On the structure of graded Lie superalgebras

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 $q \neq q'$.