We review some applications of Gröbner-Shirshov bases, including PBW theorems, linear bases of free universal algebras, normal forms for groups and semigroups, extensions of groups and algebras, embedding of algebras.

Let α be a quadratic Poisson bivector on a vector space V . Then one can also consider α as a quadratic Poisson bivector on the vector space V ∗[1]. Fixed a universal deformation quantization (prediction of some complex weights to all Kontsevich graphs [K97]), we have deformation quantization of the both algebras S(V ∗) and Λ(V ). These are graded quadratic algebras, and therefore Koszul algebr...

In [10], inspired by an idea of Gröbner, Buchberger discovered an effective algorithm for solving the reduction problem for commutative algebras, which is now called the Gröbner Basis Theory. It was generalized to associative algebras through Bergman’s Diamond Lemma [2], and the parallel theory for Lie algebras was developed by Shirshov [21]. The key ingredient of Shirshov’s theory is the Compo...

We introduce a universal R matrix for the Jordanian deformation of U (sl(2)). Using Uh (so(4)) = Uh (sl(2))U?h(sl(2)), we obtain the universal R matrix for Uh (so(4)). Applying the graded contractions on the universal R matrix of Uh (so(4)), we show that there exist three distinct R matrices for all of the contracted algebras. It is shown that Uh (sl(2)), Uh (so(4)), and all of these contracted...

W -algebras of finite type are certain finitely generated associative algebras closely related to universal enveloping algebras of semisimple Lie algebras. In this paper we prove a conjecture of Premet that gives an almost complete classification of finite dimensional irreducible modules for W -algebras. Also we get some partial results towards a conjecture by Ginzburg on their finite dimension...

We present sufficient conditions under which effective descent morphisms in a quasivariety of universal algebras are the same as regular epimorphisms and examples for which they are the same as regular epimorphisms satisfying projectivity. 1. Preliminaries A variety is a full subcategory of the category of structures for a first order (one sorted) language, closed under substructures, products ...

We develop the commutator theory for fuzzy congruence relations of general universal algebras. In particular, algebras in modular varieties, we characterize congruences using Day’s terms.

We obtain deformations of a crossed product of a polynomial algebra with a group, under some conditions, from universal deformation formulas. These formulas arise from actions of Hopf algebras generated by automorphisms and skew derivations. They are universal in the sense that they apply to deform all algebras with such Hopf algebra actions, and we give one additional example.

The purpose of this thesis is to give a new construction for central extensions of certain classes of infinite dimensional Lie algebras which include multiloop Lie algebras as motivating examples. The key idea of this construction is to view multiloop Lie algebras as twisted forms. This perspective provides a beautiful bridge between infinite dimensional Lie theory and descent theory and is cru...