Considering prime ideals invariant under the action of an algebraic torus allows us to realize a finite partially ordered set as an invariant of certain filtered algebras. We investigate an important class of such algebras, which includes many examples of quantum algebras, such as quantum affine space and the algebra of quantum matrices. Many of these quantum algebras have been realized as quantum nilpotent Lie algebras, which are subalgebras of quantized enveloping algebras determined by a word in the associated Weyl group.
We investigate an algebra we refer to as the barberae algebra which arose in attempts to relate an orthogonal version of quantum matrices to the theory of skew polynomial algebras. We give a description of the structure of this algebra and its torus-invariant prime spectrum and use this to show the barberae algebra is not a quantum nilpotent Lie algebra for any semisimple Lie algebra, but rather for an affine Lie algebra.
To this end, we consider Weyl groups as particular examples of Coxeter groups and explore the combinatorial properties of the Bruhat ordering on Coxeter groups, and relate these to the structure of the torus-invariant prime spectra of corresponding algebras.
By lining up appropriate projection maps and chains of subsets, we develop an iterative algorithm for demonstrating an isomorphism between certain Bruhat order intervals and torus-invariant prime spectra. We use this to demonstrate the invariant prime spectrum of the barberae algebra is isomorphic to the Bruhat order interval of an affine Weyl group.
|Advisor:||Goodearl, Kenneth R.|
|Commitee:||Huisgen-Zimmermann, Birge, Yakimov, Milen|
|School:||University of California, Santa Barbara|
|School Location:||United States -- California|
|Source:||DAI-B 72/01, Dissertation Abstracts International|
|Keywords:||Geometric group theory, Lie algebras, Nilpotent algebras, Noncommutative, Quantum groups, Ring theory|
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