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Down–Up Algebras and Ambiskew Polynomial Rings

✍ Scribed by David A Jordan

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
235 KB
Volume
228
Category
Article
ISSN
0021-8693

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✦ Synopsis

lie in a certain class of iterated skew polynomial rings, called ambiskew polynomial rings, in two indeterminates x and y over a commutative ring B. In such rings, commutation of the indeterminates with elements of B involve the same endomorphism σ of B, but from different sides, that is, yb = σ b y and bx = xσ b , and, for some scalar p, yx -pxy ∈ B. In previous studies of ambiskew polynomial rings, σ was required to be an automorphism but, in order to cover all down-up algebras, this requirement must be dropped. The Noetherian down-up algebras are those where σ is an automorphism and, in this case, we apply existing results on ambiskew polynomial rings to determine the finite-dimensional simple modules and the prime ideals. We adapt the methods underlying these results so as to apply to the non-Noetherian down-up algebras for which they reveal a surprisingly rich structure.

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