The coradical filtration of U Q (S(2)) Q at roots of unity

Abstract

Let u q (s[(2)) denote the quantum enveloping algebra of s((2) over a field k. In this note we show that if q is a root of unity, then the coradical filtration {H n}n ≥0 of u q (s[(2)) is the filtration by a certain degree function depending on the order of q 4 and on the characteristic of k. The degree function will be explicitly described in Section 2 for char s = 0 and in Section 3 for char k = s > 0. We use the fact that u q (g) is pointed for any finite-dimensional semisimple Lie algebra (proved in [R], [Ml] and remarked in [Tl]). The skew-primitives of u q (g[(n)) and u q (s[(n)) are described in [T2], based en [Tl]. The theorem in [T2] motivated the present result, although we did not need its full strength in the proof. The filtration of u q (g) over Q(q), where g is a finite dimensional complex simple Lie algebra, and q is an indeterminate over Q, is described in [CM]. The case of u q (s[(2)) with char k = 0 and where q is not a root of unity was first solved in [Ml].