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On the combinatorics of Kac's asymmetry function

R. M. Green (2010)

Commentationes Mathematicae Universitatis Carolinae

We use categories to recast the combinatorial theory of full heaps, which are certain labelled partially ordered sets that we introduced in previous work. This gives rise to a far simpler set of definitions, which we use to outline a combinatorial construction of the so-called loop algebras associated to affine untwisted Kac--Moody algebras. The finite convex subsets of full heaps are equipped with a statistic called parity, and this naturally gives rise to Kac's asymmetry function. The latter is...

On the ghost centre of Lie superalgebras

Maria Gorelik (2000)

Annales de l'institut Fourier

We study the invariants of the universal enveloping algebra of a Lie superalgebra with respect to a certain “twisted” adjoint action.

On the nilpotent residuals of all subalgebras of Lie algebras

Wei Meng, Hailou Yao (2018)

Czechoslovak Mathematical Journal

Let 𝒩 denote the class of nilpotent Lie algebras. For any finite-dimensional Lie algebra L over an arbitrary field 𝔽 , there exists a smallest ideal I of L such that L / I 𝒩 . This uniquely determined ideal of L is called the nilpotent residual of L and is denoted by L 𝒩 . In this paper, we define the subalgebra S ( L ) = H L I L ( H 𝒩 ) . Set S 0 ( L ) = 0 . Define S i + 1 ( L ) / S i ( L ) = S ( L / S i ( L ) ) for i 1 . By S ( L ) denote the terminal term of the ascending series. It is proved that L = S ( L ) if and only if L 𝒩 is nilpotent. In addition, we investigate the basic properties of a Lie algebra...

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