This paper introduces basic concepts describing a hierarchical algebraic structure called multisorted tree algebra. This structure is constructed by placing multisorted algebra at the bottom of a hierarchy and placing at other intermediate nodes the aggregation of algebras placed at their immediate subordinate nodes. These constructions are different from the one of subalgebras, homomorphic ima...

A perturbative quantization procedure for Lie bialgebras is introduced and used to classify all three dimensional complex quantum algebras compatible with a given coproduct. The role of elements of the quantum universal enveloping algebra that, analogously to generators in Lie algebras, have a distinguished type of coproduct is discussed, and the relevance of a symmetrical basis in the universa...

In this paper we concentrate on open problems in two directions in the development of the theory of constructive algebraic systems. The first direction deals with universal algebras whose positive open diagrams can be computably enumerated. These algebras are called positive algebras. Here we emphasize the interplay between universal algebra and computability theory. We propose a systematic stu...

A general method is presented for establishing universal factorization equalities for 2×2 and 4×4 block matrices. As applications, some universal factorization equalities for matrices over four-dimensional algebras are established, in particular, over the Hamiltonian quaternion algebra.

We suggest a formula for quantum universal R-matrices corresponding to quasitriangular classical r-matrices classified by Belavin and Drinfeld for all simple Lie algebras. The R-matrices are obtained by twisting the standard universal R-matrix.