Shapovalov determinant for restricted and quantized restricted enveloping algebras

Shapovalov Determinant for Restricted and Quantized Restricted Enveloping Algebras

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 d...

Crossed products of restricted enveloping algebras

Let K be a eld of characteristic p > 0, let L be a restricted Lie algebra and let R be an associative K-algebra. It is shown that the various constructions in the literature of crossed product of R with u(L) are equivalent. We calculate explicit formulae relating the parameters involved and obtain a formula which hints at a noncommutative version of the Bell polynomials. Crossed products of gro...

Isomorphism Invariants of Restricted Enveloping Algebras

Let L and H be finite-dimensional restricted Lie algebras over a perfect field F such that u(L) = u(H), where u(L) is the restricted enveloping algebra of L. We prove that if L is p-nilpotent and abelian, then L = H . We deduce that if L is abelian and F is algebraically closed, then L = H . We use these results to prove the main result of this paper stating that if L is p-nilpotent, then L/L +...

More Tight Monomials in Quantized Enveloping Algebras

Constructing Canonical Bases of Quantized Enveloping Algebras

Since the invention of canonical bases of quantized enveloping algebras, one of the main problems has been to establish what they look like. Explicit formulas are only known in a few cases corresponding to root systems of low rank, namely A1 (trivial), A2 ([Lusztig 90]), A3 ([Xi 99a]), and B2 ([Xi 99b]). Furthermore, there is evidence suggesting that for higher ranks the formulas become so comp...