Twisted Calabi–Yau property of Ore extensions

Abstract

Suppose that E D AOExI ; ı is an Ore extension with an automorphism. It is proved that if A is twisted Calabi-Yau of dimension d , then E is twisted Calabi-Yau of dimension d C 1. The relation between their Nakayama automorphisms is also studied. As an application, the Nakayama automorphisms of a class of 5-dimensional Artin-Schelter regular algebras are given explicitly.

Automorphisms for Some Symmetric Multiparameter Quantized Weyl Algebras and Their Localizations

“…Thus, it is of interest to further determine when A q,Λ n (K) is indeed Calabi-Yau [15]. We will compute the Nakayama automorphism for A q,Λ n (K) using the methods as developed in [25,19] and establish a necessary and sufficient condition for A q,Λ n (K) to be a Calabi-Yau algebra. We will also prove that A q,Λ n (K) is universally cancellative when none of q i is a root of unity in the sense of [6].…”

“…Since A q,Λ n (K) can be presented as an iterated skew polynomial algebra, A q,Λ n (K) is a twisted (or skew) Calabi-Yau algebra by the result in [25]. Thus, it is of interest to further determine when A q,Λ n (K) is indeed Calabi-Yau [15].…”

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

“…Since A is a quadratic algebra and yx − xy − x 2 is a principal ideal, it follows that A is 2-Koszul (see [15], page 7), A σ(K[x]) y and therefore A is a skew PBW extension. The Jordan plane A is not CalabiYau (see [30]). …”

“…The Nakayama automorphism is unique up to an inner automorphism. A ν-skew Calabi-Yau algebra A is Calabi-Yau in the sense of Ginzburg if and only if ν is an inner automorphism of A (see [30], Definition 1.1). So every Calabi-Yau algebra is skew Calabi-Yau.…”

“…Multiparameter quantum affine n−spaces O q (K n ) can be obtained by iterated Ore extensions. 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]).…”