Related papers: Vector fields on $\mathfrak{gl}_{m|n}(\mathbb C)$-…

The arXiv:2105.09738 claims several stuffs. In particular, we recall the following two. (1) Vector fields and differential forms become a Lie superalgebra structure for each manifold. (2) For an n-dimensional Euclidean space, vector fields…

This article classifies the real forms of Lie Superalgebra by Vogan diagrams, developing Borel and de Seibenthal theorem of semisimple Lie algebras for Lie superalgebras. A Vogan diagram is a Dynkin diagram of triplet…

The five simple exceptional complex Lie superalgbras of vector fields are described. One of them is new; the other four are explicitely described for the first time. All of the exceptional Lie superalgebras are obtained with the help of the…

On a given manifold M, the Nijenhuis bracket makes the superspace of vector-valued differential forms into a Lie superalgebra that can be interpreted as the centralizer of the exterior differential considered as a vector field on the…

A well-known result of A. Vaintrob characterizes Lie algebroids and their morphisms in terms of homological vector fields on supermanifolds. We give an interpretation of Lie bialgebroids and their morphisms in terms of odd symplectic…

Let $\Gamma$ be a finite group acting linearly on a vector space $V$. We compute the Lie algebra cohomology of the Lie algebra of $\Gamma$-invariant formal vector fields on $V$. We use this computation to define characteristic classes for…

We show that holomorphic vector fields on (C^3,0) have separatrices provided that they are embedded in a rank 2 representation of a two-dimensional Lie algebra. In turn, this result enables us to show that the second jet of a holomorphic…

This paper considers the multiplicative Hom-Lie superalgebra structures on infinite dimensional simple Lie superalgebras of vector fields with characteristic zero. The main result is that there is only the multiplicative Hom-Lie…

The space M_n of all isomorphism classes of n-dimensional Lie algebras over a field k has a natural non-Hausdorff topology, induced from the Segal topology by the action of GL(n). One way of studying this complicated space is by topological…

We study the cohomology of Lie superalgebra of vector fields on affine super-spaces $\mathbb{A}^{m,n}$ with trivial coefficients. In this paper we extend the methodology developed in the previous paper (arXiv:2210.16585) to perform the…

The notion of singular one-parameter deformation of a Lie algebra is introduced. It is shown that the complex infinite-dimensional Lie algebra of polynomial vector fields in C with trivial 1-jet at the origin has such singular deformation.

We classify open maximal subalgebras of all infinite-dimensional linearly compact simple Lie superalgebras. This is applied to the classification of infinite-dimensional Lie superalgebras of vector fields, acting transitively and…

In this paper we will prove a super-analogue of a well-known result by Kontsevich which states that the homology of a certain complex which is generated by isomorphism classes of oriented graphs can be calculated as the Lie algebra homology…

We study cohomology for classical Lie superalgebras $\mathfrak{g}$ (e.g. gl(m|n)) over the complex numbers. Using results from invariant theory, we show that there exist subsuperalgebras which detect the cohomology of $\mathfrak{g}.$…

The elaboration of new quantization methods has recently developed the interest in the study of subalgebras of the Lie algebra of polynomial vector fields over a Euclidean space. In this framework, these subalgebras define maximal…

A brief proof of Lie's classification of solvable algebras of vector fields on the plane is given. The proof uses basic representation theory and PDEs.

Let X be an analytic vector field on a real or complex 2-manifold, and K a compact set of zeros of X whose fixed point index is not zero. Let A denote the Lie algebra of analytic vector fields Y on M such that at every point of M the values…

In this paper we give a re-normalization of the supertrace on the category of representations of Lie superalgebras of type I, by a kind of modified superdimension. The genuine superdimensions and supertraces are generically zero. However,…

We prove the conjecture by Feigin, Fuchs and Gelfand describing the Lie algebra cohomology of formal vector fields on an $n$-dimensional space with coefficients in symmetric powers of the coadjoint representation. We also compute the…

In this paper, we study the generalized derivation of a Lie sub-algebra of the Lie algebra of polynomial vector fields on $\mathbb{R}^n$ where $n\geq1$, containing all constant vector fields and the Euler vector field, under some conditions…