Deformations of Quantum Symmetric Algebras Extended by Groups
Abstract
The study of deformations of an algebra has been a topic of interest for quite some time, since it allows us to not only produce new algebras but also better understand the original algebra. Given an algebra, finding all its deformations is, if at all possible, quite a challenging problem. For this reason, several specializations of this question have been proposed. For instance, some authors concentrate their efforts in the study of deformations of an algebra arising from an action of a Hopf algebra.
The purpose of this dissertation is to discuss a general construction of a deformation of a smash product algebra coming from an action of a particular Hopf algebra. This Hopf algebra is generated by skewprimitive and grouplike elements, and depends on a complex parameter. The smash product algebra is defined on the quantum symmetric algebra of a nitedimensional vector space and a group. In particular, an application of this result has enabled us to find a deformation of such a smash product algebra which is, to the best of our knowledge, the first known example of a deformation in which the new relations in the deformed algebra involve elements of the original vector space. Finally, using Hochschild cohomology, we show that these
deformations are nontrivial.
Subject
algebraic deformation theoryHopf algebras
Hopf module algebras
quantum symmetric algebras
smash product algebras
Hochschild cohomology
Citation
Shakalli Tang, Jeanette (2012). Deformations of Quantum Symmetric Algebras Extended by Groups. Doctoral dissertation, Texas A&M University. Available electronically from http : / /hdl .handle .net /1969 .1 /ETD TAMU 2012 05 10855.
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