The Zassenhaus variety of a reductive Lie algebra in positive characteristic

We introduce post-associative algebra structures and study their relationship to post-Lie algebra structures, Rota–Baxter operators and decompositions of associative algebras and Lie algebras. We show several results on the existence of such structures. In particular, we prove that there exists no post-Lie algebra structure on a pair [Formula: see text], where [Formula: see text] is a simple Lie algebra and [Formula: see text] is a reductive Lie algebra, which is not isomorphic to [Formula: see text]. We also show that there is no post-associative algebra structure on a pair [Formula: see text] arising from a Rota–Baxter operator of [Formula: see text], where [Formula: see text] is a semisimple associative algebra and [Formula: see text] is not semisimple. The proofs use results on Rota–Baxter operators and decompositions of algebras.

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Complexity for modules over the classical Lie superalgebra

Compositio Mathematica ◽

10.1112/s0010437x12000231 ◽

2012 ◽

Vol 148 (5) ◽

pp. 1561-1592 ◽

Cited By ~ 6

Author(s):

Brian D. Boe ◽

Jonathan R. Kujawa ◽

Daniel K. Nakano

Keyword(s):

Lie Algebra ◽

Lie Superalgebra ◽

Projective Resolution ◽

Lie Superalgebras ◽

Simple Modules ◽

Minimal Projective Resolution ◽

Finite Dimensional ◽

Reductive Lie Algebra ◽

Completely Reducible ◽

Kac Modules

AbstractLet ${\Xmathfrak g}={\Xmathfrak g}_{\zerox }\oplus {\Xmathfrak g}_{\onex }$ be a classical Lie superalgebra and let ℱ be the category of finite-dimensional ${\Xmathfrak g}$-supermodules which are completely reducible over the reductive Lie algebra ${\Xmathfrak g}_{\zerox }$. In [B. D. Boe, J. R. Kujawa and D. K. Nakano, Complexity and module varieties for classical Lie superalgebras, Int. Math. Res. Not. IMRN (2011), 696–724], we demonstrated that for any module M in ℱ the rate of growth of the minimal projective resolution (i.e. the complexity of M) is bounded by the dimension of ${\Xmathfrak g}_{\onex }$. In this paper we compute the complexity of the simple modules and the Kac modules for the Lie superalgebra $\Xmathfrak {gl}(m|n)$. In both cases we show that the complexity is related to the atypicality of the block containing the module.

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The Square-Zero Basis of Matrix Lie Algebras

Mathematics ◽

10.3390/math8061032 ◽

2020 ◽

Vol 8 (6) ◽

pp. 1032

Author(s):

Raúl Durán Díaz ◽

Víctor Gayoso Martínez ◽

Luis Hernández Encinas ◽

Jaime Muñoz Masqué

Keyword(s):

Algebraic Group ◽

Lie Algebra ◽

Lie Algebras ◽

Positive Characteristic ◽

Linear Algebraic Group ◽

Invariant Functions ◽

Arbitrary Characteristic ◽

Independent Invariant ◽

Field Of Positive Characteristic

A method is presented that allows one to compute the maximum number of functionally-independent invariant functions under the action of a linear algebraic group as long as its Lie algebra admits a basis of square-zero matrices even on a field of positive characteristic. The class of such Lie algebras is studied in the framework of the classical Lie algebras of arbitrary characteristic. Some examples and applications are also given.

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