We define the notion of a Kirby element of a ribbon category C (not necessarily semisimple). Kirby elements lead to 3-manifolds invariants. We characterize a set of Kirby elements (in terms of the structural morphisms of a Hopf algebra in C) which is sufficiently large to recover the quantum invariants of 3-manifolds of Reshetikhin-Turaev, of Hennings-Kauffman-Radford, and of Lyubashenko. The c...

In this paper we show that the center of partition algebra $\mathcal{A}_{2k}(\delta)$, in semisimple case, is given by subalgebra supersymmetric polynomials normalised Jucys-Murphy elements. For non-semisimple such a shown to be central, and particular it large enough recognise block structure $\mathcal{A}_{2k}(\delta)$. This allows one give an alternative description for when two simple $\math...

The obstructions for an arbitrary fusion algebra to be a fusion algebra of some semisimple monoidal category are constructed. Those obstructions lie in groups which are closely related to the Hochschild cohomology of fusion algebras with coefficients in the K-theory of the ground (algebraically closed) field. The special attention is devoted to the case of fusion algebra of invariants of finite...

We show that if T is an isometry (as metric spaces) from an open subgroup of the group of the invertible elements in a unital semisimple commutative Banach algebra onto an open subgroup of the group of the invertible elements in a unital Banach algebra, then T (1)T is an isometrical group isomorphism. In particular, T (1)T is extended to an isometrical real algebra isomorphism from A onto B.

For each finite semisimple tensor category, we associate a quantum group (face algebra) whose comodule category is equivalent to the original one, in a simple natural manner. To do this, we also give a generalization of the Tannaka-Krein duality, which assigns a face algebra for each tensor category equipped with an embedding into a certain kind of bimodule category.

We investigate an analogue of the Wedderburn principal theorem for associative conformal algebras with finite faithful representations. It is shown that the radical splitting property for an algebra of this kind holds if the maximal semisimple factor of this algebra is unital, but does not hold in general.

Computing the structure of a finite-dimensional algebra is a classical mathematical problem in symbolic computation with many applications such as polynomial factorization, computational group theory and differential factorization. We will investigate the computational complexity and exhibit new algorithms for this problem over the field Fq(y), where Fq is the finite field with q elements. A fi...

A relation between $\frac{1}{2}$-derivations of Lie algebras and transposed Poisson was established. Some non-trivial with a certain algebra (Witt algebra, $\mathcal{W}(a,-1)$, thin solvable abelian nilpotent radical) were constructed. In particular, we constructed an example the associative parts isomorphic to Laurent polynomials Witt algebra. On other side, it proven that there are no part se...

The aim of this work is to study a class of non-archimedean valued measures on MV-algebras. We call them hyperreal states and their definition naturally arise from (the uniform version of) Di Nola representation theorem for MV-algebras (cf [5, 6]): for any MV-algebra A = (A,⊕,¬,>,⊥) there exists a ultrafilter U on the cardinality of A such that A embeds into (∗[0, 1]U) Spec(A) (where as usual S...

The rook partition algebra RPk(x) is a semisimple algebra for all but a finite number of values of x ∈ C that arises from looking at what commutes with the action of the symmetric group Sn on U ⊗k, where U is the direct sum of the natural representation and the trivial representation of Sn. We give a combinatorial description of this algebra, construct its irreducible representations, and exhib...