J. Korean Math. Soc. 2023; 60(4): 745-777
Online first article June 14, 2023 Printed July 1, 2023
https://doi.org/10.4134/JKMS.j220200
Copyright © The Korean Mathematical Society.
Alexandru Chirvasitu, S. Paul Smith
University at Buffalo; University of Washington
The 4-dimensional Sklyanin algebras are a well-studied 2-parameter family of non-commutative graded algebras, often denoted $A(E,\tau)$, that depend on a quartic elliptic curve $E \subseteq \mathbb{P}^3$ and a translation automorphism $\tau$ of $E$. They are graded algebras generated by four degree-one elements subject to six quadratic relations and in many important ways they behave like the polynomial ring on four indeterminates except that they are not commutative. They can be seen as ``elliptic analogues'' of the enveloping algebra of $\mathfrak{gl}(2,\mathbb{C})$ and the quantized enveloping algebras $U_q(\mathfrak{gl}_2)$. Recently, Cho, Hong, and Lau conjectured that a certain 2-parameter family of algebras arising in their work on homological mirror symmetry consists of 4-dimensional Sklyanin algebras. This paper shows their conjecture is false in the generality they make it. On the positive side, we show their algebras exhibit features that are similar to, and differ from, analogous features of the 4-dimensional Sklyanin algebras in interesting ways. We show that most of the Cho-Hong-Lau algebras determine, and are determined by, the graph of a bijection between two 20-point subsets of the projective space $\mathbb{P}^3$. The paper also examines a 3-parameter family of 4-generator 6-relator algebras admitting presentations analogous to those of the 4-dimensional Sklyanin algebras. This class includes the 4-dimensional Sklyanin algebras and most of the Cho-Hong-Lau algebras.
Keywords: Sklyanin algebras, graded algebras, 4 generators and 6 relations, elliptic curve
MSC numbers: Primary 14A22, 16E65, 16S38, 16W50; Secondary 17B37, 14H52
Supported by: The work of the first author was partially supported by NSF grants DMS-1565226, DMS-1801011 and DMS-2001128.
2003; 40(6): 977-998
2008; 45(3): 695-698
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