Harer has shown that the mapping class group is a virtual duality groupmirroring the work of Borel-Serre on arithmetic groups in semisimple Q-groups. Just as the homology of the rational Tits building provides the dualizing module for any torsion-free arithmetic group, the homology of the curve complex is the dualizingmodule for any torsion-free, finite index subgroup of the mapping class group...

In this paper we show that a direct decomposition of modules M N, with N homologically independent to the inJective hull of H, is a CS-module if and only if N is injective relative to H and both of M and N are CS-modules. As an application, we prove that a direct sum of a non-singular semisimple module and a quasi-continuous module with zero socle is quasi-continuous. This result is known for q...

Let A be an associative algebra over a field K such that the dimension, [A:P], of A over K is finite. Following the terminology of [8], we shall say that A is of bounded module type if there exists an integer re such that for every indecomposable left A-module M, the inequality [M: K] zí re holds. The algebra A is of finite module type if A has only a finite number of nonisomorphic indecomposab...

Abstract We study a semisimple extension of Takiff superalgebra, which turns out to have remarkably rich representation theory. determine the blocks in both finite-dimensional and BGG module categories also classify Borel subalgebras. further compute all groups between two simple objects prove that non-principal category are Koszul.

In this paper, we introduce and investigate the notion of projection invariant semisimple modules. Some structural properties aforementioned class modules are studied. We obtain indecomposable decompositions former under some module theoretical conditions. Moreover, explore when finite exchange property implies full for Finally, that endomorphism ring a is ?- Baer ring.

For the affine vertex algebra $V_k(\mathfrak{g})$ at an admissible level $k$ of $\hat{\mathfrak{g}}$, we prove that certain subcategory weak $V_k(\mathfrak{g})$-module category is semisimple. As a consequence, show rational with respect to family Virasoro elements. We also operator superalgebras and minimal $W$-algebras are

Let G be a semisimple simply connected algebraic group over an algebraically closed field of characteristic 0. Let V be a simple finite-dimensional G-module and let y be its highest weight vector. It is a classical result of B. Kostant that the algebra of functions on the closure of G · y is quadratic. In this paper we generalize this result to the case of the quantum group Uq(g). The proof use...

We give a sufficient criterion on a highest weight module of a semisimple Lie algebra to admit a resolution in terms of sums of modules induced from a parabolic subalgebra. In particular, we show that all unitary highest weight modules admit such a resolution. As an application of our results we compute (minimal) resolutions and explicit formulas for the Hilbert series of the unitary highest we...

Let L be a finite-dimensional, semisimple Lie algebra over an algebraically closed field k of characteristic 0. Let H be a fixed Cartan subalgebra of L, and Φ be the root system. Fix a base ∆ = {α1, · · · , αl} of Φ. Let Λ denote the set of dominant, integral linear functions on H. Theorem 0.1. There is a one-to-one correspondence Λ ∼ −→ {isomorphism classes of finite-dimensional irreducible L-...