Some examples are given of finite dimensional Lie bialgebras whose brackets and cobrackets are determined by pairs of r-matrices. The aim of this Letter is to give some low-dimensional examples of classical coboundary Lie bialgebras [1, 2] with coboundary duals. Since such structures can be specified (up to automorphisms) by pairs of r-matrices, so it is natural to call them bi-r-matrix bialgeb...

A double algebra is a linear space V equipped with linear map V ⊗ V → V ⊗ V . Additional conditions on this map lead to the notions of Lie and associative double algebras. We prove that simple finite-dimensional Lie double algebras do not exist over an arbitrary field, and all simple finite-dimensional associative double algebras over an algebraically closed field are trivial. Over an arbitrary...

0. Introduction 1 0.1. The Finite-Dimensional Case SU(n) 2 0.2. The Infinite-Dimensional Case LSU(n) 4 0.3. Plan of this Dissertation 13 1. Factorization and Coordinates for Finite-Dimensional Lie Groups 16 1.1. Lie Algebras 16 1.2. Lie Groups 19 1.3. Parametrizations with Sequences of Simple Roots 22 1.4. Parametrizations with Inversion Sets 26 2. Reduced Words for Infinite Elements of Affine ...

The generalized quantum group U(ϵ) of type A is an affine analogue associated to a general linear Lie superalgebra glM|N. We prove that there exists unique R matrix on the tensor product fundamental representations for arbitrary parameter sequence ϵ corresponding non-conjugate Borel subalgebra give explicit description its spectral decomposition, and then as application, construct family finite...

The classical Tits construction provides models of the exceptional simple Lie algebras in terms of a unital composition algebra and a degree three simple Jordan algebra. A couple of actions of the symmetric group S4 on this construction are given. By means of these actions, the models provided by the Tits construction are related to models of the exceptional Lie algebras obtained from two diffe...

A certain class of partial differential equations possesses singular solutions having discontinuous first derivatives (“peakons”). The time evolution of peaks of such solutions is governed by a finite dimensional completely integrable system. Explicit solutions of this system are constructed by using algebraic-geometric method which casts it as a flow on an appropriate Riemann surface and reduc...

In this paper, the structure of parafermion vertex operator algebra associated to an integrable highest weight module for simple affine Lie superalgebra osp(1|2n) is studied. Particularly, we determine generators algebra.

The paper concerns two versions of the notion of real forms of Lie superalgebras. One is the standard approach, where a real form of a complex Lie superalgebra is a real Lie superalgebra such that its complexification is the original complex Lie superalgebra. The second is related to considering A-points of a Lie superalgebra over a commutative complex superalgebra A equipped with superconjugat...

Weyl denominator identity for the affinization of a basic Lie superalgebra with non-zero Killing form was formulated by V. Kac and M. Wakimoto and was proven by them for the defect one case. In this paper we prove this identity. 0. Introduction Let g be a basic Lie superalgebra with a non-zero Killing form. Let ĝ be the affinization of g. Let h (resp., ĥ) be the Cartan subalgebra in g (resp., i...

Abstract We study a semisimple extension of Takiff superalgebra, which turns out to have remarkably rich representation theory. determine the blocks in both finite-dimensional and BGG module categories also classify Borel subalgebras. further compute all groups between two simple objects prove that non-principal category are Koszul.