Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userExact sequences in the cohomology of a Lie superalgebra extension
76
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Let $ 0rightarrow mathfrak{a} rightarrow mathfrak{e} rightarrow mathfrak{g} rightarrow 0$ be an abelian extension of the Lie superalgebra $mathfrak{g}$. In this article we consider the problems of extending endomorphisms of $mathfrak{a}$ and lifting endomorphisms of $mathfrak{g}$ to certain endomorphisms of $mathfrak{e}$. We connect these problems to the cohomology of $mathfrak{g}$ with coefficients in $mathfrak{a}$ through construction of two exact sequences, which is our main result, involving various endomorphism groups and the second cohomology. The first exact sequence is obtained using the Hochschild-Serre spectral sequence corresponding to the above extension while to prove the second we rather take a direct approach. As an application of our results we obtain descriptions of certain automorphism groups of semidirect product Lie superalgebras.
rate research
Read More
For some exact monoidal categories, we describe explicitly a connection between topological and algebraic definitions of the Lie bracket on the extension algebra of the unit object. The topological definition, due to Schwede and Hermann, involves loops in extension categories, and the algebraic definition involves homotopy liftings as introduced by the first author. As a consequence of our description, we prove that the topological definition indeed yields a Gerstenhaber algebra structure in the monoidal category setting, answering a question of Hermann. For use in proofs, we generalize $A_{infty}$-coderivation and homotopy lifting techniques from bimodule categories to some exact monoidal categories.
For a simple Lie algebra, over $mathbb{C}$, we consider the weight which is the sum of all simple roots and denote it $tilde{alpha}$. We formally use Kostants weight multiplicity formula to compute the dimension of the zero-weight space. In type $A_r$, $tilde{alpha}$ is the highest root, and therefore this dimension is the rank of the Lie algebra. In type $B_r$, this is the defining representation, with dimension equal to 1. In the remaining cases, the weight $tilde{alpha}$ is not dominant and is not the highest weight of an irreducible finite-dimensional representation. Kostants weight multiplicity formula, in these cases, is assigning a value to a virtual representation. The point, however, is that this number is nonzero if and only if the Lie algebra is classical. This gives rise to a new characterization of the exceptional Lie algebras as the only Lie algebras for which this value is zero.
Suppose the ground field to be algebraically closed and of characteristic different from $2$ and $3$. All Heisenberg Lie superalgebras consist of two sup
We consider the extension problem for Lie algebroids over schemes over a field. Given a locally free Lie algebroid Q over a scheme (X,O), and a sheaf of finitely generated Lie O-algebras L, we determine the obstruction to the existence of extensions 0 --> L --> E --> Q --> 0, and classify the extensions in terms of a suitable Lie algebroid hypercohomology group. In the preliminary sections we study free Lie algebroids and recall some basic facts about Lie algebroid hypercohomology.
In this paper, all (super)algebras are over a field $mathbb{F}$ of characteristic different from $2, 3$. We construct the so-called 5-sequences of cohomology for central extensions of a Lie superalgebra and prove that they are exact. Then we prove that the multipliers of a Lie superalgebra are isomorphic to the second cohomology group with coefficients in the trivial module for the Lie superalgebra under consideration.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
