Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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SOME ALGEBRAS HAVING RELATIONS LIKE THOSE FOR THE 4-DIMENSIONAL SKLYANIN ALGEBRAS

  • Alexandru Chirvasitu (Department of Mathematics University at Buffalo) ;
  • S. Paul Smith (Department of Mathematics University of Washington)
  • Received : 2022.04.24
  • Accepted : 2023.04.26
  • Published : 2023.07.01
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Abstract

The 4-dimensional Sklyanin algebras are a well-studied 2-parameter family of non-commutative graded algebras, often denoted A(E, τ), that depend on a quartic elliptic curve E ⊆ ℙ3 and a translation automorphism τ 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 𝖌𝖑(2, ℂ) and the quantized enveloping algebras Uq(𝖌𝖑2). 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 ℙ3. 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

Acknowledgement

The work of the first author was partially supported by NSF grants DMS-1565226, DMS-1801011 and DMS-2001128.

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