scholarly journals Extensions and automorphisms of Lie algebras

2016 ◽  
Vol 16 (09) ◽  
pp. 1750162 ◽  
Author(s):  
Valeriy G. Bardakov ◽  
Mahender Singh

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].

Author(s):  
PHILIPPE MEYER

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.


2012 ◽  
Vol 11 (01) ◽  
pp. 1250011 ◽  
Author(s):  
MOHAMMAD REZA RISMANCHIAN ◽  
MEHDI ARASKHAN

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.


2019 ◽  
Vol 12 (02) ◽  
pp. 1950028
Author(s):  
Azam K. Mousavi ◽  
Mohammad Reza R. Moghaddam ◽  
Mehdi Eshrati

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.


1967 ◽  
Vol 19 ◽  
pp. 1250-1258 ◽  
Author(s):  
Franklin Haimo

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.


2018 ◽  
Vol 28 (01) ◽  
pp. 115-131 ◽  
Author(s):  
V. Metaftsis ◽  
A. I. Papistas ◽  
I. Sevaslidou

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].


2021 ◽  
Vol 52 ◽  
Author(s):  
Farshid Saeedi ◽  
Nafiseh Akbarossadat

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 $L$.


2020 ◽  
Vol 12 (2) ◽  
pp. 311-316
Author(s):  
O. Bezushchak

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.


2011 ◽  
Vol 08 (05) ◽  
pp. 929-935 ◽  
Author(s):  
M. SHAFIEE

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).


2016 ◽  
Vol 23 (01) ◽  
pp. 129-136
Author(s):  
Yongjie Wang ◽  
Yiqian Shi ◽  
Yun Gao

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.


2015 ◽  
Vol 22 (04) ◽  
pp. 677-686
Author(s):  
Fang Li ◽  
Dezhan Tan

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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