We study $\mathbb{Z}_{2}$ -graded identities of simple Lie superalgebras over a field of characteristic zero. We prove the existence of the graded PI-exponent for such algebras.

Given an Artinian algebra $A$ over a field $k$, there are several combinatorial objects associated to $A$. They are the diagram $D_A$ as defined by Drozd and Kirichenko, the natural quiver $\Delta_A$ defined by Li (cf. Section 2), and a generalized version of $k$-species $(A/r, r/r^2)$ with $r$ being the Jacobson radical of $A$. When $A$ is splitting over the field $k$, the diagram $D_A$ and the well-known ext-quiver $\Gamma_A$ are the same. The main objective of this paper is to investigate the relations among these combinatorial objects and in turn to use these relations to give a characterization of the algebra $A$.

For smooth compact oriented Riemannian manifolds$M$of dimension 4$k$+2,$k$≥0, with or without boundary, and a vector bundle$F$on$M$with an inner product and a flat connection, we construct a modification of the Hodge star operator on the middle-dimensional (parabolic) cohomology of$M$twisted by$F$. This operator induces a canonical complex structure on the middle-dimensional cohomology space that is compatible with the natural symplectic form given by integrating the wedge product. In particular, when$k$=0 we get a canonical almost complex structure on the non-singular part of the moduli space of flat connections on a Riemann surface, with monodromies along boundary components belonging to fixed conjugacy classes when the surface has boundary, that is compatible with the standard symplectic form on the moduli space.