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As is well known, the Shapovalov bilinear form and its determinant is an important tool in the representation theory of semisimple Lie algebras over char. 0. To our knowledge, the corresponding study of the Shapovalov bilinear form and its determinant is not available in the literature in char. p or the quantum case at roots of unity. The aim of this paper is to fully determine the Shapovalov determinant for both, the restricted enveloping algebra and its quantum analog. More precisely, let g be a semisimple Lie algebra. Fix a prime p6 which also satises p6 whenever g contains a component of type G2. This will be our tacit assumption on p through the paper. Let be a primitive p th root of unity. This paper is concerned with two algebras: a certain analog up of the restricted enveloping algebra (cf. Denition 3:1) and its quantized version u which is an algebra over the cyclotomic eld Q (cf. Denition 3:3). The main results of this paper are complete descriptions of the Shapovalov determinant for both the algebrasup andu (cf. Theorems 3:2 and 3:4).

Shapovalov determinant for restricted and quantized restricted enveloping algebras