Homological aspects of Noetherian PI Hopf algebras and irreducible modules of maximal dimension
This substantial paper shows that a Noetherian Hopf algebra of finite global dimension and satisfying a polynomial identity possesses attractive homological properties. Let \(H\) be an FBN Hopf algebra over a field \(k\) such that all irreducible \(H\)-modules are finite dimensional over \(k\). If \(H\) has finite global dimension \(n\) then \(H\) is an Auslander-regular Macaulay ring of Krull dimension \(n\). Moreover, \(H\) is a semiprime ring and, in case \(H\) is PI, a finite direct sum of prime rings. In particular, if \(\mathfrak g\) is a Lie algebra of finite dimension \(n\) over a field of positive characteristic then the enveloping algebra \(U({\mathfrak g})\) is Auslander regular and Macaulay of Krull and global dimension \(n\). On the other hand, if \(H\) has finite injective dimension \(n\) then \(H\) is Artin-Schelter Gorenstein and, in case \(H\) is PI, an Auslander-Gorenstein Macaulay ring of Krull dimension \(n\). These results are applied to quantized enveloping algebras \(U_q({\mathfrak g})\) and to quantized function algebras \({\mathcal O}_q[G]\), both in the case of a generic parameter \(q\) and in the case where \(q\) specializes to a primitive \(l\)th root of unity \(\varepsilon\). Here \(\mathfrak g\) is a finite dimensional complex semisimple Lie algebra and \(G\) is the connected simply connected semisimple Lie group with Lie algebra \(\mathfrak g\). As usual, \(l\) is odd and prime to 3 if \(\mathfrak g\) involves a factor of type \(G_2\). The second major theme in the paper is motivated by the following question: let \(\Lambda\) be a prime Noetherian ring finitely generated as a module over its centre \(Z\) where \(Z\) is an affine algebra over an algebraically closed field \(k\); for which \(m\in\max Z\) is \(\Lambda_m\) an Azumaya algebra over \(Z_m\)? The set of such \(m\) in \(Z\) is denoted by \({\mathcal A}_\Lambda\) (the Azumaya locus) and the singular locus is given by \({\mathcal S}_\Lambda=\{m\in Z:Z_m\) is not regular\}. The authors prove that \({\mathcal A}_\Lambda=\max Z\setminus{\mathcal S}_\Lambda\), in case \(\Lambda\) is Auslander-regular and Macaulay and \(\Lambda_p\) is Azumaya over \(Z_p\) for all height 1 primes \(p\) of \(Z\). This generalizes the corresponding graded result of \textit{L. Le Bruyn} [J. Algebra 177, No. 1, 142-153 (1995; Zbl 0844.17005)]. In particular, the locus of non-Azumaya points of the centre coincides with the singular locus of the centre for the algebras \(U_\varepsilon({\mathfrak g})\), \({\mathcal O}_\varepsilon[G]\) and certain enveloping algebras.
finite dimensional complex semisimple Lie algebras
scientific article; zbMATH DE number 1106639
scientific article