Related papers: Construction of Rank $2$ Indecomposable Modules in…

A differential module is a module equipped with a square-zero endomorphism. This structure underpins complexes of modules over rings, as well as differential graded modules over graded rings. We establish lower bounds on the class--a…

Let $Q$ be a Dynkin quiver and $\Pi$ the corresponding set of positive roots. For the preprojective algebra $\Lambda$ associated to $Q$ we produce a rigid $\Lambda$-module $I_Q$ with $r=|\Pi|$ pairwise non-isomorphic indecomposable direct…

In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebra $G_{2(2)}$ which is split real form of $G_2$. We give the classification of reducible…

Let $G$ be a finite group and $k$ a field of characteristic $p$. We conjecture that if $M$ is a $kG$-module with $H^*(G,M)$ finitely generated as a module over $H^*(G,k)$ then as an element of the stable module category…

Let $G$ be a group and $\Bbbk$ a commutative ring. All categories and functors are assumed to be $\Bbbk$-linear. We define a $G$-invariant bimodule ${}_SM_R$ over $G$-categories $R, S$ and a $G$-graded bimodule ${}_BN_A$ over $G$-graded…

The parafermionic cosets $C_k = \mathrm{Com} (H, L_k(\mathfrak{sl}_2) )$ are studied for negative admissible levels $k$, as are certain infinite-order simple current extensions $B_k$ of $C_k$. Under the assumption that the tensor theory…

We discuss the structure of the Motzkin algebra $M_k(D)$ by introducing a sequence of idempotents and the basic construction. We show that $\cup_{k\geq 1}M_k(D)$ admits a factor trace if and only if $D\in \{2\cos(\pi/n)+1|n\geq 3\}\cup…

A question of Bergman asks whether the adjoint of the generic square matrix over a field can be factored nontrivially as a product of square matrices. We show that such factorizations indeed exist over any coefficient ring when the matrix…

A prototypical examples of a cluster algebra is the coordinate ring of a finite Grassmannian: using the Pl\"ucker embedding the cluster algebra structure allows one to move between `maximal sets' of algebraically independent Pl\"ucker…

We explore the indecomposable submodule structure of quantum Grassmann super-algebra $\Omega_q(m|n)$ and its truncated objects $\Omega_q(m|n,\textbf{r})$ in the case when $q=\varepsilon$ is an $\ell$-th root of unity. A net-like…

In this paper we study the mod $2$ cohomology ring of the Grasmannian $\widetilde{G}_{n,3}$ of oriented $3$-planes in $\mathbb{R}^n$. We determine the degrees of the indecomposable elements in the cohomology ring. We also obtain an almost…

Let $G$ be the Klein Four-group and let $k$ be an arbitrary field of characteristic 2. A classification of indecomposable $kG$-modules is known. We calculate the relative cohomology groups $H_\{chi}^i(G,N)$ for every indecomposable…

This paper studies modular forms of rank four and level one. There are two possiblities for the isomorphism type of the space of modular forms that can arise from an irreducible representation of the modular group of rank four, and we…

For a certain family of complete modular lattices, we prove a Jordan--H\"older--Scheier-like" theorem with no assumptions on cardinality or well-orderedness. This family includes both lattices which are both join- and meet-continuous, as…

Suppose that $Q$ is a connected quiver without oriented cycles and $\sigma$ is an automorphism of $Q$. Let $k$ be an algebraically closed field whose characteristic does not divide the order of the cyclic group $\langle\sigma\rangle$. The…

It's well known that the functional Hilbert space over the unit ball in $B_{d} \in C^d$, with kernel function $K(z,w)=\frac{1}{1-z_{1}w_{1}-... -z_{d}w_{d}}$, admits a natural $A(B_{d})$-module structure. We show the rank of a nonzero…

Let R be a ring and G a group. An R-module A is said to be artinian-by-(finite rank) if TorR(A) is artinian and A/TorR(A) has finite R-rank. The authors study ZG-modules A such that A/CA(H) is artinian-by-(finite rank) (as a Z-module) for…

We investigate the permutation modules associated to the set of $k$-dimensional faces of the hyperoctahedron in dimension $n$, denoted $H^{n}.$ For any $k\leq n$ such a module can be defined over an arbitrary field $F$, it is called a face…

Inspired by Rumin's work on a subcomplex in sub-Riemannian manifolds which is cohomologically equivalent to the de Rham complex, we present a more general construction that produces subcomplexes from any filtered cochain complex of finite…

We present a construction of a 2-Hilbert space of sections of a bundle gerbe, a suitable candidate for a prequantum 2-Hilbert space in higher geometric quantisation. We introduce a direct sum on the morphism categories in the 2-category of…