arXiv subjects
Publications and source records attributed to Alexander Mazurenko.
We construct a Hopf superalgebra structure for a Drinfeld super Yangian of Lie superalgebra $B(m,n)$ relative to all possible choices of Borel subalgebras. In order to do this we introduce a simplified definition of Drinfeld super Yangians.
We construct a minimalistic presentation of Drinfeld super Yangians in the case of special linear superalgebra associated with an arbitrary Dynkin diagram. This gives us a possibility to introduce Hopf superalgebra structure on Drinfeld super Yangians. Using complete Weyl group we classify Drinfeld super Yangians endowed with mentioned Hopf superalgebra structures. Also it is constructed an isomorphism between completions of Drinfeld super Yangians and quantum loop superalgebras.
We summarize the definition of the Weyl groupoid using supercategory approach in order to investigate quantum superalgebras at roots of unity. We show how the structure of a Hopf superalgebra on a quantum superalgebra is determined by the quantum Weyl groupoid. The Weyl groupoid of $\mathfrak{sl}(m|n)$ is constructed to this end as some supercategory. We prove that in this case quantum superalgebras associated with Dynkin diagrams are isomorphic as superalgebras. It is shown how these quantum superalgebras considered as Hopf superalgebras are connected via twists and isomorphisms. We explicitly construct these twists using the Lusztig isomorphisms considered as elements of the Weyl quantum groupoid. We build a PBW basis for each quantum superalgebra, and investigate how quantum superalgebras are connected with their classical limits, i. e. Lie superbialgebras. We find explicit multiplicative formulas for universal $R$-matrices, describe relations between them for each realization and classify Hopf superalgebras and triangular structures for the quantum superalgebra $U_q(\mathfrak{sl}(m|n))$.
We summarize the definition of the Weyl groupoid using supercategory approach in order to investigate quantum superalgebras at roots of unity. We show how the structure of a Hopf superalgebra on a quantum superalgebra is determined by the quantum Weyl groupoid. The Weyl groupoid of $\mathfrak{sl}(m|n)$ is constructed to this end as some supercategory. We prove that in this case quantum superalgebras associated with Dynkin diagrams are isomorphic as superalgebras. It is shown how these quantum superalgebras considered as Hopf superalgebras are connected via twists and isomorphisms. We explicitly construct these twists using the Lusztig isomorphisms considered as elements of the Weyl quantum groupoid. We build a PBW basis for each quantum superalgebra, and investigate how quantum superalgebras are connected with their classical limits, i. e. Lie superbialgebras. We find explicit multiplicative formulas for universal $R$-matrices, describe relations between them for each realization and classify Hopf superalgebras and triangular structures for the quantum superalgebra $U_q(\mathfrak{sl}(m|n))$.
We consider fundamental facts from the theory of Hopf superalgebras. We use them to construct the quantum double of the quantum superalgebra $sl(2|1)$ at roots of unity. Thus we obtain a multiplicative formula for universal $R$-matrix. Next we construct an $R$-matrix to investigate parametrized family of centralizer algebras. We give multiplication laws in particular case and describe a structure of such algebras in the general case.