4. Integral forms of enveloping algebras of affine Lie algebras

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.

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

International Journal of Modern Physics A ◽

10.1142/s0217751x96001632 ◽

1996 ◽

Vol 11 (19) ◽

pp. 3415-3437 ◽

Cited By ~ 20

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.

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The composition Hall algebra of a weighted projective line

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle.2012.023 ◽

2013 ◽

Vol 2013 (679) ◽

pp. 75-124 ◽

Cited By ~ 4

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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