Let \((H,R)\) be a finite-dimensional quasitriangular Hopf algebra over a field \(k\) of characteristic zero. The map \(T_R\) from \(H^*\) to \(H\) which sends an \(f\) in \(H^*\) to \(f(R^2r^1)R^1r^2\) (\(R=r\) in \(H\otimes H\)) is a map of braided Hopf algebras, called the transmutation map. \((H,R)\) is called triangular if \(T_R\) is trivial, and factorizable if \(T_R\) is an isomorphism. The author studies certain quotients of \(H\). If \(C\) is a subcoalgebra of \(H^*\), let \(K_C\) be the subalgebra of \(H\) generated by \(T_R(C)\). \(K_C\) is a normal left coideal subalgebra of \(H\).
By a result of M. Takeuchi [Commun. Algebra 22, No. 7, 2503-2523 (1994; Zbl 0801.16041)], there is a Hopf ideal \(I_C\) of \(H\), and the quotient \(H/I_C\) is a quasitriangular Hopf algebra. When \(C\) is a Hopf subalgebra \(A\) of \(H^*\), then \(B=(H/I_C)^*\) can be considered as a Hopf subalgebra of \(H^*\). The author shows that \([H^*:A]\) divides \(\dim B\). Moreover, if \((H,R)\) is factorizable, then \((\dim A)(\dim B)=\dim H\), and \(A\) and \(B\) intersect in \(k1\). Results of this type are used to address the conjecture that a Hopf algebra whose dimension is square-free is semisimple. The author shows that if \((H,R)\) is quasitriangular of odd square-free dimension, then \(H\) is semisimple. In fact, \(H\) is a certain kind of group algebra and is cocommutative. The triangular and factorizable cases are considered separately, and then combined to obtain this result.