Algebra Colloquium

In [Frobenius Categories Versus Brauer Blocks, Progress in Math., 274] and in [Ordinary Grothendieck groups of a Frobenius $P$-category, Algebra Colloq. 18 (2011) 1–76], we consider suitable inverse limits of Grothendieck groups of categories of modules in characteristics $p$ and zero, obtained from a folded Frobenius $P$-category$(\mathcal{F},{\widehat{\mathfrak{a}\mathfrak{u}\mathfrak{t}}}_{{\mathcal{F}}^{\mathrm{sc}}})$, which covers the case of the Frobenius$P$-categories associated with blocks; moreover, in [Beyond a question of Markus Linckelmann, arxiv.org/abs/1507.04278] we show that a folded Frobenius $P$-categoryis actually equivalent to the choice of a regular central $k^{\ast }$-extension ${\hat{\mathcal{F}}}^{\mathrm{sc}}$ of ${\mathcal{F}}^{\mathrm{sc}}$. Here, taking advantage of the existence of a perfect ${\mathcal{F}}^{\mathrm{sc}}$ -locality ${\mathcal{P}}^{\mathrm{sc}}$, we exhibit those inverse limits as the true Grothendieck groups of the categories of ${\mathcal{K}}_{\ast }\hat{\mathcal{G}}$-and $k_{\ast }\hat{\mathcal{G}}$-modules for a suitable $k^{\ast }$-group $\hat{\mathcal{G}}$ associated to the $k^{\ast }$-category ${\hat{\mathcal{P}}}^{\mathrm{sc}}$ obtained from ${\mathcal{P}}^{\mathrm{sc}}$ and ${\hat{\mathcal{F}}}^{\mathrm{sc}}$. It depends on a vanishing cohomology result, given with more generality in the Appendix.

An abelian group $A$ is called a $TI$-group if every associative ring with the additive group $A$ is filial. The filiality of a ring $R$ means that the ring $R$ is associative and all ideals of any ideal of $R$ are ideals in $R$. In this paper, torsion-free $TI$-groups are described up to the structure of associative nil groups. It is also proved that, for torsion-free abelian groups that are not associative nil, the condition $TI$ implies the indecomposability and homogeneity. The paper contains constructions of $2^{{\aleph }_0}$ such groups of any rank from 2 to$2^{{\aleph }_0}$ which are pairwise non-isomorphic.

Let $R$ be a ring with an automorphism $𝜑$ of order two. We introduce the definition of $𝜑$-centrosymmetric matrices. Denote by $M_n(R)$ the ring of all $n\times n$ matrices over $R$, and by $S_n(𝜑,R)$ the set of all $𝜑$-centrosymmetric $n\times n$ matrices over $R$ for any positive integer $n$. We show that $S_n(𝜑,R)\subseteq M_n(R)$ is a separable Frobenius extension. If $R$ is commutative, then $S_n(𝜑,R)$ is a cellular algebra over the invariant subring $R^{𝜑}$ of $R$.

In this work we find the groups of Hochschild cohomologies for the Chinese monoid algebra and derive its Hilbert and Poincaré series. In order to obtain this result we construct the Anick resolution via the algebraic discrete Morse theory and Gröbner–Shirshov basis for the Chinese monoid.

We study a class of super W-algebras whose even part is the Virasoro type Lie algebra $W(2,2)$. We quantize the Lie superbialgebra of the super W-algebra by the Drinfeld twist quantization technique and obtain a class of noncommutative and noncocommutative Hopf superalgebras.

Let $\mathrm{\Delta }$ be a simplicial complex on $[n]$. The $\mathcal{NF}$-complex of $\mathrm{\Delta }$ is the simplicial complex ${\delta }_{\mathcal{NF}}(\mathrm{\Delta })$ on $[n]$ for which the facet ideal of $\mathrm{\Delta }$ is equal to the Stanley–Reisner ideal of ${\delta }_{\mathcal{NF}}(\mathrm{\Delta })$. Furthermore, for each $k=2,3,\dots$, we introduce the $k$th $\mathcal{NF}$-complex ${\delta }_{\mathcal{NF}}^{(k)}(\mathrm{\Delta })$, which is inductively defined as ${\delta }_{\mathcal{NF}}^{(k)}(\mathrm{\Delta })={\delta }_{\mathcal{NF}}{\delta }_{\mathcal{NF}}^{(k-1)}(\mathrm{\Delta })$ by setting ${\delta }_{\mathcal{NF}}^{(1)}(\mathrm{\Delta })={\delta }_{\mathcal{NF}}(\mathrm{\Delta })$. One can set ${\delta }_{\mathcal{NF}}^{(0)}(\mathrm{\Delta })=\mathrm{\Delta }$. The $\mathcal{NF}$-number of $\mathrm{\Delta }$ is the smallest integer $k>0$ for which ${\delta }_{\mathcal{NF}}^{(k)}(\mathrm{\Delta })\simeq \mathrm{\Delta }$. In the present paper we are especially interested in the $\mathcal{NF}$-number of a finite graph, which can be regraded as a simplicial complex of dimension one. It is shown that the $\mathcal{NF}$-number of the finite graph $K_n\bigsqcup K_m$ on $[n+m]$, which is the disjoint union of the complete graphs $K_n$ on $[n]$ and $K_m$ on $[m]$ for $n\ge 2$ and $m\ge 2$ with $(n,m)\ne (2,2)$, is equal to $n+m+2$. As a corollary, we find that the $\mathcal{NF}$-number of the complete bipartite graph $K_{n,m}$ on $[n+m]$ is also equal to $n+m+2$.

We introduce a generalization of the Gorenstein injective modules: the Gorenstein ${\mathrm{FP}}_n$-injective modules (denoted by ${\mathcal{GI}}_n$). They are the cycles of the exact complexes of injective modules that remain exact when we apply a functor $\mathrm{Hom}(A,-)$, with $A$ any ${\mathrm{FP}}_n$-injective module. Thus, ${\mathcal{GI}}_0$ is the class of classical Gorenstein injective modules, and ${\mathcal{GI}}_1$ is the class of Ding injective modules. We prove that over any ring $R$, for any $n\ge 2$, the class ${\mathcal{GI}}_n$ is the right half of a perfect cotorsion pair, and therefore it is an enveloping class. For $n=1$ we show that ${\mathcal{GI}}_1$ (i.e., the Ding injectives) forms the right half of a hereditary cotorsion pair. If moreover the ring $R$ is coherent, then the Ding injective modules form an enveloping class. We also define the dual notion, that of Gorenstein ${\mathrm{FP}}_n$-projectives (denoted by ${\mathcal{GI}}_n$). They generalize the Ding projective modules, and so, the Gorenstein projective modules. We prove that for any$n\ge 2$ the class ${\mathcal{GP}}_n$ is the left half of a complete hereditary cotorsion pair, and therefore it is special precovering.

Let $\mathfrak{g}$ be a finite dimensional complex simple Lie algebra with Cartan subalgebra $\mathfrak{h}$. Then $\mathbb{C}[\mathfrak{h}]$ has a $\mathfrak{g}$-module structure if and only if $\mathfrak{g}$ is of type $A$ or of type $C$; this is called the polynomial module of rank one. In the quantum version, the rank one polynomial modules over $U_q({\mathfrak{sl}}_2)$ have been classified. In this paper, we prove that the quantum group $U_q({\mathfrak{sl}}_3)$ has no rank one polynomial module.

We show that the second power of the cover ideal of a path graph has linear quotients. To prove our result we construct a recursively defined order on the generators of the ideal which yields linear quotients. Our construction has a natural generalization to the larger class of chordal graphs. This generalization allows us to raise some questions that are related to some open problems about powers of cover ideals of chordal graphs.

We study some homological properties of Gorenstein ${\mathrm{FP}}_{\infty }$-injective modules, and we prove (1) a ring $R$is not necessarily coherent if every Gorenstein ${\mathrm{FP}}_{\infty }$-injective $R$-module is injective, and (2) a ring $R$ is not necessarily coherent if every Gorenstein injective $R$-module is injective. In addition, we characterize $w$-Noetherian rings in terms of Gorenstein ${\mathrm{FP}}_{\infty }$-injective modules, and we prove that a ring $R$ is $w$-Noetherian if and only if every GV-torsion-free FP-injective $R$-module is Gorenstein ${\mathrm{FP}}_{\infty }$-injective, if and only if any direct sum of GV-torsion-free FP-injective $R$-modules is Gorenstein ${\mathrm{FP}}_{\infty }$-injective.

Let $p$ be a prime. In this paper, a complete classification of edge-transitive $N$-covers of a cubic symmetric graph of order $2p$ is given for the case when $N$ is a two-generator 2-group whose derived subgroup is either isomorphic to ${\mathbb{Z}}_2^3$ or generated by at most two elements. As an application, it is shown that 11 is the smallest value of $n$ for which there exist infinitely many cubic semisymmetric graphs with order of the form $2^{n}p$.