Extensions between finite-dimensional simple modules over a generalized current Lie algebra

Let Γ be a group acting on a scheme X and on a Lie superalgebra g. The corresponding equivariant map superalgebra M (g, X) Γ is the Lie superalgebra of equivariant regular maps from X to g. In this paper we complete the classification of finite-dimensional irreducible M (g, X) Γmodules when g is a finite-dimensional simple Lie superalgebra, X is of finite type, and Γ is a finite abelian group acting freely on the rational points of X. We also describe extensions between these irreducible modules in terms of extensions between modules for certain finite-dimensional Lie superalgebras. As an application, when Γ is trivial and g is of type B(0, n), we describe the block decomposition of the category of finite-dimensional M (g, X) Γ -modules in terms of spectral characters for g.

Section: Introductionsupporting

confidence: 89%

Irreducible modules for equivariant map superalgebras and their extensions

Calixto

Macedo

2019

“…Theorem 2.10 allows one to translate any reasonable question in the representation theory of finite-dimensional modules for equivariant maps algebras, where Γ is abelian and acts freely on X, to a corresponding question for untwisted map algebras (generalized current algebras). For instance, it can be used to reduce the computation of extensions between irreducible finite-dimensional (g ⊗ A) Γ -modules to the case of extensions of (g ⊗ A)modules, which were considered in [Kod10]. In this way, one can give an alternate proof of [NS, Prop.…”

Section: Lemma 23 ([Ns Lem 44])mentioning

confidence: 99%

Local Weyl modules for equivariant map algebras with free abelian group actions

Fourier¹,

Khandai²,

Kus³

et al. 2012

Suppose a finite group Γ acts on a scheme X and a finite-dimensional Lie algebra g. The associated equivariant map algebra is the Lie algebra of equivariant regular maps from X to g. Examples include generalized current algebras and (twisted) multiloop algebras.Local Weyl modules play an important role in the theory of finite-dimensional representations of loop algebras and quantum affine algebras. In the current paper, we extend the definition of local Weyl modules (previously defined only for generalized current algebras and twisted loop algebras) to the setting of equivariant map algebras where g is semisimple, X is affine of finite type, and the group Γ is abelian and acts freely on X. We do so by defining twisting and untwisting functors, which are isomorphisms between certain categories of representations of equivariant map algebras and their untwisted analogues. We also show that other properties of local Weyl modules (e.g. their characterization by homological properties and a tensor product property) extend to the more general setting considered in the current paper.

“…✷ Remark 4.9 Theorem 4.8 was first proved in the simpler setting of untwisted current algebras g ⊗ k S in [7].…”

Section: Similarly the Adjoint Action Of L Induces An Action Ofmentioning

confidence: 99%

Extensions of modules for twisted current algebras

Auger

Lau

2019

Twisted current algebras are fixed point subalgebras of current algebras under a finite group action. Special cases include equivariant map algebras and twisted forms of current algebras. Their finite-dimensional simple modules fall into two categories, those which factor through an evaluation map and those which do not. We show that there are no nontrivial extensions between finite-dimensional simple evaluation and non-evaluation modules. We then compute extensions between any pair of finite-dimensional simple modules for twisted current algebras, and use this information to determine the block decomposition for the category. In the special case of twisted forms, this decomposition can be described in terms of maps to the fundamental group of the underlying root system.