General quantum polynomials: irreducible modules and Morita equivalence. (English. Russian original) Zbl 0974.16019

Artamonov, V. A.

Izv. Math. 63, No. 5, 847-880 (1999); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 63, No. 4, 3-36 (1999).

Let \(A\) be the ring of quantum-Laurent polynomials in \(n\) variables and \(\Lambda\) the subring in which only the first \(r\) variables have inverses. The author’s aim is to describe the lattice of submodules of cyclic \(\Lambda\)-modules. If \(B\) denotes the subring of \(A\) generated by the first \(n-1\) variables, then \(A\) has the form of a skew-Laurent polynomial ring \(B[X_n,X_n^{-1};\alpha_n]\). He proves that any \(A\)-module \(M\) which is finitely generated as \(B\)-module is a projective \(B\)-module. Any finitely generated torsion-free \(A\)-module \(M\) can by change of variables be regarded as a projective \(B\)-module. Let \(H\) be the field of fractions of \(B\) and \(R=H[X_n ,X_n^{-1};\alpha_n]\). Then the embedding \(M\to R\otimes_AM\) (\(M\) as before) induces a natural isomorphism between the submodule lattices \(L_A(M)\leftrightarrow L_R(R\otimes_AM)\). It follows that \(_AM\) is irreducible if and only if \(_R(R\otimes M)\) is irreducible. After some examples the author examines irreducible modules over polynomials and the extensions to the skew fields of fractions. By studying the endomorphism rings of projective ideals, he is able to prove the following theorem. Let \(\Lambda\), \(\Lambda'\) be rings of general quantum polynomials (each in at least 2 variables of which at least one is invertible); if \(\Lambda\), \(\Lambda'\) are Morita equivalent, then they are isomorphic; moreover \(\text{Pic}(\Lambda)\) is trivial. A final result concerns Zariski’s conjecture. This states that if \(R\) is a subring of \(S=k[X_1,\dots, X_n]\) (\(k\) is a field) such that \(S=R[y]\) for some \(y\in S\), then \(R=k[X_1,\dots,X_{n-1}]\). This is still open, but the author proves that for the ring of quantum polynomials it has a positive answer.

Reviewer: Paul M.Cohn (London)

MSC:

16S34	Group rings
16D40	Free, projective, and flat modules and ideals in associative algebras
16D70	Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16W35	Ring-theoretic aspects of quantum groups (MSC2000)
16D90	Module categories in associative algebras

Keywords:

quantum-Laurent polynomials; skew-Laurent polynomial rings; projective modules; finitely generated torsion-free modules; irreducible modules; skew fields of fractions; endomorphism rings; projective ideals; Morita equivalences; Zariski’s conjecture; lattices of submodules