On the Hochschild homology of elliptic Sklyanin algebras

In this paper, we compute the Hochschild homology of elliptic Sklyanin algebras. These algebras are deformations of polynomial algebra with a Poisson bracket called the Sklyanin Poisson bracket.

💡 Research Summary

This paper provides a complete computation of the Hochschild homology of the three‑dimensional elliptic Sklyanin algebras, denoted S(E,τ), where E is a complex elliptic curve and τ∈E is a deformation parameter. The authors begin by recalling the standard presentation of S: it is generated by three degree‑one elements x, y, z subject to three quadratic relations whose coefficients (a,b,c) are functions of the points on E. These algebras are known to be Koszul, Artin‑Schelter regular of global dimension three, and they admit a natural Poisson limit.

The central observation is that S can be viewed as a deformation quantization of the commutative polynomial algebra k