scholarly journals On the Cohomology and Extensions of n-ary Multiplicative Hom-Nambu-Lie Superalgebras

2020 ◽  
Vol 2020 ◽  
pp. 1-17
Author(s):  
Lijun Tian ◽  
Baoling Guan ◽  
Yao Ma
Keyword(s):  
Lie Algebras ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Finite Dimensional

In this paper, we discuss the representations of n-ary multiplicative Hom-Nambu-Lie superalgebras as a generalization of the notion of representations for n-ary multiplicative Hom-Nambu-Lie algebras. We also give the cohomology of an n-ary multiplicative Hom-Nambu-Lie superalgebra and obtain a relation between extensions of an n-ary multiplicative Hom-Nambu-Lie superalgebra b by an abelian one a and Z1b,a0¯. We also introduce the notion of T∗-extensions of n-ary multiplicative Hom-Nambu-Lie superalgebras and prove that every finite-dimensional nilpotent metric n-ary multiplicative Hom-Nambu-Lie superalgebra over an algebraically closed field of characteristic not 2 in the case α is a surjection is isometric to a suitable T∗-extension.

Second Cohomology of the Modular Lie Superalgebra of Cartan Type K

2009 ◽  
Vol 16 (02) ◽  
pp. 309-324 ◽  
Author(s):  
Wenjuan Xie ◽  
Yongzheng Zhang
Keyword(s):  
Cohomology Group ◽  
Lie Superalgebra ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Cartan Type ◽  
Modular Lie Superalgebra ◽  
Finite Dimensional ◽  
Type K

Let 𝔽 be an algebraically closed field and char 𝔽 = p > 3. In this paper, we determine the second cohomology group of the finite-dimensional Contact superalgebra K(m,n,t).

scholarly journals Generators of Simple Modular Lie Superalgebras

2016 ◽  
Vol 23 (02) ◽  
pp. 347-360
Author(s):  
Liming Tang ◽  
Wende Liu
Keyword(s):  
Lie Superalgebras ◽  
Closed Field ◽  
Characteristic P ◽  
Algebraically Closed Field ◽  
Cartan Type ◽  
Finite Dimensional ◽  
Modular Lie Superalgebras

Let X be one of the finite-dimensional graded simple Lie superalgebras of Cartan type W, S, H, K, HO, KO, SHO or SKO over an algebraically closed field of characteristic p > 3. In this paper we prove that X can be generated by one element except the ones of type W, HO, KO or SKO in certain exceptional cases in which X can be generated by two elements. As a subsidiary result, we prove that certain classical Lie superalgebras or their relatives can be generated by one or two elements.

scholarly journals On Some Infinite Dimensional Representations of Semi-Simple Lie Algebras

1965 ◽  
Vol 25 ◽  
pp. 211-220 ◽  
Author(s):  
Hiroshi Kimura
Keyword(s):  
Tensor Product ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Irreducible Representations ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Infinite Dimensional ◽  
Complete Reducibility ◽  
Finite Dimensional

Let g be a semi-simple Lie algebra over an algebraically closed field K of characteristic 0. For finite dimensional representations of g, the following important results are known; 1) H1(g, V) = 0 for any finite dimensional g space V. This is equivalent to the complete reducibility of all the finite dimensional representations,2) Determination of all irreducible representations in connection with their highest weights.3) Weyl’s formula for the character of irreducible representations [9].4) Kostant’s formula for the multiplicity of weights of irreducible representations [6],5) The law of the decomposition of the tensor product of two irreducible representations [1].

scholarly journals Group Gradings on Finite Dimensional Lie Algebras

2013 ◽  
Vol 20 (04) ◽  
pp. 573-578 ◽  
Author(s):  
Dušan Pagon ◽  
Dušan Repovš ◽  
Mikhail Zaicev
Keyword(s):  
Finite Group ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Solvable Radical ◽  
Finite Dimensional ◽  
Levi Subalgebra ◽  
Group Gradings ◽  
Finite Dimensional Lie Algebras

We study gradings by non-commutative groups on finite dimensional Lie algebras over an algebraically closed field of characteristic zero. It is shown that if L is graded by a non-abelian finite group G, then the solvable radical R of L is G-graded and there exists a Levi subalgebra B=H1⊕ ⋯ ⊕ Hm homogeneous in G-grading with graded simple summands H1,…,Hm. All Supp Hi (i=1,…,m) are commutative subsets of G.

Cohomology of model filiform Lie superalgebras

2018 ◽  
Vol 17 (04) ◽  
pp. 1850074 ◽  
Author(s):  
Wende Liu ◽  
Yong Yang
Keyword(s):  
Characteristic Zero ◽  
Cohomology Group ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Closed Field ◽  
Algebra Structure ◽  
Ground Field ◽  
Algebraically Closed Field ◽  
One Dimensional ◽  
First Cohomology Group

Suppose the ground field [Formula: see text] is an algebraically closed field of characteristic zero. By means of spectral sequences, the computation of the first cohomology group of the model filiform Lie superalgebra [Formula: see text] with coefficients in the adjoint module is reduced to the computation of the first cohomology group of an Abel ideal and a one-dimensional subalgebra. Then, by calculating the outer derivations, the algebra structure of the first cohomology group of [Formula: see text] is completely characterized.

On indecomposable solvable Lie superalgebras having a Heisenberg nilradical

2016 ◽  
Vol 15 (10) ◽  
pp. 1650190
Author(s):  
M. C. Rodríguez-Vallarte ◽  
G. Salgado ◽  
O. A. Sánchez-Valenzuela
Keyword(s):  
Lie Algebras ◽  
Dual Space ◽  
Group Actions ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Symmetric Form ◽  
Complex Vector ◽  
Isomorphism Classes ◽  
Finite Dimensional ◽  
Linear Maps

All solvable, indecomposable, finite-dimensional, complex Lie superalgebras [Formula: see text] whose first derived ideal lies in its nilradical, and whose nilradical is a Heisenberg Lie superalgebra [Formula: see text] associated to a [Formula: see text]-homogeneous supersymplectic complex vector superspace [Formula: see text], are here classified up to isomorphism. It is shown that they are all of the form [Formula: see text], where [Formula: see text] is even and consists of non-[Formula: see text]-nilpotent elements. All these Lie superalgebras depend on an element [Formula: see text] in the dual space [Formula: see text] and on a pair of linear maps defined on [Formula: see text], and taking values in the Lie algebras naturally associated to the even and odd subspaces of [Formula: see text]; namely, if the supersymplectic form is even, the pair of linear maps defined on [Formula: see text] take values in [Formula: see text], and [Formula: see text], respectively, whereas if the supersymplectic form is odd these linear maps take values on [Formula: see text]. When the supersymplectic form is even, a bilinear, skew-symmetric form defined on [Formula: see text] is further needed. Conditions on these building data are given and the isomorphism classes of the resulting Lie superalgebras are described in terms of appropriate group actions.

scholarly journals Lie Algebras with Abelian Centralizers

2010 ◽  
Vol 17 (04) ◽  
pp. 629-636 ◽  
Author(s):  
Igor Klep ◽  
Primož Moravec
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Closed Field ◽  
Split Extension ◽  
Algebraically Closed Field ◽  
One Dimensional ◽  
Solvable Case ◽  
Finite Dimensional ◽  
Finite Dimensional Lie Algebras

We classify all finite-dimensional Lie algebras over an algebraically closed field of characteristic 0, whose nonzero elements have abelian centralizers. These algebras are either simple or solvable, where the only simple such Lie algebra is [Formula: see text]. In the solvable case, they are either abelian or a one-dimensional split extension of an abelian Lie algebra.

scholarly journals Classification of Simple Lie Superalgebras in Characteristic 2

2021 ◽  
Author(s):  
Sofiane Bouarroudj ◽  
Alexei Lebedev ◽  
Dimitry Leites ◽  
Irina Shchepochkina
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Vector Fields ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Simple Lie Algebra ◽  
Hamiltonian Vector Fields ◽  
Finite Dimensional ◽  
Characteristic 2

Abstract All results concern characteristic 2. We describe two procedures; each of which to every simple Lie algebra assigns a simple Lie superalgebra. We prove that every simple finite-dimensional Lie superalgebra is obtained as the result of one of these procedures. For Lie algebras, in addition to the known “classical” restrictedness, we introduce a (2,4)-structure on the two non-alternating series: orthogonal and Hamiltonian vector fields. For Lie superalgebras, the classical restrictedness of Lie algebras has two analogs: a $2|4$-structure, which is a direct analog of the classical restrictedness, and a novel $2|2$-structure—one more analog, a $(2,4)|4$-structure on Lie superalgebras is the analog of (2,4)-structure on Lie algebras known only for non-alternating orthogonal and Hamiltonian series.

scholarly journals Classification of Nilpotent Lie Superalgebras of Multiplier-Rank Less than or Equal to 6

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Shuang Lang ◽  
Jizhu Nan ◽  
Wende Liu
Keyword(s):  
Characteristic Zero ◽  
Lie Superalgebras ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Finite Dimensional

In this paper, we classify all the finite-dimensional nilpotent Lie superalgebras of multiplier-rank less than or equal to 6 over an algebraically closed field of characteristic zero. We also determine the covers of all the nilpotent Lie superalgebras mentioned above.

Lie Algebras with Nilpotent Centralizers

1979 ◽  
Vol 31 (5) ◽  
pp. 929-941 ◽  
Author(s):  
G. M. Benkart ◽  
I. M. Isaacs
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Simple Lie Algebras ◽  
Arbitrary Characteristic ◽  
Algebra For All ◽  
Finite Dimensional ◽  
Nilpotent Subalgebra ◽  
Finite Dimensional Lie Algebras

We consider finite dimensional Lie algebras over an algebraically closed field F of arbitrary characteristic. Such an algebra L will be called a centralizer nilpotent Lie algebra (abbreviated c.n.) provided that the centralizer C(x) is a nilpotent subalgebra of L for all nonzero x ∈ L.For each algebraically closed F, there is a unique simple Lie algebra of dimension 3 over F which we shall denote S(F). This algebra has a basis e−1, e0, e1 such that [e−1e0] = e−1, [e−1e1] = e0 and [e0e1] = e1. (If char(F) ≠ 2, then S(F) ≅ sl2(F).) It is trivial to check that S(F) is a c.n. algebra for all F.There are two other types of simple Lie algebras we consider. If char (F) = 3, construct the octonion (Cayley) algebra over F.

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