Vector fields on complex quantum groups (Q1200080)

The paper is based on the one reviewed above [\textit{B. Drabant}, \textit{M. Schlieker}, \textit{W. Weich} and \textit{B. Zumino}, Commun. Math. Phys. 147, 625-633 (1992)]. After recalling the definition of complex quantum groups of type \(A_{n-1}\), \(B_ n\), \(C_ n\), \(D_ n\) and of the \(*\)-Hopf algebra \(U_{\mathcal R}\) of regular functionals on such a quantum group \({\mathcal A}\), a certain subalgebra of \(U_{\mathcal R}\) is defined in terms of generators \(Y\), for which another equivalent set of generators \(X\) can be chosen which may be considered as \(q\)-generalizations of the left invariant vector fields on a complex Lie group. In terms of the generators \(Y\) the Casimir operators in \(U_{\mathcal R}\) are given. In the case \(A_ 1\) the \(Y\)-subalgebra of \(U_{\mathcal R}\) is explored explicitly. The commutation relations between the generators \(X\) are given which in the limit \(q\to 1\) yield the Lie algebra \(sl(2,\mathbb{C})\). It is shown that the vector fields generate a sub-\(*\)-Hopf algebra of \(U_{\mathcal R}\) and that in fact these two \(*\)-Hopf algebras are equivalent.