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Register a new userSymplectic structures on 2-step nilpotent Lie algebras
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We study symplectic structures on nilpotent Lie algebras. Since the classification of nilpotent Lie algebras in any dimension seems to be a crazy dream, we approach this study in case of 2-step nilpotent Lie algebras (in this sub-case also, the classification fo the dimension greater than 8 seems very difficult), using not a classification but a description of subfamilies associated with the characteristic sequence. We begin with the dimension $8$, first step where the classification becomes difficult.
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We investigate Lie algebras whose Lie bracket is also an associative or cubic associative multiplication to characterize the class of nilpotent Lie algebras with a nilindex equal to 2 or 3. In particular we study the class of 2-step nilpotent Lie algebras, their deformations and we compute the cohomology which parametrize the deformations in this class.
The classification of complex of real finite dimensional Lie algebras which are not semi simple is still in its early stages. For example the nilpotent Lie algebras are classified only up to the dimension 7. Moreover, to recognize a given Lie algebra in a classification list is not so easy. In this work we propose a different approach to this problem. We determine families for some fixed invariants, the classification follows by a deformation process or contraction process. We focus on the case of 2 and 3-step nilpotent Lie algebras. We describe in both cases a deformation cohomology of this type of algebras and the algebras which are rigid regarding this cohomology. Other $p$-step nilpotent Lie algebras are obtained by contraction of the rigid ones.
The symplectic structures on $3$-Lie algebras and metric symplectic $3$-Lie algebras are studied. For arbitrary $3$-Lie algebra $L$, infinite many metric symplectic $3$-Lie algebras are constructed. It is proved that a metric $3$-Lie algebra $(A, B)$ is a metric symplectic $3$-Lie algebra if and only if there exists an invertible derivation $D$ such that $Din Der_B(A)$, and is also proved that every metric symplectic $3$-Lie algebra $(tilde{A}, tilde{B}, tilde{omega})$ is a $T^*_{theta}$-extension of a metric symplectic $3$-Lie algebra $(A, B, omega)$. Finally, we construct a metric symplectic double extension of a metric symplectic $3$-Lie algebra by means of a special derivation.
Let $mathfrak{sp}_{2n}(mathbb {K})$ be the symplectic Lie algebra over an algebraically closed field of characteristic zero. We prove that for any nonzero nilpotent element $X in mathfrak{sp}_{2n}(mathbb {K})$ there exists a nilpotent element $Y in mathfrak{sp}_{2n}(mathbb {K})$ such that $X$ and $Y$ generate $mathfrak{sp}_{2n}(mathbb {K})$.
In this paper, first we introduce the notion of a quadratic Lie-Yamaguti algebra and show that the invariant bilinear form in a quadratic Lie-Yamaguti algebra induces an isomorphism between the adjoint representation and the coadjoint representation. Then we introduce the notions of relative Rota-Baxter operators on Lie-Yamaguti algebras and pre-Lie-Yamaguti algebras. We prove that a pre-Lie-Yamaguti algebra gives rise to a Lie-Yamaguti algebra naturally and a relative Rota-Baxter operator induces a pre-Lie-Yamaguti algebra. Finally we study symplectic structures on Lie-Yamaguti algebra, which give rise to relative Rota-Baxter operators as well as pre-Lie-Yamaguti algebras. As applications, we study phase spaces of Lie-Yamaguti algebras, and show that there is a one-to-one correspondence between phase spaces of Lie-Yamaguti algebras and Manin triples of pre-Lie-Yamaguti algebras.
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