scholarly journals Extensions and automorphisms of Lie algebras

2016 ◽  
Vol 16 (09) ◽  
pp. 1750162 ◽  
Author(s):  
Valeriy G. Bardakov ◽  
Mahender Singh
Keyword(s):  
Exact Sequence ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Short Exact Sequence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Nilpotent Lie Algebra ◽  
Lie Algebra Cohomology ◽  
Cohomology Of Lie Algebras ◽  
Necessary And Sufficient

Let [Formula: see text] be a short exact sequence of Lie algebras over a field [Formula: see text], where [Formula: see text] is abelian. We show that the obstruction for a pair of automorphisms in [Formula: see text] to be induced by an automorphism in [Formula: see text] lies in the Lie algebra cohomology [Formula: see text]. As a consequence, we obtain a four term exact sequence relating automorphisms, derivations and cohomology of Lie algebras. We also obtain a more explicit necessary and sufficient condition for a pair of automorphisms in [Formula: see text] to be induced by an automorphism in [Formula: see text], where [Formula: see text] is a free nilpotent Lie algebra of rank [Formula: see text] and step [Formula: see text].

scholarly journals CUBIC DIRAC OPERATORS AND THE STRANGE FREUDENTHAL–DE VRIES FORMULA FOR COLOUR LIE ALGEBRAS

2021 ◽  
Author(s):  
PHILIPPE MEYER
Keyword(s):  
Dirac Operator ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Dirac Operators ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Orthogonal Representation ◽  
Basic Colour ◽  
De Vries ◽  
Necessary And Sufficient

AbstractThe aim of this paper is to define cubic Dirac operators for colour Lie algebras. We give a necessary and sufficient condition to construct a colour Lie algebra from an ϵ-orthogonal representation of an ϵ-quadratic colour Lie algebra. This is used to prove a strange Freudenthal–de Vries formula for basic colour Lie algebras as well as a Parthasarathy formula for cubic Dirac operators of colour Lie algebras. We calculate the cohomology induced by this Dirac operator, analogously to the algebraic Vogan conjecture proved by Huang and Pandžić.

SOME PROPERTIES ON THE SCHUR MULTIPLIER OF A PAIR OF LIE ALGEBRAS

2012 ◽  
Vol 11 (01) ◽  
pp. 1250011 ◽  
Author(s):  
MOHAMMAD REZA RISMANCHIAN ◽  
MEHDI ARASKHAN
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Central Extension ◽  
Sufficient Condition ◽  
Schur Multiplier ◽  
Necessary And Sufficient Condition ◽  
Pair Of Lie Algebras ◽  
Necessary And Sufficient

The aim of this paper is to introduce the concept of the Schur multiplier [Formula: see text] of a pair of Lie algebras and to obtain some inequalities for the dimension of [Formula: see text]. Also, we consider some of the features of central extension of an arbitrary Lie algebra. Moreover, we present a necessary and sufficient condition in which the Schur multiplier of a pair of Lie algebras can be embedded into the Schur multiplier of their factor Lie algebras.

Some inequalities for the multiplier of a pair of n-Lie algebras

2019 ◽  
Vol 12 (02) ◽  
pp. 1950028
Author(s):  
Azam K. Mousavi ◽  
Mohammad Reza R. Moghaddam ◽  
Mehdi Eshrati
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Sufficient Condition ◽  
Schur Multiplier ◽  
Necessary And Sufficient Condition ◽  
Schur Multipliers ◽  
Finite Dimensional ◽  
Necessary And Sufficient

Let [Formula: see text] be a pair of [Formula: see text]-Lie algebras. Then, we introduce the concept of Schur multiplier of the pair [Formula: see text], denoted by [Formula: see text], and some inequalities for the dimension of [Formula: see text] are given. We also determine a necessary and sufficient condition, for which the Schur multiplier of a pair of [Formula: see text]-Lie algebras can be embedded into the Schur multipliers of their [Formula: see text]-Lie algebra factors. Moreover, some inequalities for the Schur multiplier of a pair of finite-dimensional nilpotent [Formula: see text]-Lie algebras are acquired.

Extensions of Endomorphisms from the Higher Centres

1967 ◽  
Vol 19 ◽  
pp. 1250-1258 ◽  
Author(s):  
Franklin Haimo
Keyword(s):  
Exact Sequence ◽  
Abelian Groups ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

If 0 → A → C → B → 0 is an exact sequence of abelian groups, if ƒ is a 2-cocyle for this extension, if α ∈ End A, and if β ∈ End B, then a necessary and sufficient condition that α extend to an endomorphism γ of C which induces β is that (M) αƒ and ƒβ be cohomologous ; see Montgomery (2). We shall extend this result to the case where 1 → A → G → B → 1 is an exact sequence of groups and A is abelian.

scholarly journals Quotient groups of IA-automorphisms of a free group of rank 3

2018 ◽  
Vol 28 (01) ◽  
pp. 115-131 ◽  
Author(s):  
V. Metaftsis ◽  
A. I. Papistas ◽  
I. Sevaslidou
Keyword(s):  
Lie Algebra ◽  
Free Group ◽  
Quotient Group ◽  
Lower Central Series ◽  
Central Series ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Graded Lie Algebra ◽  
Necessary And Sufficient ◽  
Quotient Groups

We prove that, for any positive integer [Formula: see text], the quotient group [Formula: see text] of the lower central series of the McCool group [Formula: see text] is isomorphic to two copies of the quotient group [Formula: see text] of the lower central series of a free group [Formula: see text] of rank [Formula: see text] as [Formula: see text]-modules. Furthermore, we give a necessary and sufficient condition whether the associated graded Lie algebra [Formula: see text] of [Formula: see text] is naturally embedded into the Johnson Lie algebra [Formula: see text] of the IA-automorphisms of [Formula: see text].

scholarly journals On the Dimension of Non-Abelian Tensor Squares of $n$-Lie Algebras

2021 ◽  
Vol 52 ◽  
Author(s):  
Farshid Saeedi ◽  
Nafiseh Akbarossadat
Keyword(s):  
Group Theory ◽  
Exact Sequence ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Short Exact Sequence ◽  
Upper Bound ◽  
Tensor Square ◽  
Algebra Over A Field

Let $L$ be an $n$-Lie algebra over a field $\F$. In this paper, we introduce the notion of non-abelian tensor square $L\otimes L$ of $L$ and define the central ideal $L\square L$ of it. Using techniques from group theory and Lie algebras, we show that that $L\square L\cong L^{ab}\square L^{ab}$. Also, we establish the short exact sequence\[0\lra\M(L)\lra\frac{L\otimes L}{L\square L}\lra L^2\lra0\]and apply it to compute an upper bound for the dimension of non-abelian tensor square of $L$.

scholarly journals On the Lie structure of locally matrix algebras

2020 ◽  
Vol 12 (2) ◽  
pp. 311-316
Author(s):  
O. Bezushchak
Keyword(s):  
Lie Algebra ◽  
Matrix Algebra ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Matrix Algebras ◽  
Necessary And Sufficient ◽  
Algebra Over A Field

Let $A$ be a unital locally matrix algebra over a field $\mathbb{F}$ of characteristic different from $2.$ We find a necessary and sufficient condition for the Lie algebra $A\diagup\mathbb{F}\cdot 1$ to be simple and for the Lie algebra of derivations $\text{Der}(A)$ to be topologically simple. The condition depends on the Steinitz number of $A$ only.

ON HAMILTONIAN GROUP OF MULTISYMPLECTIC MANIFOLDS

2011 ◽  
Vol 08 (05) ◽  
pp. 929-935 ◽  
Author(s):  
M. SHAFIEE
Keyword(s):  
Lie Algebra ◽  
Equivalence Classes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Hamiltonian Paths ◽  
Hamiltonian Group ◽  
Identity Component ◽  
Necessary And Sufficient ◽  
Hamiltonian Forms

In this paper the Hamiltonian group Ham (M, Ω) is defined for a compact k-plectic manifold (M, Ω) and it is shown that its Lie algebra is the space of equivalence classes of Hamiltonian forms, modulo closed forms. Also if ψ be a multisymplectomorphism in the identity component Msymp 0(M, Ω) of the group of multisymplectomorphisms Msymp (M, Ω), we obtain a necessary and sufficient condition under which ψ belongs to Ham (M, Ω). As two consequences, we show that Hamiltonian paths are generated by Hamiltonian forms and if Hk (M, ℝ) = 0, then Ham (M, Ω) is equal to Msymp 0(M, Ω).

The Steinberg Lie Algebra st2(S)

2016 ◽  
Vol 23 (01) ◽  
pp. 129-136
Author(s):  
Yongjie Wang ◽  
Yiqian Shi ◽  
Yun Gao
Keyword(s):  
Lie Algebra ◽  
Triple System ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Lie Triple System ◽  
Necessary And Sufficient

Let S be a nonassociative k-algebra. By using the Lie triple system, we study the subspace I2(S) of the Steinberg Lie algebra st2(S) and give a necessary and sufficient condition for I2(S)=0.

Differential Graded Structures on Path Categories

2015 ◽  
Vol 22 (04) ◽  
pp. 677-686
Author(s):  
Fang Li ◽  
Dezhan Tan
Keyword(s):  
Lie Algebra ◽  
Category Structure ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Graded Lie Algebra ◽  
Necessary And Sufficient ◽  
Graded Structures

In this paper, we consider the graded path category associated to a quiver. We investigate all n-differentials on such a category, and also study the associated graded Lie algebra. Moreover, a necessary and sufficient condition is given for the graded path category to admit a DG category structure.

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