Weyl modules for Lie superalgebras

Calixto, Lucas; Lemay, Joel; Savage, Alistair

doi:10.1090/proc/13146

Cited by 13 publications

(14 citation statements)

References 20 publications

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“…The next result describes local Weyl modules as universal objects. Its proof is similar to that of [CLS,Proposition 4.13].…”

Section: Local Weyl Modulesmentioning

confidence: 60%

“…(I) = {m ∈ MaxSpec(A) | I ⊆ m}.The next result generalizes[CLS, Theorem 4.15].Proposition 8.18. Let ψ, ϕ ∈ (h ⊗ A) * , ψ| h = λ, ϕ| h = µ, and suppose that λ, µ ∈ X + are such that λ + µ ∈ X + .…”

mentioning

confidence: 58%

“…In the non-super setting, this result was proved in [CP01, Theorem 1] for A = k[t ±1 ], and in [FL04, Theorem 1], for the case where A is the algebra of functions on a complex affine variety. For the case where g is either basic classical or sl(n, n) with n ≥ 2, and A is finitely generated, it was proved in [CLS,Theorem 4.12]. In our curent setting, the result is a direct consequence of Corollary 7.10 and Theorem 8.4.…”

Section: Remark 83 Recall That W Locmentioning

confidence: 71%

“…Moreover, since A is assumed to be infinite dimensional, we have that g − ⊗ A is infinite dimensional, which implies that W r (ψ) is infinite dimensional. Finally, notice that there exists a unique surjective homomorphism of g ⊗ A-modules W loc [CLS,Proposition 4.13]). In the current setting, we have proved in Proposition 8.2 that the local Weyl module W loc A (ψ) is a universal object in the category C λ A (λ = ψ| h ).…”

Section: Now Letmentioning

confidence: 99%

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Weyl Modules and Weyl Functors for Lie Superalgebras

Bagci

Calixto

Macedo

2018

Algebr Represent Theor

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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 ⊗ k A is known as a map superalgebra. Map superalgebras generalize important classes of Lie superalgebras, such as, loop superalgebras (where A = k[t, 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).2010 Mathematics Subject Classification. 17B65, 17B10.

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“…The next result describes local Weyl modules as universal objects. Its proof is similar to that of [CLS,Proposition 4.13].…”

Section: Local Weyl Modulesmentioning

confidence: 60%

mentioning

confidence: 58%

Section: Remark 83 Recall That W Locmentioning

confidence: 71%

Section: Now Letmentioning

confidence: 99%

See 2 more Smart Citations

Weyl Modules and Weyl Functors for Lie Superalgebras

Bagci

Calixto

Macedo

2018

Algebr Represent Theor

Self Cite

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“…The definition, given via generators and relations, was inspired by the similar notion in modular representation theory of algebraic groups. In the years that followed, the notion was extended to other algebras sharing some similarity with the affine Kac-Moody algebras such as algebras of the form g ⊗ A where g is a symmetrizable Kac-Moody algebra or a Lie super algebra and A is a commutative associative algebra with unit (see [2,5,14,18,20,27] and references therein). After [14], there is a distinction between two kinds of Weyl modules: the local and the global ones (both appeared in [9], but only the former under the terminology Weyl module).…”

Section: Introductionmentioning

confidence: 99%

On truncated Weyl modules

Fourier

Martins

Moura

2019

Communications in Algebra

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We study structural properties of truncated Weyl modules. A truncated Weyl module WN (λ) is a local Weyl module for g [t], where g is a finite-dimensional simple Lie algebra. It has been conjectured that, if N is sufficiently small with respect to λ, the truncated Weyl module is isomorphic to a fusion product of certain irreducible modules. Our main result proves this conjecture when λ is a multiple of certain fundamental weights, including all minuscule ones for simply laced g. We also take a further step towards proving the conjecture for all multiples of fundamental weights by proving that the corresponding truncated Weyl module is isomorphic to a natural quotient of a fusion product of Kirillov-Reshetikhin modules. One important part of the proof of the main result shows that any truncated Weyl module is isomorphic to a Chari-Venkatesh module and explicitly describes the corresponding family of partitions. This leads to further results in the case that g = sl2 related to Demazure flags and chains of inclusions of truncated Weyl modules.Part of this work was developed while the second author was a visiting Ph.D. student at the University of Cologne. He thanks Peter Littelmann and Deniz Kus for their guidance and support during that period and the University of Cologne for hospitality. He also thanks CAPES and CNPq (SWE 203324/2014-5) for the financial support.A. M. was partially supported by CNPq grant 304477/2014-1 and Fapesp grant 2014/09310-5.

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Weyl Modules for Toroidal Lie Algebras

Mukherjee

Pattanayak²,

Sharma³

2022

Algebr Represent Theor

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In this paper, we extend the notion of Weyl modules for twisted toroidal Lie algebra T (µ). We prove that the level one global Weyl modules of T (µ) are isomorphic to the tensor product of the level one representation of twisted affine Lie algebras and certain lattice vertex algebras. As a byproduct, we calculate the graded character of the level one local Weyl modules of T (µ).Notations:• The sets of complex numbers, integers, non-negative integers, and positive integers are denoted by C, Z, Z ≥0 , and Z >0 , respectively.and the sets Z n , Z n ≥0 , and Z n >0 are defined similarly. • For n ≥ 2, we denote the elements (m 1 , . . . , m n ) ∈ Z n and (m 2 , . . . , m n ) ∈ Z n−1 by m and m, respectively. • Let A n := C[t ±1 1 , t ±1 2 , . . . , t ±1 n ] be the set of Laurent polynomial ring in n variables. • For m = (m 1 , . . . , m n ) ∈ Z n and m = (m 2 , . . . , m n ) ∈ Z n−1 the notation t m and t m represents t m 1 1 • • • t mn n and t m 2 2 • • • t mn n , respectively, in A n . • Let U(g) denotes the universal enveloping algebra of g and L n (g) denotes the Lie algebra g ⊗ A n with commutator given by [x ⊗ t m , y ⊗ t n ] = [x, y] ⊗ t m+n .

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Weyl modules for Lie superalgebras

Cited by 13 publications

References 20 publications

Weyl Modules and Weyl Functors for Lie Superalgebras

Weyl Modules and Weyl Functors for Lie Superalgebras

On truncated Weyl modules

Weyl Modules for Toroidal Lie Algebras

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