Related papers: On homomorphisms between Weyl modules: The case of…

The moduli space for a flat G-bundle over the two-torus is completely determined by its holonomy representation. When G is compact, connected, and simply connected, we show that the moduli space is homeomorphic to a product of two tori mod…

We show that a matrix is a Hermitian positive semidefinite matrix whose nonzero entries have modulus 1 if and only if it similar to a direct sum of all $1's$ matrices and a 0 matrix via a unitary monomial similarity. In particular, the only…

Let $\Sigma_r$ be the symmetric group acting on $r$ letters, $K$ be a field of characteristic 2 and $\lambda$ and $\mu$ be partitions of $r$ in at most two parts. Denote the permutation module corresponding to the Young subgroup…

We study five different types of the homology of a Lie algebra over a commutative ring which are naturally isomorphic over fields. We show that they are not isomorphic over commutative rings, even over $\mathbb Z,$ and study connections…

Let $\Delta^\lambda$ be the Weyl functor for the partition $\lambda$ and let $E$ be the natural $2$-dimensional representation of $\mathrm{SL}_2(\mathbb{F})$, where $\mathbb{F}$ is an arbitrary field. We give an explicit isomorphism showing…

We study noncommutative differential structures on the group of permutations $S_N$, defined by conjugacy classes. The 2-cycles class defines an exterior algebra $\Lambda_N$ which is a super analogue of the Fomin-Kirillov algebra $\CE_N$ for…

The representations of a quiver Q over a field k have been studied for a long time. It seems to be worthwhile to consider also representations of Q over arbitrary finite-dimensional k-algebras A. Here we draw the attention to the case when…

Modular symbols for the congruence subgroup $\Gamma_0(\mathfrak{n})$ of $GL_{2}(\mathbf{F}_q[T])$ have been defined by Teitelbaum. They have a presentation given by a finite number of generators and relations, in a formalism similar to…

Let $A_g$ be an abelian variety of dimension $g$ and $p$-rank $\lambda \leq 1$ over an algebraically closed field of characteristic $p>0$. We compute the number of homomorphisms from $\pi_1^{\text{\'et}}(A_g,a)$ to $GL_n(\mathbb F_q)$,…

The present paper deals with the representation theory of the reflection equation algebra, connected with a Hecke type R-matrix. Up to some reasonable additional conditions the R-matrix is arbitrary (not necessary originated from quantum…

Let $k$ be an algebraically closed field of characteristic $p>0$, let $R$ be a commutative ring, and let $\mathbb{F}$ be an algebraically closed field of characteristic 0. We consider the $R$-linear category $\mathcal{F}^\Delta_{Rpp_k}$ of…

Let $G$ be a finite group of order $n$ and let $M$ be a $G$-module. We construct groups $H_*^\varkappa(G,M)$ for which $H_k^\varkappa (G,M^{tw}) \cong H^{n-k-1}_\lambda(G,M),$ where $M^{tw}$ is a twisting of a $G$-module $M$ defined in…

For classical Lie superalgebras of type I, we provide necessary and sufficient conditions for a Verma supermodule $\Delta(\lambda)$ to be such that every non-zero homomorphism from another Verma supermodule to $\Delta(\lambda)$ is…

We show that for the family of complex reflection groups $G=G(m,p,2)$ appearing in the Shephard--Todd classification, the endomorphism ring of the reduced hyperplane arrangement $A(G)$ is a non-commutative resolution for the coordinate ring…

This is a further investigation of our approach to group actions in homological algebra in the settings of homology of {\Gamma}-simplicial groups, particularly of {\Gamma}-equivariant homology and cohomology of {\Gamma}-groups. This…

The irreducible components of varieties parametrizing the finite dimensional representations of a finite dimensional algebra $\Lambda$ are explored, with regard to both their geometry and the structure of the modules they encode. Provided…

Let $G$ be a connected reductive complex algebraic group with a maximal torus $T$. We denote by $\Lambda$ the cocharacter lattice of $(T,G)$. Let $\Lambda^+ \subset \Lambda$ be the submonoid of dominant coweights. For $\lambda \in…

We further develop a model unifying general relativity with quantum mechanics proposed in our earlier papers (J. Math. Phys. 38, 5840 (1998); 41, 5168 (2000)). The model is based on a noncommutative algebra $A$ defined on a groupoid $\Gamma…

For a commutative ring $A$, we have the category of (bounded-below) chain complexes of $A$-modules $Ch_{+}(A\mymod)$, a closed symmetric monoidal category with a compatible stable Quillen model structure. The associated homotopy category is…

Let $\mathbb{K}$ be an algebraically closed field of characteristic $0$. We study a monoidal category $\mathbb{T}_\alpha$ which is universal among all symmetric $\mathbb{K}$-linear monoidal categories generated by two objects $A$ and $B$…