Shapovalov Determinant for Restricted and Quantized Restricted Enveloping Algebras
نویسندگان
چکیده
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 p 6= 2 which also satisfies p 6= 3 whenever g contains a component of type G2. This will be our tacit assumption on p through the paper. Let ξ be a primitive p root of unity. This paper is concerned with two algebras: a certain analog up of the restricted enveloping algebra (cf. Definition 3.1) and its quantized version uξ which is an algebra over the cyclotomic field Qξ (cf. Definition 3.3). The main results of this paper are complete descriptions of the Shapovalov determinant for both the algebras up and uξ (cf. Theorems 3.2 and 3.4).
منابع مشابه
ar X iv : 0 81 0 . 16 21 v 1 [ m at h . Q A ] 9 O ct 2 00 8 DRINFEL ’ D DOUBLES AND SHAPOVALOV DETERMINANTS
The Shapovalov determinant for a class of pointed Hopf algebras is calculated, including quantized enveloping algebras, Lusztig’s small quantum groups, and quantized Lie superalgebras. Our main tools are root systems, Weyl groupoids, and Lusztig type isomorphisms. We elaborate powerful novel techniques for the algebras at roots of unity, and pass to the general case using a density argument.
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