A Vertex Algebra Attached to the Flag Manifold and Lie Algebra Cohomology

Let [Formula: see text] be an untwisted affine Kac–Moody Lie algebra. The top of every irreducible highest weight integrable [Formula: see text]-module is the finite-dimensional irreducible [Formula: see text]-module, where the action of the simple Lie algebra [Formula: see text] is given by zeroth products arising from the underlying vertex operator algebra theory. Motivated by this fact, we consider zeroth products of level [Formula: see text] Frenkel–Jing operators corresponding to Drinfeld realization of the quantum affine algebra [Formula: see text]. By applying these products, which originate from the quantum vertex algebra theory developed by Li, on the extension of Koyama vertex operator [Formula: see text], we obtain an infinite-dimensional vector space [Formula: see text]. Next, we introduce an associative algebra [Formula: see text], a certain quantum analogue of the universal enveloping algebra [Formula: see text], and construct some infinite-dimensional [Formula: see text]-modules [Formula: see text] corresponding to the finite-dimensional irreducible [Formula: see text]-modules [Formula: see text]. We show that the space [Formula: see text] carries a structure of an [Formula: see text]-module and, furthermore, we prove that the [Formula: see text]-module [Formula: see text] is isomorphic to the [Formula: see text]-module [Formula: see text].

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Cohomology of discrete subgroups and Lie algebra cohomology

Mathematical Surveys and Monographs - Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups ◽

10.1090/surv/067/08 ◽

2000 ◽

pp. 137-150

Author(s):

A. Borel ◽

N. Wallach

Keyword(s):

Lie Algebra ◽

Discrete Subgroups ◽

Lie Algebra Cohomology

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Characteristic functions of semigroups in semi-simple Lie groups

Forum Mathematicum ◽

10.1515/forum-2018-0243 ◽

2019 ◽

Vol 31 (4) ◽

pp. 815-842

Author(s):

Luiz A. B. San Martin ◽

Laercio J. Santos

Keyword(s):

Laplace Transform ◽

Lie Groups ◽

Lie Algebra ◽

Lie Group ◽

Convex Cone ◽

Convex Set ◽

Flag Manifold ◽

Riemannian Metric ◽

Iwasawa Decomposition ◽

Characteristic Functions

Abstract Let G be a noncompact semi-simple Lie group with Iwasawa decomposition {G=KAN} . For a semigroup {S\subset G} with nonempty interior we find a domain of convergence of the Helgason–Laplace transform {I_{S}(\lambda,u)=\int_{S}e^{\lambda(\mathsf{a}(g,u))}\,dg} , where dg is the Haar measure of G, {u\in K} , {\lambda\in\mathfrak{a}^{\ast}} , {\mathfrak{a}} is the Lie algebra of A and {gu=ke^{\mathsf{a}(g,u)}n\in KAN} . The domain is given in terms of a flag manifold of G written {\mathbb{F}_{\Theta(S)}} called the flag type of S, where {\Theta(S)} is a subset of the simple system of roots. It is proved that {I_{S}(\lambda,u)<\infty} if λ belongs to a convex cone defined from {\Theta(S)} and {u\in\pi^{-1}(\mathcal{D}_{\Theta(S)}(S))} , where {\mathcal{D}_{\Theta(S)}(S)\subset\mathbb{F}_{\Theta(S)}} is a B-convex set and {\pi:K\rightarrow\mathbb{F}_{\Theta(S)}} is the natural projection. We prove differentiability of {I_{S}(\lambda,u)} and apply the results to construct of a Riemannian metric in {\mathcal{D}_{\Theta(S)}(S)} invariant by the group {S\cap S^{-1}} of units of S.

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Lie-algebra cohomology and the Osp(1,1‖2) structure in string theory

Physical Review D ◽

10.1103/physrevd.36.3731 ◽

1987 ◽

Vol 36 (12) ◽

pp. 3731-3741 ◽

Cited By ~ 5

Author(s):

Antti J. Niemi

Keyword(s):

String Theory ◽

Lie Algebra ◽

Lie Algebra Cohomology

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Vertex Lie algebras and cyclotomic coinvariants

Communications in Contemporary Mathematics ◽

10.1142/s0219199716500152 ◽

2017 ◽

Vol 19 (02) ◽

pp. 1650015 ◽

Cited By ~ 5

Author(s):

Benoît Vicedo ◽

Charles Young

Keyword(s):

Lie Algebra ◽

Lie Algebras ◽

Bethe Ansatz ◽

Riemann Sphere ◽

Marked Point ◽

Vertex Algebra ◽

The Other ◽

Gaudin Models ◽

Equivariant Functions ◽

Marked Points

Given a vertex Lie algebra [Formula: see text] equipped with an action by automorphisms of a cyclic group [Formula: see text], we define spaces of cyclotomic coinvariants over the Riemann sphere. These are quotients of tensor products of smooth modules over “local” Lie algebras [Formula: see text] assigned to marked points [Formula: see text], by the action of a “global” Lie algebra [Formula: see text] of [Formula: see text]-equivariant functions. On the other hand, the universal enveloping vertex algebra [Formula: see text] of [Formula: see text] is itself a vertex Lie algebra with an induced action of [Formula: see text]. This gives “big” analogs of the Lie algebras above. From these we construct the space of “big” cyclotomic coinvariants, i.e. coinvariants with respect to [Formula: see text]. We prove that these two definitions of cyclotomic coinvariants in fact coincide, provided the origin is included as a marked point. As a corollary, we prove a result on the functoriality of cyclotomic coinvariants which we require for the solution of cyclotomic Gaudin models in [B. Vicedo and C. Young, Cyclotomic Gaudin models: Construction and Bethe ansatz, preprint (2014); arXiv:1409.6937]. At the origin, which is fixed by [Formula: see text], one must assign a module over the stable subalgebra [Formula: see text] of [Formula: see text]. This module becomes a [Formula: see text]-quasi-module in the sense of Li. As a bi-product we obtain an iterate formula for such quasi-modules.

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