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 ${ \mathfrak L} = U + \sum\limits_{ j}I_{ j}$ 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 $j\neq k$. Under certain conditions, it is shown that ${ \mathfrak L} = (\bigoplus\limits_{ k \in K} I_k) \oplus (\bigoplus\limits_{ q \in Q} I_q),$ 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 $[I_q,I_{ q'}] = 0$ for any $q'\in Q$ with $q \neq q'$.

Categories: math.RT

On the structure of graded Lie superalgebras