On central elements in the universal enveloping algebras of the orthogonal Lie algebras (Q2747675)

The main result of this paper is the following relation between the Pfaffian and the determinant of the alternating matrix \(A=(A_{ij})_{i,j=1}^{2m}\) consisting of standard generators \(A_{ij}=E_{ij}-E_{ji}\) of the universal enveloping algebra of the orthogonal Lie algebra \(\mathfrak{o}_{2m}\): \( \text{Pf}(A)^2=\det(A+\text{diag}(m-1,m-2,\dots,-m)).\) This identity clarifies the relation between the two sets of central elements in the universal enveloping algebra of an orthogonal Lie algebra, the first being expressed by column determinant and the other one by Sklyanin determinant.