Automorphism theorem - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T04:27:31Z http://mathoverflow.net/feeds/question/23806 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/23806/automorphism-theorem Automorphism theorem Mamur 2010-05-07T01:45:46Z 2010-07-26T03:22:16Z <p>Help me please to find reference for the proof of the following theorem:</p> <p>Theorem. Let $\theta$ be a Leibniz cocycle on the Leibniz algebra L with values in $V,$ and assume $\theta^{\bot}\cap C(L)=(0).$ Then the automorphism group $Aut(L_{\theta})$ of the extension algebra $L_{\theta}$ consists of all linear operators of the matrix form <code>$$\left[ \begin{array}{lr} \alpha_0 &amp; 0 \\ \\ \phi &amp; \psi \\ \end{array} \right],$$</code></p> <p>where $\alpha_0 \in Aut(L), \ \psi = \alpha|{C(L_{\theta})} \in GL(k),$ and $\phi \in Hom(L, V),$ such that $\theta (\alpha_0x, \alpha_0y) = \phi[x,y] + \psi\theta(x,y),$ all $x, y \in L.$</p> <p>(L, [.,.]) is a Leibniz algebra over F </p> http://mathoverflow.net/questions/23806/automorphism-theorem/24895#24895 Answer by Pasha Zusmanovich for Automorphism theorem Pasha Zusmanovich 2010-05-16T15:57:24Z 2010-05-16T15:57:24Z <p>I doubt the reference for exactly such statement exists. The closest reference I am aware of is: D. Liu, L. Lin, On the toroidal Leibniz algebras, Acta Math. Sinica 24 (2008), N2, 227-240 DOI:10.1007/s10114-007-1003-z . There, in Propostion 5.1, they establish a relationship between automorphisms of a perfect Leibniz algebra and automorphisms of its universal central extension, and Corollary 5.3 appears to be a particular case of your statement. The arguments are quite straitforward, repeating almost verbatim the corresponding arguments in the case of Lie algebras due to A. Pianzola (see references in that paper), and, if true, it seems not to be difficult to extend them to your statement.</p>