Algebra Colloquium

We first show that if there is a counterexample to the two-dimensional Jacobian conjecture, then there exists a nonempty closed smooth affine surface in ${\mathbb{C}}^2\times {\mathbb{C}}^2$ satisfying various properties. Next, we conjecture that such a surface does not exist.

Motivated by the concept of matching Rota-Baxter algebras arising from polarized associative Yang-Baxter equations and Volterra integral equations, we introduce the notion of a matching Rota-Baxter system, which generalizes the Rota-Baxter system proposed by Brzeziński. We show that this notion is also related to Yang-Baxter pairs and to matching pre-Lie algebras. We then modify the definition of matching Rota-Baxter systems by adding a curvature term, and make a connection with matching pre-Lie algebras and with compatible associative algebras. Furthermore, we study matching Rota-Baxter systems on a dendriform algebra and show how they induce matching quadri-algebra structures. Finally, we give a linear basis of free matching Rota-Baxter system by Gröbner-Shirshov bases methods.

We present a new generalized version of Cline's formula and Jacobson's lemma for g-Drazin inverses in a ring. These generalized results extend many known results, e.g., Chen and Abdolyousefi [Generalized Jacobson's lemma in a Banach algebra, Comm. Algebra 49 (2021) 3263–3272], and Yan and Zeng [The generalized inverses of the products of two elements in a ring, Turkish J. Math. 44 (2020) 1744–1756].

We introduce the doubled Hecke algebra, which is an infinite-dimensional algebra generated by two Hecke algebras. This concept originates from the degenerate doubled Hecke algebra in the theory of Schur-Weyl duality related to enhanced reductive algebraic groups. We study the finite-dimensional natural representation of the doubled Hecke algebra on tensor space and prove that the doubled Hecke algebra forms a duality with the quantum group of Levi type.

A finite group $G$ is said to be a $B(n,k)$ group if for any $n$-element subset $\{a_1,\dots ,a_n\}$ of $G$, $|\{a_i{a}_j | 1\le i,j\le n\}|\le k$. It is of interest to characterize the structure of $B(n,k)$ groups for $n(n+1)/2\le k\le n^2-1$. The $B(5,k)$ groups for $15\le k\le 19$ have been investigated by several authors. In this paper, we give a complete characterization of $B(5,20)$ 2-groups by showing there are five classes of such groups which are nontrivial and nonabelian.

We find an explicit expression of the associated primes of monomial ideals as a colon by an element $v$, using the unique irredundant irreducible decomposition whose irreducible components are monomial ideals. An algorithm to compute $v$ is given using Macaulay2. For squarefree monomial ideals the problem is related to the combinatorics of the underlying clutter or graph. For ideals of Borel type, the monomial $v$ takes a simpler form, and we classify when $v$ is unique.

Let $R$ be a ring (not necessarily commutative) with identity 1 and let $\mathcal{CL}(R)$ be its clean graph. In this paper, we investigate the genus number of the compact Riemann surface in which $\mathcal{CL}(R)$ can be embedded and explicitly determine all commutative rings $R$ (up to isomorphism) such that $\mathcal{CL}(R)$ has genus at most two. It is shown that for any Artinian ring $R$, $\mathcal{CL}(R)$ is a projective graph if and only if $R$ is isomorphic to ${\mathbb{F}}_4$. Furthermore, we determine all isomorphism classes of commutative rings whose clean graphs have crosscap two.

In this paper, we give a sufficient condition under which an involution monoid generates a variety with continuum many subvarieties. According to this result, several involution $J$-trivial monoids are shown to generate varieties with continuum many subvarieties. These examples include Rees quotients of free involution monoids, Lee monoids with involution, and Straubing monoids with involution.

Let $R$ be a ring and $n$, $k$ be two non-negative integers. As an extension of several known notions, we introduce and study $(n,k)$-weak cotorsion modules using the class of right $R$-modules with $n$-weak flat dimensions at most $k$. Various examples and applications are also given.

For a nilpotent group $G$ without $\pi$-torsion, and $x$, $y\in G$, if $x^n=y^n$ for a $\pi$-number $n$, then $x=y$; if $x^m{y}^n=y^n{x}^m$ for $\pi$-numbers $m$, $n$, then $xy=yx$. This is a well-known result in group theory. In this paper, we prove two analogous theorems on matrices, which have independence significance. Specifically, let $m$ be a given positive integer and $A$ a complex square matrix satisfying that (i) all eigenvalues of $A$ are nonnegative, and (ii) $\mathrm{rank}{A}^2=\mathrm{rank}A$; then $A$ has a unique $m$-th root $X$ with $\mathrm{rank}{X}^2=\mathrm{rank}X$, all eigenvalues of $X$ are nonnegative, and moreover there is a polynomial $f(\lambda )$ with $X=f(A)$. In addition, let $A$ and $B$ be complex $n\times n$ matrices with all eigenvalues nonnegative, and $\mathrm{rank}{A}^2=\mathrm{rank}A$, $\mathrm{rank}{B}^2=\mathrm{rank}B$; then (i) $A=B$ when $A^r=B^r$ for some positive integer $r$, and (ii) $AB=BA$ when $A^s{B}^t=B^t{A}^s$ for two positive integers $s$ and $t$.

We consider stable representations of non-Dynkin quivers with respect to a central charge. These attract a lot of interest in mathematics and physics since they can be identified with so-called BPS states. Another motivation is the work of Dimitrov et al. on the phases of stable representations of the generalized Kronecker quiver. One aim is to explain for general Euclidean and wild quivers the behavior of phases of stable representations well known in some examples. In addition, we study especially the behavior of preinjective, postprojective and regular indecomposable modules. We show that the existence of a stable representation with self-extensions implies the existence of infinitely many stables without self-extensions for rigid central charges. In this case the phases of the stable representations approach one or two limit points. In particular, the phases are not dense in two arcs. The category of representations of acyclic quivers is a special case of rigid Abelian categories which show this behavior for rigid central charges.

A graph $G$ is called triangle-free if $G$ does not contain any triangle as its induced subgraph. Let ${\mathcal{G}}_n$ be the set of triangle-free graphs of order $n$ each of which has three positive eigenvalues. In this paper, we find 20 specific graphs in ${\mathcal{G}}_n$, each of which has nullity no more than 2, and we show that in terms of three graph transformations all the other graphs of ${\mathcal{G}}_n$ can be constructed from these 20 specific graphs. Hence, we completely characterize the triangle-free graphs with exactly three positive eigenvalues.