Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 066, 81 pages      arXiv:2009.05993      https://doi.org/10.3842/SIGMA.2021.066

Manin Matrices for Quadratic Algebras

Alexey Silantyev ab
a) Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia
b) State University ''Dubna'', 141980 Dubna, Moscow region, Russia

Received December 11, 2020, in final form June 26, 2021; Published online July 12, 2021

Abstract
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.

Key words: Manin matrices; quantum groups; quadratic algebras.

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