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Quotients of degenerate Sklyanin algebras

Banff International Research Station for Mathematical Innovation and Discovery

The 3-dimensional Sklyanin algebras degenerate to algebras isomorpic to either $\frac{\mathbb{C} < x,y,z > }{(x^2,y^2,z^2)}$ or $\frac{\mathbb{C} < x,y,z > }{ (xy,yz,zx)} $ due to a result of S. Paul Smith. The Heisenberg group of order 27, $H_3$ acts on these algebras as gradation preserving automorphisms. I will show that, using the representation theory of $H_3$, there exists a 1-dimensional family of quotients $A_t$, $t \in \mathbb{C}$ of these algebras such that $A_t \cong S(V)$ as graded $H_3$-module, where $S(V)$ is the polynomial ring in 3 variables with $V$ the Schrödinger representation of $H_3$. In addition, these quotients are noetherian and have a central element of degree 3, as in the regular case.

Mathematics; Associative rings and algebras; Algebraic geometry; Algebraic structures

video/mp4

Author affiliation: University of Antwerp

2017-03-14

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/60877

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/