Vertex algebras and integral bases for the enveloping algebras of affine Lie algebras

1992 ◽  
Vol 96 (466) ◽  
pp. 0-0 ◽  
Author(s):  
Shari A. Prevost
Keyword(s):  
Lie Algebras ◽  
Vertex Algebras ◽  
Affine Lie Algebras ◽  
Enveloping Algebras ◽  
Integral Bases

scholarly journals Trigonometric Lie algebras, affine Lie algebras, and vertex algebras

2020 ◽  
Vol 363 ◽  
pp. 106985
Author(s):  
Haisheng Li ◽  
Shaobin Tan ◽  
Qing Wang
Keyword(s):  
Lie Algebras ◽  
Vertex Algebras ◽  
Affine Lie Algebras

scholarly journals Principal subspaces of higher-level standard $\widehat{\mathfrak{sl}(n)}$-modules

2015 ◽  
Vol 26 (08) ◽  
pp. 1550053 ◽  
Author(s):  
Christopher Sadowski
Keyword(s):  
Lie Algebras ◽  
Operator Algebras ◽  
Intertwining Operators ◽  
Natural Setting ◽  
Affine Lie Algebras ◽  
Universal Enveloping Algebras ◽  
Enveloping Algebras ◽  
Defining Relations ◽  
Standard Formula ◽  
Exact Sequences

Using completions of certain universal enveloping algebras, we provide a natural setting for families of defining relations for the principal subspaces of standard modules for untwisted affine Lie algebras. We also use the theory of vertex operator algebras and intertwining operators to construct exact sequences among principal subspaces of certain standard [Formula: see text]-modules, n ≥ 3. As a consequence, we obtain the multigraded dimensions of the principal subspaces W(k1Λ1 + k2Λ2) and W(kn-2Λn-2 + kn-1Λn-1). This generalizes earlier work by Calinescu on principal subspaces of standard [Formula: see text]-modules.

Vertex algebras associated to the affine Lie algebras of abelian polynomial current algebras

2016 ◽  
Vol 27 (05) ◽  
pp. 1650046 ◽  
Author(s):  
Jinwei Yang
Keyword(s):  
Differential Equations ◽  
Lie Algebras ◽  
Tensor Category ◽  
Extension Property ◽  
Vertex Algebras ◽  
Affine Lie Algebras ◽  
Systems Of Differential Equations ◽  
Convergence And Extension ◽  
Logarithmic Intertwining Operators ◽  
Current Algebras

We construct a family of vertex algebras associated to the affine Lie algebra of polynomial current algebras of finite-dimensional abelian Lie algebras, along with their modules and logarithmic modules. These vertex algebras and their (logarithmic) modules are strongly [Formula: see text]-graded and quasi-conformal. We then show that matrix elements of products and iterates of logarithmic intertwining operators among these logarithmic modules satisfy certain systems of differential equations. Using these systems of differential equations, we verify the convergence and extension property needed in the logarithmic tensor category theory developed by Huang, Lepowsky and Zhang.

scholarly journals TWISTED QUANTUM AFFINE ALGEBRAS AND SOLUTIONS TO THE YANG-BAXTER EQUATION

1996 ◽  
Vol 11 (19) ◽  
pp. 3415-3437 ◽  
Author(s):  
GUSTAV W. DELIUS ◽  
MARK D. GOULD ◽  
YAO-ZHONG ZHANG
Keyword(s):  
Lie Algebras ◽  
Spectral Parameter ◽  
Baxter Equation ◽  
Affine Lie Algebras ◽  
Quantum Affine Algebras ◽  
Enveloping Algebras ◽  
Quantized Enveloping Algebras ◽  
Affine Algebras ◽  
Parameter Dependent

We construct spectral-parameter-dependent R matrices for the quantized enveloping algebras of twisted affine Lie algebras. These give new solutions to the spectral-parameter-dependent quantum Yang-Baxter equation.

Vertex algebras and extended affine Lie algebras coordinated by rational quantum tori

2021 ◽  
Vol 569 ◽  
pp. 111-142
Author(s):  
Fulin Chen ◽  
Xiaoling Liao ◽  
Shaobin Tan ◽  
Qing Wang
Keyword(s):  
Lie Algebras ◽  
Vertex Algebras ◽  
Affine Lie Algebras ◽  
Quantum Tori

The composition Hall algebra of a weighted projective line

2013 ◽  
Vol 2013 (679) ◽  
pp. 75-124 ◽  
Author(s):  
Igor Burban ◽  
Olivier Schiffmann
Keyword(s):  
Lie Algebras ◽  
Projective Line ◽  
Affine Lie Algebras ◽  
Hall Algebra ◽  
Weighted Projective Line ◽  
Drinfeld Double ◽  
Coherent Sheaves ◽  
Enveloping Algebras ◽  
Quantized Enveloping Algebras

Abstract In this article, we deal with properties of the reduced Drinfeld double of the composition subalgebra of the Hall algebra of the category of coherent sheaves on a weighted projective line. This study is motivated by applications in the theory of quantized enveloping algebras of some Lie algebras. We obtain a new realization of the quantized enveloping algebras of affine Lie algebras of simply-laced types as well as some new embeddings between them. Moreover, our approach allows to derive new results on the structure of the quantized enveloping algebras of the toroidal algebras of types D4(1, 1), E6(1, 1), E7(1, 1) and E8(1, 1). In particular, our method leads to a construction of a modular action and allows to define a PBW-type basis for that classes of algebras.

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