A bicovariant differential algebra of a quantum group
In the framework of the general concept of quantum (\(q\)-deformed) Lie groups it is shown in which way the Hopf-algebraic structure of the noncommutative bicovariant differential calculus (1989, Woronowicz) can be used to build an associative noncommutative bicovariant algebra. Such an algebra on \(\text{GL}_q(N)\) involving the quantum analogues of the classical coordinate functions on Lie groups, differential forms, Lie derivatives along vector fields and inner derivations on Hopf algebras is explicitly constructed. As distinct from known investigations, a direct application is made to the Woronowicz differential complex of the cross-products of mutually dual Hopf algebras providing the bicovariance of the noncommutative differential algebra under consideration. A correspondence with classical differential calculus as well as the (co)module properties of the obtained bicovariant algebra on \(\text{GL}_q(N)\) are discussed. The present paper gives a reformulation of previous more extended work (1997, Radko, Vladimirov).
scientific article; zbMATH DE number 1342959