Algebra Colloquium

Let $G$ be a classical group over an arbitrary field $F$, acting on an $n$-dimensional $F$-space $V=V(n,F)$. All those maximal subgroups of $G$ are classified each of which normalizes a solvable subgroup $N$ of $\mathrm{GL}(V/F)$ not lying in $F^{\ast }{1}_V$.

We study the quadratic quotients of the incidence category of the Young lattice defined by the zero relations corresponding to adding two boxes to the same row, or to the same column, or both. We show that the last quotient corresponds to the Koszul dual of the original incidence category, while the first two quotients are, in a natural way, Koszul duals of each other and hence they are in particular Koszul self-dual. Both of these two quotients are known to be basic representatives in the Morita equivalence class of the category of injections between finite sets. We also present a new, rather direct, argument establishing this Morita equivalence.

We introduce the notions of a four-angle Hopf quasimodule and an adjoint quasiaction over a Hopf quasigroup $H$ in a symmetric monoidal category $\mathcal{C}$. If $H$ possesses an adjoint quasiaction, we show that symmetric Yetter-Drinfeld categories are trivial, and hence we obtain a braided monoidal category equivalence between the category of right Yetter-Drinfeld modules over $H$ and the category of four-angle Hopf modules over $H$ under some suitable conditions.

A graph $\Gamma$ is said to be symmetric if its automorphism group $\mathrm{Aut}(\Gamma )$ acts transitively on the arc set of $\Gamma$. We show that if $\Gamma$ is a finite connected heptavalent symmetric graph with solvable stabilizer admitting a vertex-transitive non-abelian simple group $G$ of automorphisms, then either $G$ is normal in $\mathrm{Aut}(\Gamma )$, or $\mathrm{Aut}(\Gamma )$ contains a non-abelian simple normal subgroup $T$ such that $G\le T$ and $(G,T)$ is explicitly given as one of 11 possible exceptional pairs of non-abelian simple groups. If $G$ is arc-transitive, then $G$ is always normal in $\mathrm{Aut}(\Gamma )$, and if $G$ is regular on the vertices of $\Gamma$, then the number of possible exceptional pairs $(G,T)$ is reduced to 5.

We provide some exact formulas for the projective dimension and regularity of edge ideals associated to some vertex-weighted oriented cyclic graphs with a common vertex or edge. These formulas are functions in the weight of the vertices, and the numbers of edges and cycles. Some examples show that these formulas are related to direction selection and the assumption that $w(x)\ge 2$ for any vertex $x$ cannot be dropped.

Let $\mathbb{k}$ be a fixed algebraically closed field of arbitrary characteristic, let $\Lambda$ be a finite dimensional self-injective $\mathbb{k}$-algebra, and let $V$ be an indecomposable non-projective left $\Lambda$-module with finite dimension over $\mathbb{k}$. We prove that if ${\tau }_{\Lambda }V$ is the Auslander–Reiten translation of $V$, then the versal deformation rings $R(\Lambda ,V)$ and $R(\Lambda ,{\tau }_{\Lambda }V)$ (in the sense of F.M. Bleher and the second author) are isomorphic. We use this to prove that if $\Lambda$ is further a cluster-tilted $\mathbb{k}$-algebra, then $R(\Lambda ,V)$ is universal and isomorphic to $\mathbb{k}$.

We describe all degenerations of the variety ${\mathfrak{J}\mathfrak{o}\mathfrak{r}\mathfrak{d}}_3$ of Jordan algebras of dimension three over $\mathbb{C}$. In particular, we describe all irreducible components in ${\mathfrak{J}\mathfrak{o}\mathfrak{r}\mathfrak{d}}_3$. For every $n$ we define an $n$-dimensional rigid “marginal” Jordan algebra of level one. Moreover, we discuss marginal algebras in associative, alternative, left alternative, non-commutative Jordan, Leibniz and anticommutative cases.

Let $F$ be a field, and let $e,k$ be integers such that $1\le e\le |F\setminus \{0\}|$ and $k\ge 0$. We show that for any subset $\{a_1,\dots ,a_e\}\subseteq F\setminus \{0\}$, the curious identity

A Cayley graph $𝛤=\mathrm{Cay}(G,S)$ is said to be normal if $G$ is normal in $\mathrm{Aut}𝛤$. In this paper, we investigate the normality problem of the connected 11-valent symmetric Cayley graphs $𝛤$ of finite nonabelian simple groups $G$, where the vertex stabilizer ${\mathrm{A}}_v$ is soluble for $A=\mathrm{Aut}𝛤$ and $v\in V𝛤$. We prove that either $𝛤$ is normal or $G={\mathrm{A}}_5$, ${\mathrm{A}}_{10}$, ${\mathrm{A}}_{54}$, ${\mathrm{A}}_{274}$, ${\mathrm{A}}_{549}$ or ${\mathrm{A}}_{1099}$. Further, 11-valent symmetric nonnormal Cayley graphs of ${\mathrm{A}}_5$, ${\mathrm{A}}_{54}$ and ${\mathrm{A}}_{274}$ are constructed. This provides some more examples of nonnormal 11-valent symmetric Cayley graphs of finite nonabelian simple groups after the first graph of this kind (of valency 11) was constructed by Fang, Ma and Wang in 2011.

Let $W(\Gamma )$ be a class of not-finitely graded Lie algebras related to generalized Virasoro algebras with basis $\{L_{\alpha ,i},C | \alpha \in \Gamma , i\in {\mathbb{Z}}_{+}\}$, which satisfies relations $[L_{\alpha ,i},L_{\beta ,j}]=(\beta -\alpha )L_{\alpha +\beta ,i+j}+(j-i)L_{\alpha +\beta ,i+j+1}+{\delta }_{\alpha +\beta ,0}{\delta }_{i+j,0}\frac{{\alpha }^3-\alpha }{12}C$ and $[C,L_{\alpha ,i}]=0$. In this paper, $W(\Gamma )$-modules of the intermediate series satisfying some conditions are constructed and classified. We also obtain modules of the intermediate series over the related Lie superalgebra.

In this article we investigate the relations between the Gorenstein projective dimensions of $\Lambda$-modules and their socles for $n$-minimal Auslander–Gorenstein algebras $\Lambda$. First we give a description of projective-injective $\Lambda$-modules in terms of their socles. Then we prove that a $\Lambda$-module $N$ has Gorenstein projective dimension at most $n$ if and only if its socle has Gorenstein projective dimension at most $n$ if and only if $N$ is cogenerated by a projective $\Lambda$-module. Furthermore, we show that $n$-minimal Auslander–Gorenstein algebras can be characterised by the relations between the Gorenstein projective dimensions of modules and their socles.

Let $H$ be a finite-dimensional pointed Hopf algebra of rank one over an algebraically closed field of characteristic zero. In this paper we show that any finite-dimensional indecomposable $H$-module is generated by one element. In particular, any indecomposable submodule of $H$ under the adjoint action is generated by a special element of $H$. Using this result, we show that the Hopf algebra $H$ is a principal ideal ring, i.e., any two-sided ideal of $H$ is generated by one element. As an application, we give explicitly the generators of ideals, primitive ideals, maximal ideals and completely prime ideals of the Taft algebras.