Algebra Colloquium

A semigroup (S, ·) is called right (left) quasiresiduated if for any a, b in S there exists x in S such that ax ≤_S b (xa ≤_S b) with respect to the natural partial order ≤_S of S. This concept has its origin in the theory of residuated semigroups, but can also be seen as a generalization of the right (left) simplicity of semigroups. It is first studied for totally-, resp., trivially-ordered semigroups, and then for semigroups with idempotents. In particular, the cases when (S, ≤_S) is directed downwards and when S contains a zero (with respect to a more restrictive definition) are dealt with. Throughout, examples are given; in total, 30 classes of (often well-known) semigroups of this kind are specified.

In this paper, we construct some $1\frac{1}{2}$-designs, which are also known as partial geometric designs, using totally isotropic subspaces of the unitary space. Furthermore, these $1\frac{1}{2}$-designs yield six infinite families of directed strongly regular graphs.

The main purpose of this paper is to study generalized derivations in rings with involution which behave like strong commutativity preserving mappings. In fact, we prove the following result: Let R be a noncommutative prime ring with involution of the second kind such that char $\left(R\right)\ne 2$. If R admits a generalized derivation $F:R\to R$ associated with a derivation $d:R\to R$ such that $\left[F\right(x),F(x^{*}\left)\right]-[x,x^{*}]=0$ for all $x\in R$, then $F\left(x\right)=x$ for all $x\in R$ or $F\left(x\right)=-x$ for all $x\in R$. Moreover, a related result is also obtained.

We characterize pure lexsegment complexes which are Cohen-Macaulay in arbitrary codimension. More precisely, we prove that any lexsegment complex is Cohen-Macaulay if and only if it is pure and its 1-dimensional links are connected, and that a lexsegment flag complex is Cohen-Macaulay if and only if it is pure and connected. We show that any non-Cohen-Macaulay lexsegment complex is a Buchsbaum complex if and only if it is a pure disconnected flag complex. For $t\ge 2$, a lexsegment complex is strictly Cohen-Macaulay in codimension t if and only if it is the join of a lexsegment pure disconnected flag complex with a $(t-2)$-dimensional simplex. When the Stanley-Reisner ideal of a pure lexsegment complex is not quadratic, the complex is Cohen-Macaulay if and only if it is Cohen-Macaulay in some codimension. Our results are based on a characterization of Cohen-Macaulay and Buchsbaum lexsegment complexes by Bonanzinga, Sorrenti and Terai.

In this paper, rational extensions of affine vertex operator algebras L_sl3 (k, 0) with $k\in {\mathrm{\mathbb{Z}}}_{+}$ are classified by modular invariants.

We study a family of “symmetric” multiparameter quantized Weyl algebras ${\mathcal{A}}_n^{\overline{q},\text{Λ}}(\mathbb{K})$ and some related algebras. We compute the Nakayama automorphism of ${\mathcal{A}}_n^{\overline{q},\text{Λ}}(\mathbb{K})$, give a necessary and sufficient condition for ${\mathcal{A}}_n^{\overline{q},\text{Λ}}(\mathbb{K})$ to be Calabi-Yau, and prove that ${\mathcal{A}}_n^{\overline{q},\text{Λ}}(\mathbb{K})$ is cancellative. We study the automorphisms and isomorphism problem for ${\mathcal{A}}_n^{\overline{q},\text{Λ}}(\mathbb{K})$ and ${\mathcal{A}}_n^{\overline{q},\text{Λ}}\left(\mathbb{K}\right[t\left]\right)$. Similar results are established for the Maltsiniotis multiparameter quantized Weyl algebra ${\mathcal{A}}_n^{\overline{q},\text{Γ}}\left(\mathbb{K}\right)$ and its polynomial extension. We prove a quantum analogue of the Dixmier conjecture for a simple localization ${\left({\mathcal{A}}_n^{\overline{q},\text{Λ}}\right(\mathbb{K}\left)\right)}_{Ƶ}$ and determine its automorphism group.

We give a lower bound of the Loewy length of the projective cover of the trivial module for the group algebra kG of a finite group G of Lie type defined over a finite field of odd characteristic p, where k is an arbitrary field of characteristic p. The proof uses Auslander-Reiten theory.

A graph is symmetric if its automorphism group acts transitively on the set of arcs of the graph. We classify connected heptavalent symmetric graphs of order 16p for each prime p. As a result, there are two such sporadic graphs with p = 3 and 7, and an infinite family of 1-regular normal Cayley graphs on the group ${\mathbb{Z}}_2^3\times D_{2p}$ with 7|(p – 1).

In this paper we determine all tetravalent Cayley graphs of a non-abelian group of order 3p², where p is a prime number greater than 3, and with a cyclic Sylow p-subgroup. We show that all of these tetravalent Cayley graphs are normal. The full automorphism group of these Cayley graphs is given and the half-transitivity and the arc-transitivity of these graphs are investigated. We show that this group is a 5-CI-group.

In this article, we define the nil clean graph of a ring R. The vertex set is the ring R, and two ring elements a and b are adjacent if and only if a + b is nil clean in R. Graph theoretic properties like the girth, dominating sets, diameter, etc., of the nil clean graph are studied for finite commutative rings.

In this paper, by using Gröbner-Shirshov bases theories, we prove that each countably generated associative differential algebra (resp., associative Ω-algebra, associative λ-differential algebra) can be embedded into a simple 2-generated associative differential algebra (resp., associative Ω-algebra, associative λ-differential algebra).

Let (R, m) be a Cohen–Macaulay local ring of dimension d, C a canonical R-module and M an almost Cohen–Macaulay R-module of dimension n and of depth t. We prove that dim ${\mathrm{Ext}}_R^{d-n}(M\mathit{,}\mathit{ }C)=n$ and if $n\le 3$ then ${\mathrm{Ext}}_R^{d-n}(M\mathit{,}\mathit{ }C)$ is an almost Cohen–Macaulay R-module. In particular, if $n=d\le 3$ then Hom_R(M, C) is an almost Cohen–Macaulay R-module. In addition, with some conditions, we show that ${\mathrm{Ext}}_R^1(M\mathit{,}\mathit{ }C)$ is also almost Cohen–Macaulay. Finally, we study the vanishing ${\mathrm{Ext}}_R^i\left({\mathrm{Ext}}_R^{d\mathit{-}n}\right(M,C),C)$ and ${\mathrm{Ext}}_R^i\left({\mathrm{Ext}}_R^{d\mathit{-}t}\right(M,C),C)$.

q-Deformations of 3-Lie algebras and representations of q-3-Lie algebras are discussed. A q-3-Lie algebra $(A, [,,]q,[,,{]}_q^{\prime },J_q)$, where $q\in \mathbb{F}$ and $q\ne 0$, is a vector space A over a field 𝔽 with 3-ary linear multiplications [ , , ]_q and $[,,{]}_q^{\prime }$ from $A^{\otimes 3}$ to A, and a map $J_q:A^{\otimes 5}\to \mathbb{F}$ satisfying the q-Jacobi identity $J_q(x_1,x_2,x_3,x_4,x_5)[x_1,x_2,[x_3,x_4,x_5{]}_q{]}_q^{\prime }=J_q(x_4,x_5,x_1,x_2,x_3)[x_4,x_5,[x_1,x_2,x_3{]}_q{]}_q^{\prime }$ for all $x_i\in A$. If the multiplications satisfy that $[,,{]}_q=[,,{]}_q^{\prime }$ and $[,,{]}_q$ is skew-symmetry, then $(A,[,,{]}_q,J_q)$ is called a type (I)-q-3- Lie algebra. From q-Lie algebras, group algebras and commutative associative algebras, q-3-Lie algebras and type (I)-q-3-Lie algebras are constructed. Also, the non-trivial onedimensional central extension of q-3-Lie algebras is investigated, and new q-3-Lie algebras $\overline{({\mathrm{Der}}_{\mathit{q}} \mathrm{ℂ}[x,x^{\mathit{-}\mathit{1}}]},[,,{]}_{\mathit{q}\mathit{,}\mathit{ }}[,,{]}_q^{\prime },J_{\mathit{q}})$, and $\overline{({\mathrm{Der}}_{{\mathit{\delta }}_{\mathit{q}}} \mathrm{ℂ}[x,x^{\mathit{-}\mathit{1}}]},[,,{]}_{\mathit{q}\mathit{,}\mathit{ }}[,,{]}_q^{\prime },J_{\mathit{q}})$ are obtained.