one-dimensional. ∎
Note that the algebra of endomorphisms of an irreducible module over a Lie superalgebra is not always isomorphic to . Indeed, in some cases this algebra of endomorphisms is spanned (over ) by the identity and an involution interchanging the even and odd parts of the module (see [Kac78, p. 609]).
Proposition 4.9 (Density Theorem).
Suppose is an irreducible highest weight -module, and let be the corresponding representation of the universal enveloping superalgebra. Let and . Then there exists such that for . If is finite dimensional, then .
By Lemma 4.7, we have . The result then follows from the Jacobson Density Theorem (see, for example, [Lan02, Thm. XVII.3.2]), where we consider only homogeneous elements. See also [Che95, Prop. 8.2]. ∎
Suppose are reductive Lie algebras, basic classical Lie superalgebras, or , , and are commutative associative algebras. If is an irreducible highest weight -module for , then is an irreducible highest weight module for .
For , we have the triangular decomposition as described at the beginning of this section. For each weight of , let be a basis for . Let be an arbitrary nonzero element of . Then can be written as
where the first sum is over the weights of and the are weight vectors with only a finite number of them nonzero. Now, let be a minimal weight among the weights of the (nonzero) and fix a nonzero of weight . Proposition 4.9 and the PBW Theorem (Lemma 2.7) imply that there exists a weight vector such that is a highest weight vector of . Then it follows from our choices that, for all , a weight of , the vector is a either zero or a highest weight vector of . Therefore, if is a highest weight vector of , we have
for some , not all zero (but with only finitely many nonzero), and a nonempty finite set of weights of . An argument similar to the one above shows that there exists such that
where is a highest weight vector of . Since generates as a module for , so does . It follows that is irreducible. It is also clearly highest weight, with highest weight vector . ∎
Suppose are reductive Lie algebras, basic classical Lie superalgebras, or , , and are commutative associative algebras. Then any irreducible finite dimensional module for is of the form for irreducible finite dimensional -modules , .
Suppose is an irreducible finite dimensional module for . Then it is also an irreducible module for (tensor product as superalgebras). By [Che95, Prop. 8.4], there exist irreducible -modules , , such that is either isomorphic to or a proper submodule of this tensor product. Since this tensor product is irreducible by Lemma 4.10, the result follows. ∎
The tensor product of two irreducible highest weight -modules with disjoint supports is irreducible.
Suppose and are irreducible highest weight -modules with disjoint supports. For , let and let be the representation corresponding to . The representation factors through the composition
where is the diagonal embedding and the second map is the obvious projection on each summand. Since the supports of and are disjoint, we have by Lemma 2.3. Therefore . We thus have the following commutative diagram: