On the structure of graded Lie superalgebras

Abstract

We study the structure of graded Lie superalgebras with arbitrary dimension and over an arbitrary field $\mathbb{K}$. We show that any of such algebras $\mathfrak{L}$ with a symmetric $G$-support is of the form $\mathfrak{L}=U+\sum _j{I}_j$ with $U$ a subspace of ${\mathfrak{L}}_1$ and any $I_j$ a well described graded ideal of $\mathfrak{L}$, satisfying $[I_j,I_k]=0$ if $j\ne k$. Under certain conditions, it is shown that $\mathfrak{L}=(\underset{k\in K}{⨁}{I}_k)\oplus (\underset{q\in Q}{⨁}{I}_q),$ where any $I_k$ is a gr-simple graded ideal of $\mathfrak{L}$ and any $I_q$ a completely determined low dimensional non gr-simple graded ideal of $\mathfrak{L}$, satisfying $[I_q,I_{q^{\prime }}]=0$ for any $q^{\prime }\in Q$ with $q\ne q^{\prime }$.

References in corpus (2)

• Quasifinite representations of the Lie superalgebra of quantum pseudo differential operators
• Embedding of the Lie superalgebra $D(2,1;\alpha )$ into the Lie superalgebra of pseudodifferential symbols on $S^{1|2}$