English

An Elliptic Generalization of $A_1$ Spherical DAHA at $K=2$

Quantum Algebra 2023-06-06 v1 Mathematical Physics math.MP Rings and Algebras Representation Theory

Abstract

We construct an algebra that is an elliptic generalization of A1A_1 spherical DAHA acting on its finite-dimensional module at t=qK/2t=-q^{-K/2} with K=2K=2. We prove that PSL(2,Z)PSL(2,\mathbb Z) acts by automorphisms of the algebra we constructed, and provide an explicit representation of automorphisms and algebra operators alike by 3×33\times 3 matrices of elliptic functions. A relation of this construction to the K-theory character of affine Laumon space is conjectured. We point out two potential applications, respectively to SL(3,Z)SL(3,\mathbb Z) symmetry of Felder-Varchenko functions and to new elliptic invariants of torus knots and Seifert manifolds.

Keywords

Cite

@article{arxiv.2306.00215,
  title  = {An Elliptic Generalization of $A_1$ Spherical DAHA at $K=2$},
  author = {S. Arthamonov and Sh. Shakirov},
  journal= {arXiv preprint arXiv:2306.00215},
  year   = {2023}
}

Comments

32 pages