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...

In this paper, we study conjugacy classes for pivotal fusion categories. particular, prove a Burnside type formula the structure constants concerning product of two class sums such category. For braided weakly integral category C, show that these multiplied by dim ( C ) are non-negative integers, extending some results obtained Zhou and Zhu (see Preprint, 2019, arXiv:1912.07831v1) in settings s...

There are some new developments on Plancherel formula and growth of matrix coefficients for unitary representations of nilpotent Lie groups. These have several consequences for the geometry of weakly symmetric spaces and analysis on parabolic subgroups of real semisimple Lie groups, and to (infinite dimensional) locally nilpotent Lie groups. Many of these consequences are still under developmen...

Different versions of Wiener’s Tauberian theorem are discussed for the generalized group algebra LI(G,A) (of integrable functions on a locally compact abelian group G taking values in a commutative semisimple regular Banach algebra A) using A-valued Fourier transforms. A weak form of Wiener’s Tauberian property is introduced and it is proved that LI(G,A) is weakly Tauberian if and only if A is....

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...