This paper studies Lie superalgebras graded by an arbitrary set [Formula: see text] (set grading). We show that the set-graded Lie superalgebra [Formula: see text] decomposes as the sum of well-described set-graded ideals plus a certain linear subspace. Under certain conditions, the simplicity of [Formula: see text] is characterized and it is shown that the above decomposition is exactly the direct sum of the family of its minimal set-graded ideals, each one being a simple set-graded Lie superalgebra.
We study the fixed point subalgebra of a centerless irreducible Lie torus under a certain finite order automorphism. We investigate which axioms of a Lie torus hold for the fixed points and which do not. We relate our study to some recent results about the fixed points of extended affine Lie algebras under a class of finite order automorphisms.
In this paper we study the structure of arbitrary split involutive regular Hom-Lie color algebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split involutive regular Hom-Lie color algebra L is of the form L = U ⊕[α]∈Π/∼ I [α] , with U a subspace of the involutive abelian subalgebra H and any I [α] , a well-described involutive ideal of L, satisfying [I [α] , I [β] . Under certain conditions, in the case of L being of maximal length, the simplicity of the algebra is characterized and it is shown that L is the direct sum of the family of its minimal involutive ideals, each one being a simple split involutive regular Hom-Lie color algebra. Finally, an example will be provided to characterise the inner structure of split involutive Hom-Lie color algebras.2010 Mathematics Subject Classification. 17B40, 17B55, 17B75.
We state the notion of an extension datum of a root system. This concept generalizes the root system extended by an abelian group. We classify them, and show that there is a one-to-one correspondence between (reduced) root systems extended by an abelian group and (reduced) tame irreducible extension data. We also show that an extension datum, modulo some isotropic roots, is the union of a finite number of tame irreducible extension data.
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