arXiv subjects

Alexey Silantyev

Publications and source records attributed to Alexey Silantyev.

4 recordsLinked to original sources

arXiv2022

Quantum Representation Theory and Manin matrices II: super case

We construct super-version of Quantum Representation Theory. The quadratic super-algebras and operations on them are described. We also describe some important monoidal functors. We proved that the monoidal category of graded super-algebras with Manin product is coclosed relative to the subcategory of finitely generated quadratic super-algebras. The super-version of the $(A,B)$-Manin matrices is introduced and related with the quadratic super-algebras. We define a super-version of quantum representations and of quantum linear actions, relate them to each other and describe them by the super-Manin matrices. Some operations on quantum representations/quantum linear actions are described. We show how the classical representations lift to the quantum level.

math.QA↗

arXiv2021

Manin Matrices for Quadratic Algebras

We give a general definition of Manin matrices for arbitrary quadratic algebras in terms of idempotents. We establish their main properties and give their interpretation in terms of the category theory. The notion of minors is generalised for a general Manin matrix. We give some examples of Manin matrices, their relations with Lax operators and obtain the formulae for some minors. In particular, we consider Manin matrices of the types $B$, $C$ and $D$ introduced by A. Molev and their relation with Brauer algebras. Infinite-dimensional Manin matrices and their connection with Lax operators are also considered.

math.QA↗

arXiv2015

KP hierarchy for the cyclic quiver

We introduce a generalisation of the KP hierarchy, closely related to the cyclic quiver and the Cherednik algebra $H_k(\mathbb Z_m)$. This hierarchy depends on $m$ parameters (one of which can be eliminated), with the usual KP hierarchy corresponding to the $m=1$ case. Generalising the result of G. Wilson, we show that our hierarchy admits solutions parameterised by suitable quiver varieties. The pole dynamics for these solutions is shown to be governed by the classical Calogero-Moser system for the wreath-product $\mathbb Z_m\wr S_n$ and its new spin version. These results are further extended to the case of the multi-component hierarchy.

math.QA↗

arXiv2009

Manin Matrices, Quantum Elliptic Commutative Families and Characteristic Polynomial of Elliptic Gaudin Model

In this paper we construct the quantum spectral curve for the quantum dynamical elliptic gl(n) Gaudin model. We realize it considering a commutative family corresponding to the Felder's elliptic quantum group and taking the appropriate limit. The approach of Manin matrices here suits well to the problem of constructing the generation function of commuting elements which plays an important role in SoV and Langlands concept.

math-ph↗