Representations of the restricted Lie algebras of Cartan type

Abstract

Let L be a restricted simple Lie algebra of Cartan type. In this paper, we construct the simple L-modules having nonsingular characters and some simple modules with singular characters.  2005 Elsevier Inc. All rights reserved. IntroductionLet L be a simple restricted Lie algebra of Cartan type over an algebraically closed field F with characteristic p 5. Let χ ∈ L * = Hom F (L, F ). A L-module M has character χ provided that D p m − D [p] m = χ(D) p m for D ∈ L and m ∈ M. By [7, Theorem 2.5, p. 207], every simple module has a character. If M is a L-module with χ = 0, then we call M a restricted L-module; if χ = 0, then M is called a nonrestricted L-module.Let L = i L i be the standard grading on L and put L i = j i L j . The height ht(χ) of the character χ is defined by: ht(χ) = min i −1 χ L i = 0 .

IRREDUCIBLE REPRESENTATIONS OF THE HAMILTONIAN ALGEBRAH(2r;n)

“…R 4.4. For n = 1, that is, the restricted case, the result of Theorem 4.3 can be deduced by combining [26,Theorem 2.5,Proposition 2.6]. In this case, the result also coincides with a recent theorem of Wu, Jiang and Pu (see [23,Theorem 1]).…”

“…This work follows lines similar to Shen's work on graded modules (see [3,4,25]). Furthermore, Zhang and Steffensen studied irreducible modules of L and the rank-two Witt algebra W(2, 1) for general χ which are either nonsingular or 'nice', respectively (see [6,26]).…”

Support varieties of semisimple-character representations for Cartan type Lie algebras

“…Then we have the following direct corollary. Proof When the character χ satisfies the condition stated in the corollary, all irreducible non-exceptional U χ (L)-modules are baby Kac modules by [5][6][7]13,20] (see also [15,18,19]). Therefore, the statement is a direct consequence of Corollary 1.…”

“…One can generalize the result of Corollary 3 to any finitedimensional U χ (L)-module M, where χ is a nonsingular character in the sense of the definition given in [20].…”

Representations of the Restricted Cartan Type Lie Superalgebra W (m, n, 1)

“…We call it the Kac module of type I. By applying a similar argument as that for [20,Th. 2.5 ], one gets Applying a similar as that for [20, 2.6], one can show that χ is nonsingular, if ht(χ) ≤ p − 1 and χ(W [s−1] ∩ W (m, 1)) = 0.…”