Related papers: The factorization problem for Jordan algebras. App…
Let $K$ be a field (finite or infinite) of char$(K)\neq 2$ and let $UT_n=UT_n(K)$ be the $n\times n$ upper triangular matrix algebra over $K$. If $\cdot $ is the usual product on $UT_n$ then with the new product $a\circ b=(1/2)(a\cdot b…
We compute the quantum double, braiding and other canonical Hopf algebra constructions for the bicrossproduct Hopf algebra $H$ associated to the factorization of a finite group into two subgroups. The representations of the quantum double…
A conjecture for the dimension and the character of the homogenous components of the free Jordan algebras is proposed. As a support of the conjecture, some numerical evidences are generated by a computer and some new theoretical results are…
For two families of Yetter-Drinfel'd Hopf algebras considered earlier by the authors, we determine which of them are isomorphic. We also determine which of their Radford biproducts are isomorphic.
We classify all linearly compact simple Jordan superalgebras over an algebraically closed field of characteristic zero. As a corollary, we deduce the classification of all linearly compact unital simple generalized Poisson superalgebras.
We consider the factorisation problem for bialgebras: when a bialgebra $K$ factorises as $K=HL$, where $H$ and $L$ are algebras and coalgebras (but not necessarly bialgebras). Given two maps $R: H\ot L\to L\ot H$ and $W:L\ot H\to H\ot L$,…
We introduce the notion of a matched pair of fusion rings and fusion categories, generalizing the one for groups. Using this concept, we define the bicrossed product of fusion rings and fusion categories and we construct exact…
We apply the quaternionic Jordan form to classify the hypercomplex nilpotent almost abelian Lie algebras in all dimensions and to carry out the complete classification of 12-dimensional hypercomplex almost abelian Lie algebras. Moreover, we…
We prove that any twisted generalized Weyl algebra satisfying certain consistency conditions can be embedded into a crossed product. We also introduce a new family of twisted generalized Weyl algebras, called multiparameter twisted Weyl…
We define Jordan quadruple systems by the polynomial identities of degrees 4 and 7 satisfied by the Jordan tetrad {a,b,c,d} = abcd + dcba as a quadrilinear operation on associative algebras. We find further identities in degree 10 which are…
Several classes of baric algebras studied by different authors will be given a unified treatment, using the technique of gametization introduced by Mallol et al. Many of these algebras will be shown to be either Jordan algebras or to be…
We prove that every Jordan left $\alpha$-centralizer from an algebra $A$ with a right identity into an arbitrary algebra $B$ is a left $\alpha$-centralizer. This implies all Jordan homomorphisms between such algebras are homomorphisms. We…
Any rational number can be factored into a product of several rationals whose sum vanishes. This simple but nontrivial fact was suggested as a problem on a maths olympiad for high-school students. We completely solve similar questions in…
The notion of f-derivations was introduced by Beidar and Fong to unify several kinds of linear maps including derivations, Lie derivations and Jordan derivations. In this paper we introduce the notion of f-biderivations as a natural…
We introduce a new approach to an enumerative problem closely linked with the geometry of branched coverings; that is, we study the number of ways a permutation can be decomposed into a product of a given number of 2-cycles, 3-cycles, etc.…
In this paper, we characterize Jordan derivable mappings in terms of Peirce decomposition and determine Jordan all-derivable points for some general bimodules. Then we generalize the results to the case of Jordan higher derivable mappings.…
In this paper, we first explore the extending structures problem by the unified product for anti-dendriform algebras. In particular,the crossed product and non-abelian extension are studied. Furthermore, we explore the inducibility problem…
An algebra with identities $a(bc)=b(ac),$ $(ab)c=(ac)b$ is called bicommutative. We construct list of identities satisfied by commutator and anti-commutator products in a free bicommutative algebra. We give criterions for elements of a free…
Jordan Normal Forms serve as excellent representatives of conjugacy classes of matrices over closed fields. Once we knows normal forms, we can compute functions of matrices, their main invariant, etc. The situation is much more complicated…
We take an algorithmic and computational approach to a basic problem in abstract algebra: determining the correct generalization to dialgebras of a given variety of nonassociative algebras. We give a simplified statement of the KP algorithm…