On the dimension of \(H\)-strata in quantum algebras. (English) Zbl 1226.16024

This paper addresses the dimensions of the strata of prime ideals in an algebra \(A\) over an infinite field \(\mathbb K\), with a \(\mathbb K\)-torus \(H\) acting rationally on \(A\). There is a stratification of \(\text{spec\,}A\) indexed by the \(H\)-invariant prime ideals of \(A\), and E. S. Letzter and the reviewer showed that each stratum is homeomorphic to the prime spectrum of a Laurent polynomial ring [Trans. Am. Math. Soc. 352, No. 3, 1381-1403 (2000; Zbl 0978.16040)].
Here the authors give a formula for the Krull dimension of the strata \(\text{spec}_JA\) for a class of iterated skew polynomial algebras known as uniparameter CGL extensions, which include the standard quantizations \(\mathcal O_q(M_{m,n})\) of the coordinate rings of matrix varieties. The formula expresses \(\dim\text{spec}_JA\) as the dimension of the null space of an antisymmetric integer matrix whose entries come from commutation relations in \(A\) and whose row and column indices are determined by the Cauchon diagram (or Le-diagram) associated to the \(H\)-invariant prime \(J\) by G. Cauchon’s deleting-derivations algorithm [J. Algebra 260, No. 2, 476-518 (2003; Zbl 1017.16017)]. Applications to \(\mathcal O_q(M_{m,n})\) and the algebras \(U_q[w]\) introduced by C. De Concini, V. G. Kac and C. Procesi [in Geometry and analysis. Stud. Math., Tata Inst. Fundam. Res. 13, 41-65 (1995; Zbl 0878.17014)] are given. In particular, the dimensions of the \((0)\)-strata of \(\mathcal O_q(M_{m,n})\), computed by S. Launois and T. H. Lenagan [in Algebr. Represent. Theory 10, No. 4, 339-365 (2007; Zbl 1124.16037)], are recovered.
For the case \(A=\mathcal O_q(M_{m,n})\), the authors prove that the dimensions of the \(H\)-strata of \(\text{spec\,}A\) are bounded above by \(\min\{m,n\}\), and that each nonnegative integer \(d\leq\min\{m,n\}\) occurs as the dimension of such a stratum. Moreover, if \(P\) is an \(H\)-invariant prime of \(A\) such that \(\dim\text{spec}_PA=d\), there exists a chain \(P_0=P\subset P_1\subset\cdots\subset P_d\) of \(H\)-invariant primes of \(A\) such that \(\dim\text{spec}_{P_i}A=d-i\).


16T20 Ring-theoretic aspects of quantum groups
16S36 Ordinary and skew polynomial rings and semigroup rings
17B37 Quantum groups (quantized enveloping algebras) and related deformations
16D25 Ideals in associative algebras
16P60 Chain conditions on annihilators and summands: Goldie-type conditions
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