Please let me know if you find any mistakes! Friday, October 3, 2014. Grading will be based on an exam on Friday, December 12. Groups of general interest often have some additional natural structure: Topological groups, Lie groups. From Lie groups, we get Lie algebras via a standard procedure in differential topology. There’s also an association in the reverse direction, from Lie algebras to Li...

We complete the study of Manin triples of real simple Lie algebras. In the Part 2 of the article we classify the Manin triples (g(R), W, g(R) ⊕ g(R)) (case 2 of the doubles) up to weak and gauge equivalence. First we recall the main definitions of the Part 1. Definition 1. Let g 1 , g 2 , d be Lie algebras over a field K and let Q be a symmetric nondegenerate bilinear form on d. A triple (g 1 ,...

In this paper we give explicit descriptions of complete sets of mutually unbiased bases (MUBs) and orthogonal decompositions of special Lie algebras sln(C) obtained from commutative and symplectic semifields, and from some other non-semifield symplectic spreads. Relations between various constructions are studied as well. We showed that automorphism groups of complete sets of MUBs and correspon...

The aim of this paper is to explain the bialgebra structure of the positive part of the quantized universal enveloping algebra (Drinfeld-Jimbo quantum group) of type Anusing the Lie algebra theory concepts. Recently has been introduced a generalization of Lie algebras, the basic T -Lie algebras [1]. Using the T -Lie algebra concept some new (we think) quantum groups of type Ancan be constructed...

Vertex operators discovered by physicists in string theory have turned out to be important objects in mathematics. One can use vertex operators to construct various realizations of the irreducible highest weight representations for affine Kac-Moody algebras. Two of these, the principal and homogeneous realizations, are of particular interest. The principal vertex operator construction for the a...

Let Pn = K[x1, . . . , xn] be a polynomial algebra over a field K of characteristic zero and div0n (respectively, div c n) be the Lie algebra of derivations of Pn with zero (respectively, constant) divergence. We prove that AutLie(div 0 n) ≃ AutK−alg(Pn) (n ≥ 2) and AutLie(div c n) ≃ AutK−alg(Pn). The Lie algebra div c n is a maximal Lie subalgebra of DerK(Pn). Minimal finite sets of generators...

In this paper we study degenerations of nilpotent Lie algebras. If λ, μ are two points in the variety of nilpotent Lie algebras, then λ is said to degenerate to μ , λ→deg μ , if μ lies in the Zariski closure of the orbit of λ . It is known that all degenerations of nilpotent Lie algebras of dimension n < 7 can be realized via a one-parameter subgroup. We construct degenerations between characte...

In a paper by Xu, some simple Lie algebras of generalized Cartan type were constructed, using the mixtures of grading operators and down-grading operators. Among them, are the simple Lie algebras of generalized Witt type, which are in general nongraded and have no torus. In this paper, some representations of these simple Lie algebras of generalized Witt type are presented. 1. INTRODUCTION The ...