Some Homological Properties of SkewPBWExtensions

Abstract

We prove that if R is a left Noetherian and left regular ring then the same is true for any bijective skew P BW extension A of R. From this we get Serre's Theorem for such extensions. We show that skew P BW extensions and its localizations include a wide variety of rings and algebras of interest for modern mathematical physics such as P BW extensions, well known classes of Ore algebras, operator algebras, diffusion algebras, quantum algebras, quadratic algebras in 3-variables, skew quantum polynomials, among many others. We estimate the global, Krull and Goldie dimensions, and also Quillen's K-groups.

Some Remarks About the Cyclic Homology of Skew PBW Extensions / Algunas observaciones sobre la homología cíclica de extensiones PBW torcidas

“…, z n with the relations z i r = c i,r z i , z j z i = c i, j z i z j , for 1 ≤ i ≤ n, where c i,r , c i, j are the same constants that define A. See [13], Proposition 2.1 for a proof of this assertion.…”

“…This relationship shows that cyclic homology can be considered as a Lie analogue of algebraic K-theory and it is sometimes referred to as non-commutative differential geometry. With this in mind, in the first part of this section we recall the higher algebraic K-theory of skew PBW extensions following [13], while the second part treats with the cyclic homology of skew PBW extensions.…”

“…In Section 2 we describe the skew PBW extensions. In Section 3 we recall the higher algebraic K-theory of these extensions following [13], and then we present the key results of this paper concerning about cyclic homology of these rings. Finally, in Section 4 we illustrate the results about cyclic homology of Section 3 with several examples such as Ore extensions, diffusion algebras, operator algebras and 3-dimensional skew polynomial algebras.…”

“…The results presented are new for skew PBW extensions and all they are similar to others existing in the literature. In this way, we continue the task of studying several properties of skew PBW extensions and its relationship with other noncommutative rings (see [1], [8], [12], [13], [17], [18], [19], [20], [22], [24] and others).…”

Algunas relaciones entre álgebras N-Koszul, Artin-Schelter regular y Calabi-Yau con extensiones PBW torcidas

“…Let n ≥ 1 and q be a matrix (q i j ) n×n whit entries in a field K where q ii = 1 y q i j q ji = 1 for all 1 ≤ i, j ≤ n. Then quantum affine n-space O q (K n ) is defined to be K−algebra generated by x 1 , · · · , x n with the relations x j x i = q i j x i x j for all 1 ≤ i, j ≤ n. The K−algebra O q (K n ) is skew Calabi-Yau whit the Nakayama automorphism ν such that ν(x i ) = ( n j=1 q ji )x i (see [30], Proposition 4.1). This K−algebra is a skew PBW extension (see [29]). …”