arXiv subjects

Pavel Sultanich

Publications and source records attributed to Pavel Sultanich.

3 recordsLinked to original sources

arXiv2020

On explicit realization of algebra of complex powers of generators of $U_{q}(\mathfrak{sl}(3))$

In this note we prove an integral identity involving complex powers of generators of quantum group $U_{q}(\mathfrak{sl}(3))$ considered as certain positive operators in the setting of positive principal series representations. This identity represents a continuous analog of one of the Lusztig's relations between divided powers of generators of quantum groups, which play an important role in the study of irreducible modules \cite{Lu 1}. We also give definitions of arbitrary functions of $U_{q}(\mathfrak{sl}(3))$ generators and give another proofs for some of the known results concerning positive principal series representations of $U_{q}(\mathfrak{sl}(3))$.

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arXiv2019

On explicit realization of algebra of complex divided powers of $U_{q}(\mathfrak{sl}(2))$

In this note we prove that the explicit realization of arbitrary complex powers of generators of quantum group $U_{q}(\mathfrak{sl}(2))$ satisfies all the commutation relations of the algebra of complex powers, including the generalized Kac's identity which was announced in our previous paper. It turns out that the latter identity in this realization is equivalent to $6-9$ integral identity on quantum dilogarithm.

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arXiv2018

On modular double of semisimple quantum groups

In this note we propose a construction of the Hopf algebra of a complex analog of devided powers of the Weyl generators of a semisimple simply-laced quantum group. Here we consider the generators as positive, self-adjoint operators. In particular, we generalize the Lusztig relations on the usual divided powers of generators of a quantum group to the case of complex devided powers of generators. These relations, some of which were known present the complete set of defining relations. As a by-product result, the pure algebraic definition of the Faddeev modular double in the case of semisimple simply-laced quantum groups is formulated. Finally, we introduce an infinite dimension version of the Gelfand-Zetlin finite-dimensional representation of the modular double $M_{q,\tilde{q}}(\mathfrak{gl}(N))$.

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