According to the classification by Kac, there are eight Cartan series and five exceptional Lie superalgebras in infinite-dimensional simple linearly compact Lie superalgebras of vector fields. In this paper, the Hom-Lie superalgebra structures on the five exceptional Lie superalgebras of vector fields are studied. By making use of the ℤ-grading structures and the transitivity, we prove that there is only the trivial Hom-Lie superalgebra structures on exceptional simple Lie superalgebras. This is achieved by studying the Hom-Lie superalgebra structures only on their 0-th and (−1)-th ℤ-components.
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Let G be the Banach-Lie group of all holomorphic automorphisms of the open unit ball $$B_\mathfrak{A} $$ in a J*-algebra $$\mathfrak{A}$$ of operators. Let $$\mathfrak{F}$$ be the family of all collectively compact subsets W contained in $$B_\mathfrak{A} $$ . We show that the subgroup F ⊂ G of all those g ∈ G that preserve the family $$\mathfrak{F}$$ is a closed Lie subgroup of G and characterize its Banach-Lie algebra. We make a detailed study of F when $$\mathfrak{A}$$ is a Cartan factor.
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The Banach-Lie algebras ℌκ of all holomorphic infinitesimal isometries of the classical symmetric complex Banach manifolds of compact type (κ = 1) and non compact type (κ = −1) associated with a complex JB*-triple Z are considered and the Lie ideal structure of ℌκ is studied.
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