Quantized Rank R Matrices

Abstract

First some old as well as new results about P.I. algebras, Ore extensions, and degrees are presented. Then quantized n = r matrices as well as certain quantized rq 1 Ž . Ž . rq 1 Ž . factor algebras M n of M n are analyzed. For r s 1, . . . , n y 1, M n is q q q the quantized function algebra of rank r matrices obtained by working modulo the Ž . Ž . ideal generated by all r q 1 = r q 1 quantum subdeterminants and a certain localization of this algebra is proved to be isomorphic to a more manageable one. In almost all cases, the quantum parameter is a primitive mth root of unity. The degrees and centers of the algebras are determined when m is a prime and the general structure is obtained for arbitrary m. ᮊ 2001 Elsevier Science

PI degree parity in q-skew polynomial rings

“…When k has characteristic zero and q is a primitive mth root of unity for m odd, Jakobsen and Zhang found in [15] that PIdeg O q (M n (k)) = m n(n−1) 2 by using De Concini and Procesi's tool given in Theorem 1.3. This result is reproved in [14] using results of De Concini and Procesi and also Jøndrup's work from [16]. Now we can recover PIdeg O q (M n (k)) without the assumption that k has characteristic zero.…”

“…by using De Concini's and Procesi's tool given in Theorem 1.2. This result is reproved in [19] using results of De Concini and Procesi and also Jøndrup's work from [21]. Now we can recover PIdeg O q (M n (k) without the assumption that k has characteristic zero.…”

Quantized matrix algebras and quantum seeds

“…In case of a regular matrix x, y must have opposite parities. According to Corollary 3.2, the determinant of D is given by The number f was only determined in a few special cases in [11]. We can now use Corollary 3.2 to determine it.…”

The Geometry of Noncommutative Plane Curves

“…For a semiprime PI-algebra A, the degree d = deg(A) of A is the least integer such that A can be embedded in a ring of d × d-matrices over a commutative kalgebra R. The degree d is also the maximum of dim k V , where V runs through all simple A-modules (cf. [6]), and finally, d is one half of the least possible degree of a non-zero polynomial f ∈ k x 1 , . .…”