Homogeneous Construction of the Toroidal Lie Algebra of Type A1

2007 ◽  
Vol 14 (03) ◽  
pp. 425-442
Author(s):  
Haifeng Lian ◽  
Cui Chen ◽  
Qinzhu Wen
Keyword(s):  
Lie Algebra ◽  
Type A ◽  
Vertex Operators ◽  
Moody Algebra ◽  
Completely Reducible ◽  
Toroidal Lie Algebra

In this paper, we consider an analogue of the level two homogeneous construction of the affine Kac–Moody algebra [Formula: see text] by vertex operators. We construct modules for the toroidal Lie algebra and the extended toroidal Lie algebra of type A1. We also prove that the module is completely reducible for the extended toroidal Lie algebra.

scholarly journals Polynomial ring representations of endomorphisms of exterior powers

2021 ◽  
Author(s):  
Ommolbanin Behzad ◽  
André Contiero ◽  
Letterio Gatto ◽  
Renato Vidal Martins
Keyword(s):  
Lie Algebra ◽  
Polynomial Ring ◽  
Vertex Operators ◽  
Exterior Power ◽  
Infinite Dimensional ◽  
Symmetric Structure ◽  
Rational Polynomials ◽  
Exterior Powers ◽  
Vertex Representation ◽  
Countably Infinite

AbstractAn explicit description of the ring of the rational polynomials in r indeterminates as a representation of the Lie algebra of the endomorphisms of the k-th exterior power of a countably infinite-dimensional vector space is given. Our description is based on results by Laksov and Throup concerning the symmetric structure of the exterior power of a polynomial ring. Our results are based on approximate versions of the vertex operators occurring in the celebrated bosonic vertex representation, due to Date, Jimbo, Kashiwara and Miwa, of the Lie algebra of all matrices of infinite size, whose entries are all zero but finitely many.

scholarly journals RELATIVE COMPLETE REDUCIBILITY AND NORMALIZED SUBGROUPS

2020 ◽  
Vol 8 ◽  
Author(s):  
MAIKE GRUCHOT ◽  
ALASTAIR LITTERICK ◽  
GERHARD RÖHRLE
Keyword(s):  
Automorphism Group ◽  
Algebraic Group ◽  
Lie Algebra ◽  
Reductive Algebraic Group ◽  
Identity Component ◽  
Complete Reducibility ◽  
Closed Fields ◽  
Completely Reducible ◽  
Reductive Subgroup ◽  
The Absolute

We study a relative variant of Serre’s notion of $G$ -complete reducibility for a reductive algebraic group $G$ . We let $K$ be a reductive subgroup of $G$ , and consider subgroups of $G$ that normalize the identity component $K^{\circ }$ . We show that such a subgroup is relatively $G$ -completely reducible with respect to $K$ if and only if its image in the automorphism group of $K^{\circ }$ is completely reducible. This allows us to generalize a number of fundamental results from the absolute to the relative setting. We also derive analogous results for Lie subalgebras of the Lie algebra of $G$ , as well as ‘rational’ versions over nonalgebraically closed fields.

scholarly journals Fermionic realization of twisted toroidal Lie algebras

2020 ◽  
pp. 2150143
Author(s):  
Naihuan Jing ◽  
Chad R. Mangum ◽  
Kailash C. Misra
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Type Formula ◽  
Toroidal Lie Algebra

In this paper, we construct a fermionic realization of the twisted toroidal Lie algebra of type [Formula: see text] and [Formula: see text] based on the newly found Moody–Rao–Yokonuma-like presentation.

scholarly journals Complexity for modules over the classical Lie superalgebra

2012 ◽  
Vol 148 (5) ◽  
pp. 1561-1592 ◽  
Author(s):  
Brian D. Boe ◽  
Jonathan R. Kujawa ◽  
Daniel K. Nakano
Keyword(s):  
Lie Algebra ◽  
Lie Superalgebra ◽  
Projective Resolution ◽  
Lie Superalgebras ◽  
Simple Modules ◽  
Minimal Projective Resolution ◽  
Finite Dimensional ◽  
Reductive Lie Algebra ◽  
Completely Reducible ◽  
Kac Modules

AbstractLet ${\Xmathfrak g}={\Xmathfrak g}_{\zerox }\oplus {\Xmathfrak g}_{\onex }$ be a classical Lie superalgebra and let ℱ be the category of finite-dimensional ${\Xmathfrak g}$-supermodules which are completely reducible over the reductive Lie algebra ${\Xmathfrak g}_{\zerox }$. In [B. D. Boe, J. R. Kujawa and D. K. Nakano, Complexity and module varieties for classical Lie superalgebras, Int. Math. Res. Not. IMRN (2011), 696–724], we demonstrated that for any module M in ℱ the rate of growth of the minimal projective resolution (i.e. the complexity of M) is bounded by the dimension of ${\Xmathfrak g}_{\onex }$. In this paper we compute the complexity of the simple modules and the Kac modules for the Lie superalgebra $\Xmathfrak {gl}(m|n)$. In both cases we show that the complexity is related to the atypicality of the block containing the module.

scholarly journals Vertex Representations for the ν+1-Toroidal Lie Algebra of Type Bl

2001 ◽  
Vol 246 (2) ◽  
pp. 564-593 ◽  
Author(s):  
Jiang Cuipo ◽  
Meng Daoji
Keyword(s):  
Lie Algebra ◽  
Toroidal Lie Algebra

scholarly journals Twisted Vertex Operators and Unitary Lie Algebras

2015 ◽  
Vol 67 (3) ◽  
pp. 573-596 ◽  
Author(s):  
Fulin Chen ◽  
Yun Gao ◽  
Naihuan Jing ◽  
Shaobin Tan
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Polynomial Ring ◽  
Central Extension ◽  
Laurent Polynomial ◽  
New Method ◽  
Vertex Operators ◽  
Irreducible Decomposition ◽  
Laurent Polynomial Ring

AbstractA representation of the central extension of the unitary Lie algebra coordinated with a skew Laurent polynomial ring is constructed using vertex operators over an integral ℤ2–lattice. The irreducible decomposition of the representation is explicitly computed and described. As a by–product, some fundamental representations of affine Kac–Moody Lie algebra of type A(2)n are recovered by the new method.

Lie algebra extensions of current algebras on S3

2015 ◽  
Vol 12 (09) ◽  
pp. 1550087 ◽  
Author(s):  
Tosiaki Kori ◽  
Yuto Imai
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Lie Algebra ◽  
Current Algebra ◽  
Central Extension ◽  
Mathematical Tool ◽  
Two Dimensional ◽  
Root Space ◽  
Moody Algebra ◽  
Conformal Field

An affine Kac–Moody algebra is a central extension of the Lie algebra of smooth mappings from S1 to the complexification of a Lie algebra. In this paper, we shall introduce a central extension of the Lie algebra of smooth mappings from S3 to the quaternization of a Lie algebra and investigate its root space decomposition. We think this extension of current algebra might give a mathematical tool for four-dimensional conformal field theory as Kac–Moody algebras give it for two-dimensional conformal field theory.

scholarly journals Vertex representations for toroidal Lie algebra of type G2

2005 ◽  
Vol 198 (1-3) ◽  
pp. 257-279 ◽  
Author(s):  
Dong Liu ◽  
Naihong Hu
Keyword(s):  
Lie Algebra ◽  
Toroidal Lie Algebra
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