On the Consistency of Twisted Generalized Weyl Algebras

Abstract

A twisted generalized Weyl algebra A of degree n depends on a base algebra R, n commuting automorphisms s_i of R, n central elements t_i of R and on some additional scalar parameters. In a paper by V.Mazorchuk and L.Turowska (1999) it is claimed that certain consistency conditions for s_i and t_i are sufficient for the algebra to be nontrivial. However, in this paper we give an example which shows that this is false. We also correct the statement by finding a new set of consistency conditions and prove that the old and new conditions together are necessary and sufficient for the base algebra R to map injectively into A. In particular they are sufficient for the algebra A to be nontrivial. We speculate that these consistency relations may play a role in other areas of mathematics, analogous to the role played by the Yang-Baxter equation in the theory of integrable systems.

References in corpus (1)

• Locally finite simple weight modules over twisted generalized Weyl algebras

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• Multiparameter Twisted Weyl Algebras
• Simplicity and maximal commutative subalgebras of twisted generalized Weyl algebras
• Noncommutative fiber products and lattice models
• Twisted generalized Weyl algebras and primitive quotients of enveloping algebras
• Cylindrical Dyck paths and the Mazorchuk-Turowska equation
• Classification of twisted generalized Weyl algebras over polynomial rings
• Fixed rings of twisted generalized Weyl algebras
• Clifford and Weyl superalgebras and spinor representations
• On rational twisted generalized Weyl algebra
• Simple weight modules over weak Generalized Weyl algebras