Abstract
Given an algebraically closed field 𝕜 of characteristic zero, a Lie superalgebra 𝔤 over 𝕜 and an associative, commutative 𝕜-algebra A with unit, a Lie superalgebra of the form 𝔤 ⊗𝕜A is known as a map superalgebra. Map superalgebras generalize important classes of Lie superalgebras, such as, loop superalgebras (where A = 𝕜[t±1]), and current superalgebras (where A = 𝕜[t]). In this paper, we define Weyl functors, global and local Weyl modules for all map superalgebras where 𝔤 is either 𝔰𝔩(n, n) with n ≥ 2, or a finite-dimensional simple Lie superalgebra not of type 𝔮(n). Under certain conditions on the triangular decomposition of these Lie superalgebras we prove that global and local Weyl modules satisfy certain universal and tensor product decomposition properties. We also give necessary and sufficient conditions for local (resp. global) Weyl modules to be finite dimensional (resp. finitely generated).
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Acknowledgments
We would like to thank M. Brito, T. Brons, A. Moura and A. Savage for useful discussions. L. C. and T. M. would also like to thank IMPA for the wonderful research environment and CNPq for the support during the Summer of 2017. L. C. was also supported by FAPESP grant 2013/08430-4 and PRPq grant ADRC - 05/2016.
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Presented by: Vyjayanthi Chari
Appendix: Homological Properties of Weyl Functors
Appendix: Homological Properties of Weyl Functors
The results of this section show that the Weyl functors defined in the current paper satisfy properties similar to the ones satisfied by Weyl functors defined in the non-super setting. Since the proofs of ther results of this appendix are very similar to those in the non-super setting, we refer to [1, §3.7] and [9, §4] for the details.
Throughout this appendix, we assume that 𝔤 is either 𝔰𝔩(n, n) with n ≥ 2, or a finite-dimensional simple Lie superalgebra not of type 𝔮(n), endowed with a triangular decomposition satisfying ℭ1). We will also assume that A, B and C are associative, commutative 𝕜-algebras with unit.
Proposition A.1
Let λ ∈ X+.
-
(a)
For every A λ -module M, there is an isomorphism of A λ -modules \(\mathbf {R}_{A}^{\lambda } \mathbf {W}_{A}^{\lambda } M \cong M\) that is functorial in M.
-
(b)
\(\mathbf {W}_{A}^{\lambda } \colon \mod {\mathbf {A}_{\lambda }} \to \mathcal {C}_{A}^{\lambda }\)isleft adjoint to\(\mathbf {R}_{A}^{\lambda } \colon \mathcal {C}_{A}^{\lambda } \to \mod {\mathbf {A}_{\lambda }}\).
-
(c)
\(\mathbf {W}_{A}^{\lambda }\) is fully faithful.
-
(d)
IfM is a projectiveAλ-module,then\(\mathbf {W}_{A}^{\lambda } M\)isa projective object in\(\mathcal {C}_{A}^{\lambda }\).
Corollary A.2
For each \(\lambda \in X^{+}\), the module\(W_{A}(\lambda )\)isprojective in\(\mathcal {C}_{A}^{\lambda }\)andthe moduleK(λ) isprojective in\(\mathcal {C}^{\lambda }_{A}\). Moreover, there is an isomorphism of algebras\(\text {Hom}_{\mathcal {C}^{\lambda }_{A}} \left (W_{A} (\lambda ), W_{A} (\lambda ) \right ) \cong \mathbf {A}_{\lambda }\).
Notice that, despite \(\mathbf {W}_{A}^{\lambda }\) being a fully faithful functor (by Proposition A.1(c)), it is not an equivalence of categories, as it is not essentially surjective. In fact, if μ < λ, then \(W_{A} (\mu )\) is an object of \(\mathcal {C}_{A}^{\lambda }\) for which there exists no \(\mathbf {A}_{\lambda }\)-module N satisfying \(\mathbf {W}_{A}^{\lambda } N \cong W_{A}(\mu )\). (If \(\mathbf {W}_{A}^{\lambda } N \cong W_{A}(\mu )\), then \(N \cong \mathbf {R}_{A}^{\lambda } \mathbf {W}_{A}^{\lambda } N \cong \mathbf {R}_{A}^{\lambda } W_{A}(\mu ) = 0\).) Theorem A.3 describes for which objects M of \(\mathcal {C}_{A}^{\lambda }\) there exists an \(\mathbf {A}_{\lambda }\)-module N satisfying \(\mathbf {W}_{A}^{\lambda } N \cong M\).
Theorem A.3
LetM be an object of\(\mathcal {C}_{A}^{\lambda }\). Then\(M \cong \mathbf {W}_{A}^{\lambda } \mathbf {R}_{A}^{\lambda } M\)if and only if, foreach objectN of\(\mathcal {C}_{A}^{\lambda }\)thatsatisfies\(N_{\lambda } = 0\), wehave
Corollary A.4
The functor\(\mathbf {W}_{A}^{\lambda }\)isexact if and only if, for each objectN of\(\mathcal {C}_{A}^{\lambda }\)thatsatisfies\(N_{\lambda } = 0\), we have
Similar to the definition of \(\mathbf {A}_{\lambda }\) in Section 6, for each \(\lambda \in X^{+}\), let
Remark A.5
-
(a)
Every homomorphism of 𝕜-algebras π : C → A induces a unique (even) homomorphism of Lie superalgebras (which we denote by the same symbol) π : 𝔤 ⊗ C → 𝔤 ⊗ A satisfying
$$\pi (x \otimes c) = x \otimes \pi(c) \quad \textup{ for all } x \in \mathfrak{g} \textup{ and } c \in C. $$This latter homomorphism induces an action of 𝔤 ⊗ C on any 𝔤 ⊗ A-module M via the pull-back. Let π∗M denote such a 𝔤 ⊗ C-module.
-
(b)
Let \(\lambda \in X^{+}\) and π : C → A be a homomorphism of 𝕜-algebras. Using item (a), we see that π also induces a homomorphism of associative superalgebras (which we keep denoting by the same symbol), \(\pi \colon \mathbf {U}{(\mathfrak {g} \otimes C}) \to \mathbf {U}{(\mathfrak {g} \otimes A})\). Notice that by construction, \(\pi (\mathfrak {n}^{+} \otimes C) \subseteq \mathfrak {n}^{+} \otimes A\), \(\pi (h) - \lambda (h) = h - \lambda (h)\) for all h ∈ 𝔥, and \(\pi (x_{\alpha }^{-})^{k} = (x_{\alpha }^{-})^{k}\) for all \(\alpha \in {\Delta }_{\mathfrak r}\) and k ≥ 0. Hence
$$\pi \left( \text{Ann}_{\mathfrak{g} \otimes C} (w_{\lambda}) \right) \subseteq \text{Ann}_{\mathfrak{g} \otimes A} (w_{\lambda}) \quad \textup{ and } \quad \pi \left( \text{Ann}_{\mathfrak{h} \otimes C} (w_{\lambda}) \right) \subseteq \text{Ann}_{\mathfrak{h} \otimes A} (w_{\lambda}). $$Thus π induces a homomorphism of 𝕜-algebras \(\overline \pi \colon \mathbf {C}_{\lambda } \to \mathbf {A}_{\lambda }\), and every \(\mathbf {A}_{\lambda }\)-module V admits a structure of \(\mathbf {C}_{\lambda }\)-module via the pull-back along \(\overline \pi \). Denote this \(\mathbf {C}_{\lambda }\)-module by \(\overline \pi ^{*}V\).
-
(c)
Let \(\lambda , \mu \in X^{+}\) be such that \(\lambda +\mu \in X^{+}\), and recall that the action of the superalgebra \(\mathbf {U}{(\mathfrak {g} \otimes A})\) on \(W_{A}(\lambda ) \otimes W_{A} (\mu )\) is induced by the comultiplication \({\Delta } \colon \mathbf {U}(\mathfrak {g} \otimes A) \to \mathbf {U}(\mathfrak {g} \otimes A) \otimes \mathbf {U}(\mathfrak {g} \otimes A)\). In particular, we have x(wλ ⊗ wμ) = (xwλ) ⊗ wμ + wλ ⊗ (xwμ) for all \(x \in \mathfrak {g} \otimes A\), and thus \(w_{\lambda } \otimes w_{\mu }\) is a highest-weight vector in WA(λ) ⊗ WA(μ). Hence, there exists a unique surjective homomorphism of \(\mathfrak {g} \otimes A\)-modules \(\xi \colon W_{A}(\lambda + \mu ) \twoheadrightarrow W_{A}(\lambda ) \otimes W_{A}(\mu )\) satisfying \(\xi (w_{\lambda +\mu } ) = w_{\lambda } \otimes w_{\mu }\). Now, notice that \(\mathbf {R}_{A}^{\lambda +\mu } \xi \) is a surjective homomorphism of \(\mathbf {U}(\mathfrak {h} \otimes A)\)-modules:
$$\mathbf{R}_{A}^{\lambda+\mu} \xi \colon \mathbf{U}(\mathfrak{h} \otimes A) w_{\lambda+\mu} \twoheadrightarrow \mathbf{U}(\mathfrak{h} \otimes A)w_{\lambda} \otimes \mathbf{U}(\mathfrak{h} \otimes A)w_{\mu}. $$Moreover, since \(\mathbf {U}(\mathfrak {h} \otimes A) w_{\nu } \cong \mathbf {A}_{\nu }\) for all \({\nu } \in X^{+}\), \(\mathbf {R}_{A}^{\lambda +\mu } \xi \) induces a homomorphism of commutative 𝕜-algebras \({\Delta }_{\lambda , \mu } \colon \mathbf {A}_{\lambda +\mu } \to \mathbf {A}_{\lambda } \otimes \mathbf {A}_{\mu }\). Thus, given an \(\mathbf {A}_{\lambda }\)-module M and an \(\mathbf {A}_{\mu }\)-module N, their tensor product M ⊗ N admits an \(\mathbf {A}_{\lambda + \mu }\)-module structure via the pull-back along \({\Delta }_{\lambda , \mu }\). Denote this \(\mathbf {A}_{\lambda +\mu }\)-module by \({\Delta }_{\lambda ,\mu }^{*} (M\otimes N)\).
Theorem A.6
Let 𝔤 bea finite-dimensional simple Lie superalgebra not oftype 𝔮(n), with a fixedtriangular decomposition satisfying (ℭ2). Suppose also thatA andB are finite-dimensional commutative,associative 𝕜-algebras with unit andlet\(\pi _{A}\colon A\oplus B\twoheadrightarrow A\)and \(\pi _{B}\colon A\oplus B\twoheadrightarrow B\)bethe canonical projections.Let\(\lambda ,\mu \in X^{+}\)besuch that \(\lambda +\mu \in X^{+}\). If\(M \in \mod {\mathbf {A}_{\lambda }}\), \(N \in \mod {\mathbf {B}_{\mu }}\)arefinite dimensional, then there is an isomorphismof 𝔤 ⊗ (A ⊕ B)-modules
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Bagci, I., Calixto, L. & Macedo, T. Weyl Modules and Weyl Functors for Lie Superalgebras. Algebr Represent Theor 22, 723–756 (2019). https://doi.org/10.1007/s10468-018-9796-2
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DOI: https://doi.org/10.1007/s10468-018-9796-2