Rota-Baxter algebras appeared in both the physics and mathematics literature. It is of great interest to have a simple construction of the free object of this algebraic structure. For example, free commutative Rota-Baxter algebras relate to double shuffle relations for multiple zeta values. The interest in the non-commutative setting arised in connection with the work of Connes and Kreimer on t...

These notes form Lecture Notes of a short course which I will give at 1st School on Universal Logic in Montreux. They cannot be recommended for self studies because, although all definitions and main ideas are included, there are no proofs and examples. I’m going to provide some of them during my lectures, leaving easy ones as exercises. In the first part we discuss some of the most important n...

We show how the Connes–Moscovici bialgebroid construction naturally provides universal objects for Lie algebras acting on non-commutative algebras. As a supplement, we see these interact with study of flat bimodule connections.

In this text we use the quantum double construction to show that a suitable category of finite dimensional left U -modules is braided, where U is the quantized universal enveloping algebra of sl2(k). We produce the expression of the corresponding braiding as the action of a formal universal R-matrix for U . We indicate the generalization to sln(k) and the relation to Hecke algebras and Temperle...

We prove that any two homotopic projections in certain C*-algebras can be connected by a rectifiable path of projections whose length is bounded by a universal constant. In comparison, N.C. Phillips (1992) proved that there are C*algebras in which such a universal constant does not exist. Our techniques are to estimate the number of symmetries needed to conjugate any two homotopic projections a...

We define some new algebraic structures, termed coloured Hopf algebras, by combining the coalgebra structures and antipodes of a standard Hopf algebra set H, corresponding to some parameter set Q, with the transformations of an algebra isomorphism group G, herein called colour group. Such transformations are labelled by some colour parameters, taking values in a colour set C. We show that vario...

We construct a universal envelope for any Poissonand Gerstenhaber algebra. While the deformation theory of Poisson algebras seems to be partially trivial, results from string and M -theory suggest a rich deformation theory of Gerstenhaber algebras. We apply our construction in this case to well known questions on the topological closed string BRST-complex. Finally, we find a simliar algebraic s...

We present several embedding results for 3-graded Lie algebras and KKT algebras that are generated by two homogeneous elements of degrees 1 and −1. We also propose the canonical kernel function for a “universal Bergman kernel” which extends the usual Bergman kernel on a bounded symmetric domain to a group-valued function or, in terms of formal series, to an element in the formal completion of t...

Representation theory is developed for the class of Galois algebras introduced recently by the authors. In particular, categories of Harish-Chandra modules are studied for integral Galois algebras which include generalized Weyl algebras, the universal enveloping algebra of gl n , the quantization and Yangians for gl 2 .

Here is discussed application of the Weyl pair to construction of universal set of quantum gates for high-dimensional quantum system. An application of Lie algebras (Hamiltonians) for construction of universal gates is revisited first. It is shown next, how for quantum computation with qubits can be used two-dimensional analog of this Cayley-Weyl matrix algebras, i.e. Clifford algebras, and dis...