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Let $(𝔄,{ϵ}_u)$ and $(𝔅,{ϵ}_b)$ be two pointed sets. Given a family of three maps $ℱ=\{f_1:𝔄\to 𝔄;f_2:𝔄\times 𝔄\to 𝔄;f_3:𝔄\times 𝔄\to 𝔅\}$, this family provides an adequate decomposition of $𝔄\setminus \left\{{ϵ}_u\right\}$ as the orthogonal disjoint union of well-described $ℱ$-invariant subsets. This decomposition is applied to the structure theory of graded involutive algebras, graded quadratic algebras and graded weak $H^{*}$-algebras.

We introduce the class of split regular Hom-Poisson algebras formed by those Hom-Poisson algebras whose underlying Hom-Lie algebras are split and regular. This class is the natural extension of the ones of split Hom-Lie algebras and of split Poisson algebras. We show that the structure theorems for split Poisson algebras can be extended to the more general setting of split regular Hom-Poisson algebras. That is, we prove that an arbitrary split regular Hom-Poisson algebra is of the form $=U+{\sum }_j{I}_j$ with U...

We study the Banach-Lie group $Aut(A^{-})$ of Lie automorphisms of a complex associative $H^{*}$-algebra. Also some consequences about its Lie algebra, the algebra of Lie derivations of $A$, are obtained. For a topologically simple $A$, in the infinite-dimensional case we have $Aut{(A^{-})}_0=Aut(A)$ implying $Der(A)=Der(A^{-})$. In the finite dimensional case $Aut{(A^{-})}_0$ is a direct product of $Aut(A)$ and a certain subgroup of Lie derivations $\delta$ from $A$ to its center, annihilating commutators.