New realizations of deformed double current algebras and Deligne categories

Creators: Etingof, P.; Kalinov, D.; Rains, E.

Abstract

In this paper, we propose an alternative construction of a certain class of Deformed Double Current Algebras. We construct them as spherical subalgebras of symplectic reection algebras in the Deligne category. They can also be thought of as ultraproducts of the corresponding spherical subalgebras in finite rank. We also provide new presentations of DDCA of types A and B by generators and relations.

Additional Information

© The Author(s). This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Received May 29, 2020. Accepted June 23, 2021. To the memory of Ernest Borisovich Vinberg. This paper owes its existence to Victor Ginzburg, who proposed to study deformed double current algebras in the spring of 2001 and suggested, around the same time, some of the important ideas explored below. We are very grateful to Victor for sharing these ideas and initiating this research. We are also grateful to N. Guay, V. Ostrik, and T. Schedler for useful discussions. The work of P. E. and D. K. was partially supported by the NSF grant DMS-1502244. The computer calculations for this paper were done using MAGMA ([BCP97]). Open Access funding provided by the MIT Libraries.

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