KERNELS OF REPRESENTATIONS AND COIDEAL SUBALGEBRAS OF HOPF ALGEBRAS

We define left and right kernels of representations of Hopf algebras. In the case of group algebras, left and right kernels coincide and they are the usual kernels of modules. In the general case, we show that these kernels coincide with the categorical left and right Hopf kernels of morphisms of Hopf algebras defined in Andruskiewitsch and Devoto [Extensions of Hopf algebras, Algebra i Analiz7 (1995), 22–69]. Brauer's theorem for kernels over group algebras is generalised to Hopf algebras.

WIGNER–ECKART THEOREM FOR TENSOR OPERATORS OF HOPF ALGEBRAS

We prove Wigner–Eckart theorem for the irreducible tensor operators for arbitrary Hopf algebras, provided that tensor product of their irreducible representations is completely reducible. The proof is based on the properties of the irreducible representations of Hopf algebras, in particular on Schur lemma. Two classes of tensor operators for the Hopf algebra Ut(su(2)) are considered. The reduced matrix elements for the class of irreducible tensor operators are calculated. A construction of some elements of the center of Ut(su(2)) is given.