Introduction 1 1. Preliminaries 6 2. Generic stabilisers (centralisers) for the adjoint representation 9 3. Generic stabilisers for the coadjoint representation 10 4. Semi-direct products of Lie algebras and modules of covariants 12 5. Generic stabilisers and rational invariants for semi-direct products 14 6. Reductive semi-direct products and their polynomial invariants 21 7. Takiff Lie algebr...

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...

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...

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...

This paper begins by introducing the concept of a quasi-hom-Lie algebra, or simply, a qhl-algebra, which is a natural generalization of hom-Lie algebras introduced in a previous paper [14]. Quasi-hom-Lie algebras include also as special cases (color) Lie algebras and superalgebras, and can be seen as deformations of these by homomorphisms, twisting the Jacobi identity and skew-symmetry. The nat...

In paper by I.T. Habibullin and our joint the algorithm for classification of integrable equations with three independent variables was proposed. This method is based on requirement existence an infinite set Darboux reductions notion characteristic Lie-Rinehart algebras. The applied cases different subclasses $u_{n,xy} = f(u_{n+1},u_n,u_{n-1}, u_{n,x},u_{n,y})$ special forms. Under this approac...

Using fixed point method, we prove some new stability results for Lie $(alpha,beta,gamma)$-derivations and Lie $C^{ast}$-algebra homomorphisms on Lie $C^{ast}$-algebras associated with the Euler-Lagrange type additive functional equation begin{align*} sum^{n}_{j=1}f{bigg(-r_{j}x_{j}+sum_{1leq i leq n, ineq j}r_{i}x_{i}bigg)}+2sum^{n}_{i=1}r_{i}f(x_{i})=nf{bigg(sum^{n}_{i=1}r_{i}x_{i}bigg)} end{...