The Clifford algebra of a reductive Lie algebra

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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Matrix 3-Lie superalgebras and BRST supersymmetry

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887817501602 ◽

2017 ◽

Vol 14 (11) ◽

pp. 1750160 ◽

Cited By ~ 10

Author(s):

Viktor Abramov

Keyword(s):

Clifford Algebra ◽

Lie Algebra ◽

Vector Fields ◽

Jacobi Identity ◽

Lie Superalgebras ◽

Infinite Dimensional ◽

Algebra Approach ◽

The Matrix ◽

Infinite Dimensional Lie Algebra

Given a matrix Lie algebra one can construct the 3-Lie algebra by means of the trace of a matrix. In the present paper, we show that this approach can be extended to the infinite-dimensional Lie algebra of vector fields on a manifold if instead of the trace of a matrix we consider a differential 1-form which satisfies certain conditions. Then we show that the same approach can be extended to matrix Lie superalgebras [Formula: see text] if instead of the trace of a matrix we make use of the supertrace of a matrix. It is proved that a graded triple commutator of matrices constructed with the help of the graded commutator and the supertrace satisfies a graded ternary Filippov–Jacobi identity. In two particular cases of [Formula: see text] and [Formula: see text], we show that the Pauli and Dirac matrices generate the matrix 3-Lie superalgebras, and we find the non-trivial graded triple commutators of these algebras. We propose a Clifford algebra approach to 3-Lie superalgebras induced by Lie superalgebras. We also discuss an application of matrix 3-Lie superalgebras in BRST-formalism.

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