Derivations of Skew Polynomial Rings

The first Hochschild cohomology group of quantum matrices and the quantum special linear group

“…Hence Q Â is a central derivation of P .ƒ/, in the terminology of [12]. Thus we deduce from [12] As a corollary of Theorem 2.9, we obtain some new information on the twisted homology of quantum matrices.…”

“…Thus we deduce from [12] As a corollary of Theorem 2.9, we obtain some new information on the twisted homology of quantum matrices. We refer to [7] and references therein for definition and properties of the twisted homology.…”

“…However, the idea is relatively easy to follow. The starting point is a result of Osborn and Passman 282 S. Launois and T. H. Lenagan [12] that describes the derivations of a quantum torus. In particular, they show that the first Hochschild cohomology group of the quantum torus with n 2 generators is a free module of rank n 2 over the centre of the quantum torus.…”

Derivations and invariant forms of Jordan and alternative tori

“…We learned the concept of central derivations from the paper [24] by Osborn and Passman where they have been introduced for twisted group algebras and where (1) is proven for these algebras. That the derivation algebra is a semidirect product of the ideal of inner derivations and a subalgebra had been proven before in [4] for quantum tori and in [5] for Cayley tori, see section 4.3.…”

Periodic modules over general quantum Laurent polynomials

“…In the present paper it is proved that if M is a finitely generated periodic module over the algebra of general quantum Laurent polynomials, then we can choose variables X1,... , Xn such that M is a finitely generated projective module over the subalgebra generated by X1, ..., X~-I (see Theorem 2.4). It is shown that the converse assertion also holds.Among studies dealing with the same subject, we can mention the book [3] and the papers [4][5][6], and also other recent papers of these authors.Note that quantum polynomials play an important role in the development of noncommutative geom- …”