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.

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.

scholarly journals ON TYPE I QUANTUM AFFINE SUPERALGEBRAS

1995 ◽  
Vol 10 (23) ◽  
pp. 3259-3281 ◽  
Author(s):  
GUSTAV W. DELIUS ◽  
MARK D. GOULD ◽  
JON R. LINKS ◽  
YAO-ZHONG ZHANG
Keyword(s):  
Spectral Parameter ◽  
Lie Superalgebras ◽  
Baxter Equation ◽  
Type I ◽  
Free Fermion ◽  
General Technique ◽  
Finite Dimensional ◽  
Quantum Deformations ◽  
Free Fermion Model ◽  
Parameter Dependent

The type I simple Lie superalgebras are sl(m|n) and osp(2|2n). We study the quantum deformations of their untwisted affine extensions Uq[sl(m|n)(1)] and Uq[osp(2|2n)(1)]. We identify additional relations between the simple generators (“extra q Serre relations”) which need to be imposed in order to properly define Uq[sl(m|n)(1)] and Uq[osp(2|2n)(1)]. We present a general technique for deriving the spectral-parameter-dependent R matrices from quantum affine superalgebras. We determine the R matrices for the type I affine superalgebra Uq[sl(m|n)(1)] in various representations, thereby deriving new solutions of the spectral-parameter-dependent Yang-Baxter equation. In particular, because this algebra possesses one-parameter families of finite-dimensional irreps, we are able to construct R matrices depending on two additional spectral-parameter-like parameters, providing generalizations of the free fermion model.

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.

scholarly journals Generalized Rogers–Ramanujan identities for twisted affine algebras

2017 ◽  
Vol 32 (21) ◽  
pp. 1750110
Author(s):  
Arel Genish ◽  
Doron Gepner
Keyword(s):  
Lie Algebras ◽  
Low Rank ◽  
Field Theories ◽  
Complete Agreement ◽  
Affine Lie Algebras ◽  
Conformal Field Theories ◽  
Low Level ◽  
Conformal Field ◽  
Affine Algebras

The characters of parafermionic conformal field theories are given by the string functions of affine algebras, which are either twisted or untwisted algebras. Expressions for these characters as generalized Rogers–Ramanujan algebras have been established for the untwisted affine algebras. However, we study the identities for the string functions of the twisted affine Lie algebras. A conjecture for the string functions was proposed by Hatayama et al., for the unit fields, which expresses the string functions as Rogers–Ramanujan type sums. Here we propose to check the Hatayama et al. conjecture, using Lie algebraic theoretic methods. We use Freudenthal’s formula, which we computerized, to verify the identities for all the algebras at low rank and low level. We find complete agreement with the conjecture.

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