Periodic modules over general quantum Laurent polynomials

Abstract

ABSTRACT. We construct, by induction on the number of variables, periodic finitely generated modules over general quantum Laurent polynomials.KEY WORDS: quantum Laurent polynomial, finitely generated module.The present paper continues the investigation of the structure of finitely generated modules over general quantum Laurent polynomials initiated in [1,2]. In [1] it was shown that the finitely generated projective modules of rank at least two over the algebra of general quantum Laurent polynomials are free. In [2] it was shown that finitely generated modules over general quantum Laurent polynomials can be decomposed into the direct sum of the periodic part, which is a cyclic submodule, and a finitely generated projective submodule. 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- [7]. w Main constructionsLet D be a skew field over a central subfield k, and let al,..., a,~ be k-antomorphisms of D, where n _> 2. Assume that elements qij E D*, i,j = 1, ... ,n, such that qii = qijqji = Qij,.QjriQrij = 1, where Qijr = qijaj(qir) are given. Consider the associative k-algebra generated by the elements of the skew field D and by the elements X1,..., Xn, X~ "1 , ..., X~ -1 with the defining relations . . . , n, x~x i = q~jXjX~, i, j = 1,..., n. This algebra is the crossed product A = DhkG, where G is a free Abelian group with base X1, ..., Xn and t: kG | kG --, D* is a 2-cocycle such that t(Xi, Xj)t(Xj, X~) -1 = qij [1,8,3]. Moreover, there is a weak action of G on D such that