INTERNAL NATURAL TRANSFORMATIONS AND FROBENIUS ALGEBRAS IN THE DRINFELD CENTER

For ℳ and $$ \mathcal{N} $$ N finite module categories over a finite tensor category $$ \mathcal{C} $$ C , the category $$ \mathrm{\mathcal{R}}{ex}_{\mathcal{C}} $$ ℛ ex C (ℳ, $$ \mathcal{N} $$ N ) of right exact module functors is a finite module category over the Drinfeld center $$ \mathcal{Z} $$ Z ($$ \mathcal{C} $$ C ). We study the internal Homs of this module category, which we call internal natural transformations. With the help of certain integration functors that map $$ \mathcal{C} $$ C -$$ \mathcal{C} $$ C -bimodule functors to objects of $$ \mathcal{Z} $$ Z ($$ \mathcal{C} $$ C ), we express them as ends over internal Homs and define horizontal and vertical compositions. We show that if ℳ and $$ \mathcal{N} $$ N are exact $$ \mathcal{C} $$ C -modules and $$ \mathcal{C} $$ C is pivotal, then the $$ \mathcal{Z} $$ Z ($$ \mathcal{C} $$ C )-module $$ \mathrm{\mathcal{R}}{ex}_{\mathcal{C}} $$ ℛ ex C (ℳ, $$ \mathcal{N} $$ N ) is exact. We compute its relative Serre functor and show that if ℳ and $$ \mathcal{N} $$ N are even pivotal module categories, then $$ \mathrm{\mathcal{R}}{ex}_{\mathcal{C}} $$ ℛ ex C (ℳ, $$ \mathcal{N} $$ N ) is pivotal as well. Its internal Ends are then a rich source for Frobenius algebras in $$ \mathcal{Z} $$ Z ($$ \mathcal{C} $$ C ).

On the Radical of a Hecke–Kiselman Algebra

The Hecke-Kiselman algebra of a finite oriented graph Θ over a field K is studied. If Θ is an oriented cycle, it is shown that the algebra is semiprime and its central localization is a finite direct product of matrix algebras over the field of rational functions K(x). More generally, the radical is described in the case of PI-algebras, and it is shown that it comes from an explicitly described congruence on the underlying Hecke-Kiselman monoid. Moreover, the algebra modulo the radical is again a Hecke-Kiselman algebra and it is a finite module over its center.

An Application of Collapsing Levels to the Representation Theory of Affine Vertex Algebras

We discover a large class of simple affine vertex algebras $V_{k} ({\mathfrak{g}})$, associated to basic Lie superalgebras ${\mathfrak{g}}$ at non-admissible collapsing levels $k$, having exactly one irreducible ${\mathfrak{g}}$-locally finite module in the category ${\mathcal O}$. In the case when ${\mathfrak{g}}$ is a Lie algebra, we prove a complete reducibility result for $V_k({\mathfrak{g}})$-modules at an arbitrary collapsing level. We also determine the generators of the maximal ideal in the universal affine vertex algebra $V^k ({\mathfrak{g}})$ at certain negative integer levels. Considering some conformal embeddings in the simple affine vertex algebras $V_{-1/2} (C_n)$ and $V_{-4}(E_7)$, we surprisingly obtain the realization of non-simple affine vertex algebras of types $B$ and $D$ having exactly one nontrivial ideal.

On FGDF-modules

Let R be a unital ring and M a unitary module not necessary over R. The FGDF-module is a generalization of FGDF-rings (Touré, Diop, Mohamed and Sangharé, 2014). In this work, we first give some properties of FGDF-modules. After that, we show that for a finitely generated module M, M is a FGDF-module if and only if M is of finite representation type module. Finally, we show that M is a finitely generated FGDF-module if and only if every Dedekind finite module of $\sigma[M]$ is noetherian.

A Note on the Vanishing of Certain Local Cohomology Modules

For a finite module M over a local, equicharacteristic ring (R, m), we show that the well-known formula cd(m,M) = dim M becomes trivial if ones uses Matlis duals of local cohomology modules together with spectral sequences. We also prove a new ring-theoretic vanishing criterion for local cohomology modules.

Gröbner bases for quadratic algebras of skew type

Non-degenerate monoids of skew type are considered. This is a class of monoids S defined by n generators and $\binom{n}{2}$ quadratic relations of certain type, which includes the class of monoids yielding set-theoretic solutions of the quantum Yang–Baxter equation, also called binomial monoids (or monoids of I-type with square-free defining relations). It is shown that under any degree-lexicographic order on the associated free monoid FMn. of rank n the set of normal forms of elements of S is a regular language in FMn. As one of the key ingredients of the proof, it is shown that an identity of the form xN yN = yN xN holds in S. The latter is derived via an investigation of the structure of S viewed as a semigroup of matrices over a field. It also follows that the semigroup algebra K[S] is a finite module over a finitely generated commutative subalgebra of the form K[A] for a submonoid A of S.