Degrees of quantum function algebras at roots of 1 (Q1911793)

Consider a complex, semisimple simply connected algebraic group \(G\) and denote its Lie algebra by \(g\). The quantized universal enveloping algebra \(U_q(g)\), introduced by \textit{V. Drinfel'd} [Proc. Int. Congr. Math., Berkeley 1986, Vol. 1, 798-820 (1987; Zbl 0476.16008)] and \textit{M. Jimbo} [Lett. Math. Phys. 11, 247-252 (1986; Zbl 0602.17005)] is a Hopf algebra, generated over the field \(\mathbb{C} (q)\) by elements \(E_i\), \(F_i\) and \(K_\alpha\), \(K_\alpha^{-1}\) satisfying certain relations \((i\) runs from 1 to \(n\), \(n\) being the dimension of the vector space spanned by the root system and \(\alpha\) is a root of \(g)\). The function algebra \(F_q (G)\) is a suitable subalgebra of the dual algebra of \(U_q (g)\), chosen in such a way that it is also made into a Hopf algebra by the comultiplication, dual to the multiplication in \(U_q (g)\). By taking the restrictions of the functions in \(F_q (G)\) to the subalgebras of \(U_q (g)\) generated respectively by (1) the elements \(K_\alpha\) and \(K_\alpha^{-1}\), (2) the elements \(K_\alpha\), \(K^{-1}_\alpha\) and \(E_i\) and (3) the elements \(K_\alpha\), \(K_\alpha^{-1}\) and \(F_i\), one obtains three quotient Hopf algebras. It turns out to be possible to specialize the parameter \(q\) to a root of unity \(\varepsilon\) and obtain the associated Hopf algebras \(F_\varepsilon (G)\) and its corresponding quotients [see \textit{C. De Concini} and \textit{V. Lyubashenko}, Adv. Math. 108, 205-262 (1994; Zbl 0846.17008)]. The representation theory of the algebra \(F_\varepsilon(G)\) turns out to be very rich and an important rôle in developing this theory seems to be the degree of the algebras \(F_\varepsilon (G)\) and its quotients described above. The degree of \(F_\varepsilon (G)\) is the square root of the dimension of \(F_\varepsilon (G)\) over the field of fractions \(Q(Z)\) of the center of \(Z\) of \(F_\varepsilon (G)\). In this paper, the author calculates the degree of these algebras using a combinatorial technique and careful investigation of the root systems and the Weyl group.