A quantum analogue of the Capelli identity and an elementary differential calculus on \(GL_ q(n)\)
The aim of this paper is to prove an analog of the classical Capelli identity for the quantum group \(GL_q (n)\). The authors first review the definitions of \(A_q (GL (n))\) and \(U_q ({\mathfrak {gl}} (n))\) and the pairing between these two algebras in the \(R\)-matrix formalism. Next, they define quantum analogs of the partial derivatives along the coordinate functions by solving an analog of the classical relation between these partial derivatives and the generators of the Lie algebra \({\mathfrak {gl}} (n)\) viewed as left invariant vector fields on \(GL (n)\). They also show that the partial derivatives along the coordinate functions can be viewed as an appropriate classical limit of these operators. Then they state and prove their analog of the Capelli identity, which expresses the central element \(z(x)\in U_q ({\mathfrak {gl}} (n))\) introduced by \textit{M. Jimbo} [Lett. Math. Phys 11, 247-252 (1986; Zbl 0602.17005)]\ as a product of two \(q\)-determinants. Finally, the authors study commutation relations involving the quantum partial derivatives in more detail and they relate these operators to certain \(q\)-difference operators. In two appendices they reprove the fact that the element \(z(x)\) from above is central and discuss a second possible choice of quantum partial derivatives.
scientific article; zbMATH DE number 727836
scientific article