A class of quadratic matrix algebras arising from the quantized enveloping algebra \(\mathcal U_q(A_{2n-1})\) (Q2737933)

The authors introduce a natural family of quantized matrix algebras, which consists of quadratic algebras with the same Hilbert series as polynomials in \(n^2\) variables, including the two best studied such. Their view is that the underlying classical space should be a Hermitian symmetric space rather than a reductive Lie group. So, in this paper, they only consider the Hermitian symmetric space corresponding to \(\text{SU}(p, q)\) and thus end up by quantized \(p\times q\) matrices, specially, for the case \(p = q = n\).NEWLINENEWLINENEWLINEIn \S\ II, the quadratic matrix algebras are introduced and are proved to be iterated Ore extensions. Briefly, some major results about the degree of an algebra are listed due to De Concini and Procesi. In \S\ III, the Poisson structures are studied. In \S\ V, the specific algebras \(D_q(n)\), \(J^n_q(n)\), \(J^0_q(n)\) and \(J^z_q(n)\) are introduced and their canonical forms are found. In \S\ VI, the associated varieties (in the terminology of quadratic algebras) are determined. Moreover, in \S\ VII, the symplectic leaves are discussed. The centers are determined in \S\ VIII. The quantum algebra \(C[L^\pm_1,\dots, L^\pm_{2n-1}]\times_s M^p_q(n)\) is analyzed in \S\ IX. Finally, in \S\ X, the rank \(r\) matrices are considered.