Abstract
It is shown that for the restricted Zassenhaus algebra \({\mathfrak{W} = \mathfrak{W}(1; n)}\), n > 1, defined over an algebraically closed field \({\mathbb{F}}\) of characteristic 2, any projective indecomposable restricted \({\mathfrak{W}}\) -module has maximal possible dimension \({2^{2^n-1}}\), and thus is isomorphic to some induced module \({{\rm idn}^{\mathfrak{W}}_{\mathfrak{t}}(\mathbb{F}(\mu))}\) for some torus of maximal dimension \({\mathfrak{t}}\). This phenomenon is in contrast to the behavior of finite-dimensional non-solvable restricted Lie algebras in characteristic p > 3 (cf. Feldvoss et al. Restricted Lie algebras with maximal 0-pim, 2014, Theorem 6.3).
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References
G. M. Benkart, I. M. Isaacs, and J. M. Osborn, Albert-Zassenhaus Lie algebras and isomorphisms, J. Algebra 57 (1979), 310–338. MR 533801 (80i:17007)
A. Dzhumadil’daev, Simple Lie algebras with a subalgebra of codimension one, Russian Math. Surveys 40 (1985), 215–216. MR 783616 (86g:17011)
J. Feldvoss, On the cohomology of modular Lie algebras, Lie algebras, vertex operator algebras and their applications, Contemp. Math., vol. 442, Amer. Math. Soc., Providence, RI, 2007, pp. 89–113. MR 2372558 (2008m:17037)
J. Feldvoss, S. Siciliano, and Th. Weigel, Restricted Lie algebras with maximal 0-pim, preprint, available at http://arxiv.org/pdf/1407.1902v1, 2014.
B. Lancellotti, Lie algebras of Cartan type, Tesi di laurea magistrale, Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, 2014.
H. Strade, Simple Lie algebras over fields of positive characteristic. I, de Gruyter Expositions in Mathematics, vol. 38, Walter de Gruyter & Co., Berlin, 2004, Structure theory. MR 2059133 (2005c:17025)
H. Strade and R. Farnsteiner, Modular Lie algebras and their representations, Monographs and Textbooks in Pure and Applied Mathematics, vol. 116, Marcel Dekker, Inc., New York, 1988. MR 929682 (89h:17021)
C. A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324 (95f:18001)
B. Yu. Veisfeiler and V. G. Kats, Irreducible representations of Lie p-algebras, Functional Analysis and its Applications 5 (1971), 111–117. MR 0285575 (44 #2793)
H. Zassenhaus, Über Lie’sche Ringe mit Primzahlcharakteristik, Abh. Math. Sem. Univ. Hamburg 13 (1939), 1–100. MR 3069699
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Lancellotti, B., Weigel, T. The p.i.m.s for the restricted Zassenhaus algebras in characteristic 2. Arch. Math. 104, 333–340 (2015). https://doi.org/10.1007/s00013-015-0748-3
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DOI: https://doi.org/10.1007/s00013-015-0748-3