Journal of Algebra最新文献

$\mathrm{GT}$-shadows [8] are tantalizing objects that can be thought of as approximations of elements of the mysterious Grothendieck-Teichmueller group $\widehat{\mathrm{GT}}$ introduced by V. Drinfeld in 1990. $\mathrm{GT}$-shadows form a groupoid $\mathrm{GTSh}$ whose objects are finite index subgroups of the pure braid group ${\mathrm{PB}}_4$, that are normal in $B_4$. The goal of this paper is to describe the action of $\mathrm{GT}$-shadows on Grothendieck's child's drawings and show that this action agrees with that of $\widehat{\mathrm{GT}}$. We discuss the hierarchy of orbits of child's drawings with respect to the actions of $\mathrm{GTSh}$, $\widehat{\mathrm{GT}}$, and the absolute Galois group $G_Q$ of rationals. We prove that the monodromy group and the passport of a child's drawing are invariant with respect to the action of the subgroupoid ${\mathrm{GTSh}}^{♡}$ of charming $\mathrm{GT}$-shadows. We use the action of $\mathrm{GT}$-shadows on child's drawings to prove that every Abelian child's drawing admits a Belyi pair defined over $Q$. Finally, we describe selected examples of non-Abelian child's drawings.

Let R be a commutative noetherian ring. Denote by $D^b\left(R\right)$ the bounded derived category of finitely generated R-modules. Extending the notion of a proxy small object of $D^b\left(R\right)$ in the sense of Dwyer, Greenlees, Iyengar and Pollitz, we introduce the notion of a proxy small thick subcategory of $D^b\left(R\right)$. When R is a locally dominant ring, we give a complete classification of the proxy small thick subcategories of $D^b\left(R\right)$ in terms of pairs of specialization-closed subsets of Spec R and Sing R.

We first generalize the cohomology of $O$-operators on BiHom-associative algebras by construct a graded Lie-algebra, in which the Maurer-Cartan elements are characterized by the given $O$-operator, and show that the cohomology represents the Hochschild cohomology of a certain BiHom-associative algebra with coefficients in a bimodule. Next, we study the linear and formal deformations of $O$-operators on BiHom-associative algebras, which are controlled by the Hochschild cohomology. Finally, as applications, we introduce the deformations of BiHom-associative r-matrices and infinitesimal BiHom-bialgebras on certain regular BiHom-associative algebras.

We show that any graph product of residually finite monoids is residually finite. As a special case we obtain that any free product of residually finite monoids is residually finite. The corresponding results for graph products of semigroups follow.

In this paper, we first establish relationships between Gorenstein projective modules linked by the separable equivalence of rings, and prove that Gorenstein, CM-finite and CM-free algebras are invariant under separable equivalences. Secondly, we provide a new method to produce separable equivalences. As applications, the following results are obtained. Let Λ and Γ be Artin algebras such that Λ is separably equivalent to Γ. (1) For representation-finite algebras Λ and Γ, their Auslander algebras are separably equivalent; (2) For CM-finite algebras Λ and Γ, the endomorphism algebras of their representative generators are separably equivalent. Finally, we discuss when tilted algebras are invariant under separable equivalences, and give an example to illustrate it.

It is well known that for all $n\ge 3$, the group $\mathrm{SL}(n,Z)$ has a finite presentation given by its $n^2-n$ transvections, subject to the Steinberg relations. Also by a 1962 theorem of Trott, if n is odd then $\mathrm{SL}(n,Z)$ is generated by two elements, one of infinite order, and by the combined work of Tamburini, J.S. Wilson and Vsemirnov and others (from 1993 to 2021), it is now known that $\mathrm{SL}(n,Z)$ is generated by two elements of orders 2 and 3 precisely when $n\ge 5$. On the other hand, little appears to be known about 2-generator presentations for $\mathrm{SL}(n,Z)$ for $n\ge 3$. In this paper, some finite 2-generator presentations are given for $\mathrm{SL}(3,Z)$, which as far as the authors are aware, are the only 2-generator finite presentations known for $\mathrm{SL}(3,Z)$. Also some new generating pairs are given for $\mathrm{SL}(n,Z)$ for $n\ge 3$. In particular, some of these extend Trott's 1962 theorem by showing that $\mathrm{SL}(n,Z)$ is generated by two elements, one of order 2 and the other of infinite order, for all $n>2$.

Let f be a homogeneous polynomial of even degree d. We study the decompositions $f={\sum }_{i=1}^r{f}_i^2$ where $\mathrm{deg}{f}_i=d/2$. The minimal number of summands r is called the 2-rank of f, so that the polynomials having 2-rank equal to 1 are exactly the squares. Such decompositions are never unique and they are divided into $O\left(r\right)$-orbits, the problem becomes counting how many different $O\left(r\right)$-orbits of decomposition exist. We say that f is $O\left(r\right)$-identifiable if there is a unique $O\left(r\right)$-orbit. We give sufficient conditions for generic and specific $O\left(r\right)$-identifiability. Moreover, we show the generic $O\left(r\right)$-identifiability of ternary forms.

Axial algebras are a class of commutative non-associative algebras which have a natural group of automorphisms, called the Miyamoto group. The motivating example is the Griess algebra which has the Monster sporadic simple group as its Miyamoto group. Previously, using an expansion algorithm, about 200 examples of axial algebras in the same class as the Griess algebra have been constructed in dimensions up to about 300. In this list, we see many reoccurring dimensions which suggests that there may be some unexpected isomorphisms. Such isomorphisms can be found when the full automorphism groups of the algebras are known. Hence, in this paper, we develop methods for computing the full automorphism groups of axial algebras and apply them to a number of examples of dimensions up to 151.

We construct Young wall models for the crystal bases of level 1 irreducible highest weight representations and Fock space representations of quantum affine algebras in types $E_6^{\left(1\right)}$, $E_7^{\left(1\right)}$ and $E_8^{\left(1\right)}$. In each case, Young walls consist of coloured blocks stacked inside the relevant Young wall pattern which satisfy a certain combinatorial condition. Moreover the crystal structure is described entirely in terms of adding and removing blocks.

In this note we prove linear independence of the combinatorial spanning set for standard $C_{\ell }^{\left(1\right)}$-module $L\left(k{\Lambda }_0\right)$ by establishing a connection with the combinatorial basis of Feigin-Stoyanovsky's type subspace $W\left(k{\Lambda }_0\right)$ of $C_{2\ell }^{\left(1\right)}$-module $L\left(k{\Lambda }_0\right)$. It should be noted that the proof of linear independence for the basis of $W\left(k{\Lambda }_0\right)$ is obtained by using simple currents and intertwining operators in the vertex operator algebra $L\left(k{\Lambda }_0\right)$.