Cohomology of simple modules for sl3(k) in characteristic 3

In this paper we calculate cohomology of a classical Lie algebra of type A2 over an algebraically field k of characteristic p = 3 with coefficients in simple modules. To describe their structure we will consider them as modules over an algebraic group SL3(k). In the case of characteristic p = 3, there are only two peculiar simple modules: a simple that module isomorphic to the quotient module of the adjoint module by the center, and a one-dimensional trivial module. The results on the cohomology of simple nontrivial module are used for calculating the cohomology of the adjoint module. We also calculate cohomology of the simple quotient algebra Lie of A2 by the center.

Homology and Cohomology of the Super Schrödinger Algebra S(1/1) with Coefficients in the Trivial Module

In this paper we study the homology and cohomology groups of the super Schrödinger algebra [Formula: see text] in (1 + 1)-dimensional spacetime. We explicitly compute the homology groups of [Formula: see text] with coefficients in the trivial module. Then using duality, we finally obtain the dimensions of the cohomology groups of [Formula: see text] with coefficients in the trivial module.

Gröbner-Shirshov Basis and Minimal Projective Resolution of Uq+(B2)

In this paper, by using the Anick resolution and Gröbner-Shirshov basis for quantized enveloping algebra of type B2, we compute the minimal projective resolution of the trivial module of [Formula: see text], and as an application we compute the global dimension of [Formula: see text].

The Projective Cover of the Trivial Representation for a Finite Group of Lie Type in Defining Characteristic

We give a lower bound of the Loewy length of the projective cover of the trivial module for the group algebra kG of a finite group G of Lie type defined over a finite field of odd characteristic p, where k is an arbitrary field of characteristic p. The proof uses Auslander-Reiten theory.

The MCM-approximation of the trivial module over a category algebra

For a finite free EI category, we construct an explicit module over its category algebra. If in addition the category is projective over the ground field, the constructed module is a maximal Cohen–Macaulay approximation of the trivial module and is the tensor identity of the stable category of Gorenstein-projective modules over the category algebra. We give conditions on when the trivial module is Gorenstein-projective.

Ghosts and Strong Ghosts in the Stable Category

Suppose that G is a finite group and k is a field of characteristic p > 0. A ghost map is a map in the stable category of finitely generated kG-modules which induces the zero map in Tate cohomology in all degrees. In an earlier paper we showed that the thick subcategory generated by the trivial module has no nonzero ghost maps if and only if the Sylow p-subgroup of G is cyclic of order 2 or 3. In this paper we introduce and study variations of ghost maps. In particular, we consider the behavior of ghost maps under restriction and induction functors. We find all groups satisfying a strong form of Freyd’s generating hypothesis and show that ghosts can be detected on a finite range of degrees of Tate cohomology. We also consider maps that mimic ghosts in high degrees.

On cohomology of a class of nonclassical restricted simple Lie algebras

In this note, we study the cohomology of the nonclassical restricted Lie algebras [Formula: see text], [Formula: see text] over a field [Formula: see text] of characteristic [Formula: see text], which are by definition the Lie algebras of derivations on the truncated polynomial algebras [Formula: see text] (called the Jacobson–Witt algebras), and simple unless [Formula: see text] and [Formula: see text]. We show a vanishing theorem for the cohomology of [Formula: see text] with coefficients in the trivial module [Formula: see text], which says that [Formula: see text] for all [Formula: see text]. Moreover, we compute the cohomology of [Formula: see text] with coefficients in its defining truncated polynomial algebra under a certain assumption on the characteristic of the ground field.

GENERIC JORDAN TYPE OF THE SYMMETRIC AND EXTERIOR POWERS

We prove a result relating the stable generic Jordan types of the symmetric and exterior powers of the Heller translations of a module for a finite elementary abelian p-group. In the case of the trivial module, the stable generic Jordan types of the symmetric and exterior powers of its Heller translations are completely described.