Large commutative subalgebras of quantum algebras. (Q2488373)

Let \(Q_{n,r}\) be an associative algebra generated over a field \(k\) by elements \(x_1^{\pm 1},\dots,x_r^{\pm 1},x_{r+1},\dots,x_n\) with defining relations \(x_ix_j=q_{ij}x_jx_i\), where \(q_{ij}\in k^*\). Suppose that \(A\) is an \(\mathbb{N}^n_0\)-filtered associative algebra with respect to some semigroup order on \(\mathbb{N}^n_0\) such that the associated graded algebra is isomorphic to \(Q_{n,0}\). Then the transcendence degree of \(A\) does not exceed the maximal cardinality of a system of independent elements of a specific base of \(A\) whose images commute in \(Q_{n,0}\). It follows that the maximal cardinalities of commuting independent elements in \(Q_{n,0}\) and of similar monomials coincide. The center of \(Q_{n,0}\) is generated by finitely many monomials. There are given applications to the estimation of the transcendence degrees of some quantum algebras.