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.

In this paper we use a homological approach to obtain upper bounds for a few homological invariants of FI_G-modules V. These upper bounds are expressed in terms of the generating degree and torsion degree, which measure the top and socle of V under actions of non-invertible morphisms in the category respectively.

In this paper we describe several characterizations of basic finite-dimensional k-algebras A stratified for all linear orders, and classify their graded algebras as tensor algebras satisfying some extra property. We also discuss whether for a given preorder ≼, F(≼Δ), the category of A-modules with ≼Δ-filtrations, is closed under cokernels of monomorphisms, and classify quasi-hereditary algebras satisfying this property.