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preprint2020arXiv

Weyl modules for Lie superalgebras

We define global and local Weyl modules for Lie superalgebras of the form $\mathfrak{g} \otimes A$, where $A$ is an associative commutative unital $\mathbb{C}$-algebra and $\mathfrak{g}$ is a basic Lie superalgebra or $\mathfrak{sl}(n,n)$, $n \ge 2$. Under some mild assumptions, we prove universality, finite-dimensionality, and tensor product decomposition properties for these modules. These properties are analogues of those of Weyl modules in the non-super setting. We also point out some features that are new in the super case.

preprint2020arXiv

Tate-Vogel and relative cohomologies of complexes with respect to cotorsion pairs

We study Tate-Vogel and relative cohomologies of complexes by applying the model structure induced by a complete hereditary cotorsion pair ($\A$, $\B$) of modules. We show first that the class of complexes admitting a complete $\A$ resolution is exactly the class of complexes with finite Gorenstein $\A$ dimension. This lets us give general techniques for computing Tate-Vogel cohomoloies of complexes with finite Gorenstein $\A$ dimension. As a consequence, relative cohomology groups for complexes with finite Gorenstein $\A$ dimension are investigated. Finally, the relationships between Gorenstein $\A$ dimensions and $\A$ dimensions for complexes are given.

preprint2020arXiv

On Steinberg algebras of Hausdorff ample groupoids over commutative semirings

We investigate the algebra of a Hausdorff ample groupoid, introduced by Steinberg, over a commutative semiring S. In particular, we obtain a complete characterization of congruence-simpleness for such Steinberg algebras, extending the well-known characterizations when S is a field or a commutative ring. We also provide a criterion for the Steinberg algebra of the graph groupoid associated to an arbitrary graph to be congruence-simple. Motivated by a result of Clark and Sims, we show that, over the Boolean semifield, the natural homomorphism from the Leavitt path algebra to the Steinberg algebra is an isomorphism if and only if the associated graph is row-finite. Moreover, we establish the Reduction Theorem and Uniqueness Theorems for Leavitt path algebras of row-finite graphs over the Boolean semifield.

preprint2020arXiv

Multiplier Hopf coquasigroups with faithful integrals

Let $A$ be a multiplier Hopf coquasigroup. If the faithful integrals exist, then they are unique up to scalar. Furthermore, if $A$ is of discrete type, then its integral duality $\widehat{A}$ is a Hopf quasigroup, and the biduality $\widehat{\widehat{A}}$ is isomorphic to the original $A$ as multiplier Hopf coquasigroups. This biduality theorem also holds for a class of Hopf quasigroups with faithful integrals.

preprint2020arXiv

Structures of BiHom-Poisson algebras

This paper gives some constructions results and examples of BiHom-Poisson algebras. Next, BiHom-flexible algebras are defined and it is shown that admissible BiHom-Poisson algebras are BiHom-flexible. Furthermore, generalized derivations of Bihom-Poisson algebras are introduced and some their basic properties are given. Finally, BiHom-Poisson modules and several constructions of these notions are obtained.

preprint2020arXiv

2-local derivations on the Jacobson-Witt algebras in prime characteristic

This paper initiates the study of 2-local derivations on Lie algebras over fields of prime characteristic. Let $\mathfrak{g}$ be a simple Jacobson-Witt algebra $W_n$ over a field of prime characteristic $p$ with cardinality no less than $p^n$. In this paper, we study properties of 2-local derivations on $\mathfrak{g}$, and show that every 2-local derivation on $\mathfrak{g}$ is a derivation.

preprint2020arXiv

The ideal of Lesieur-Croisot elements of Jordan pairs

We study the sets of elements of Jordan pairs whose local Jordan algebras are Lesieur-Croisot algebras, that is, classical orders in nondegenerate Jordan algebras with finite capacity. It is then proved that, if the Jordan pair is nondegenerate, the set of its Lesieur-Croisot elements is an ideal of the Jordan pair.

preprint2020arXiv

Equivalence of slice semi-regular functions via Sylvester operators

The aim of this paper is to study some features of slice semi-regular functions $\mathcal{RM}(Ω)$ on a circular domain $Ω$ contained in the skew-symmetric algebra of quaternions $\mathbb{H}$ via the analysis of a family of linear operators built from left and right $*$-multiplication on $\mathcal{RM}(Ω)$; this class of operators includes the family of Sylvester-type operators $\mathcal{S}_{f,g}$. Our strategy is to give a matrix interpretation of these operators as we show that $\mathcal{RM}(Ω)$ can be seen as a $4$-dimensional vector space on the field $\mathcal{RM}_{\mathbb{R}}(Ω)$. We then study the rank of $\mathcal{S}_{f,g}$ and describe its kernel and image when it is not invertible. By using these results, we are able to characterize when the functions $f$ and $g$ are either equivalent under $*$-conjugation or intertwined by means of a zero divisor, thus proving a number of statements on the behaviour of slice semi-regular functions. We also provide a complete classification of idempotents and zero divisors on product domains of $\mathbb{H}$.

preprint2020arXiv

Minimal abelian varieties of algebras, I

We show that any abelian variety that is not affine has a nontrivial strongly abelian subvariety. In later papers in this sequence we apply this result to the study of minimal abelian varieties.

preprint2020arXiv

2-Local derivations on the Super Virasoro algebra and Super W(2,2) algebra

The present paper is devoted to study 2-local superderivations on the super Virasoro algebra and the super W(2,2) algebra. We prove that all 2-local superderivations on the super Virasoro algebra as well as the super W(2,2) algebra are (global) superderivations.

preprint2020arXiv

Chains in evolution algebras

In this work we approach three-dimensional evolution algebras from certain constructions performed on two-dimensional algebras. More precisely, we provide four different constructions producing three-dimensional evolution algebras from two-dimensional algebras. Also we introduce two parameters, the annihilator stabilizing index and the socle stabilizing index, which are useful tools in the classification theory of these algebras. Finally, we use moduli sets as a convenient way to describe isomorphism classes of algebras.

preprint2020arXiv

Universal tensor categories generated by dual pairs

Let $V_*\otimes V\rightarrow\mathbb{C}$ be a non-degenerate pairing of countable-dimensional complex vector spaces $V$ and $V_*$. The Mackey Lie algebra $\mathfrak{g}=\mathfrak{gl}^M(V,V_*)$ corresponding to this paring consists of all endomorphisms $φ$ of $V$ for which the space $V_*$ is stable under the dual endomorphism $φ^*: V^*\rightarrow V^*$. We study the tensor Grothendieck category $\mathbb{T}$ generated by the $\mathfrak{g}$-modules $V$, $V_*$ and their algebraic duals $V^*$ and $V^*_*$. This is an analogue of categories considered in prior literature, the main difference being that the trivial module $\mathbb{C}$ is no longer injective in $\mathbb{T}$. We describe the injective hull $I$ of $\mathbb{C}$ in $\mathbb{T}$, and show that the category $\mathbb{T}$ is Koszul. In addition, we prove that $I$ is endowed with a natural structure of commutative algebra. We then define another category $_I\mathbb{T}$ of objects in $\mathbb{T}$ which are free as $I$-modules. Our main result is that the category ${}_I\mathbb{T}$ is also Koszul, and moreover that ${}_I\mathbb{T}$ is universal among abelian $\mathbb{C}$-linear tensor categories generated by two objects $X$, $Y$ with fixe

preprint2020arXiv

Functional equations in polynomials

We determine all F,G in C[X] of degree at least 2 for which the semigroup generated by F and G under composition is not the free semigroup on the letters F and G. We also solve the same problem for F,G in X^2 C[[X]], and prove partial analogues over arbitrary fields.

preprint2020arXiv

On the rational relationships among pseudo-roots of a non-commutative polynomial

For a non-commutative ring R, we consider factorizations of polynomials in R[t] where t is a central variable. A pseudo-root of a polynomial p(t) is an element x in R, for which there exist polynomials q(t) and s(t) such that p(t)=q(t)(t-x)s(t). We investigate the rational relationships that hold among the pseudo-roots of p(t) by using the diamond operations for cover graphs of modular lattices.

preprint2020arXiv

Decomposition algebras and axial algebras

We introduce decomposition algebras as a natural generalization of axial algebras, Majorana algebras and the Griess algebra. They remedy three limitations of axial algebras: (1) They separate fusion laws from specific values in a field, thereby allowing repetition of eigenvalues; (2) They allow for decompositions that do not arise from multiplication by idempotents; (3) They admit a natural notion of homomorphisms, making them into a nice category. We exploit these facts to strengthen the connection between axial algebras and groups. In particular, we provide a definition of a universal Miyamoto group which makes this connection functorial under some mild assumptions. We illustrate our theory by explaining how representation theory and association schemes can help to build a decomposition algebra for a given (permutation) group. This construction leads to a large number of examples. We also take the opportunity to fix some terminology in this rapidly expanding subject.

preprint2020arXiv

Centrally Essential Torsion-Free Rings of Finite Rank

It is proved that centrally essential rings, whose additive groups of finite rank are torsion-free groups of finite rank, are quasi-invariant but not necessarily invariant. Torsion-free Abelian groups of finite rank with centrally essential endomorphism rings are faithful. The paper will appear in Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry.

preprint2020arXiv

Blockwise relations between triples, and derived equivalences for wreath products

Motivated by the reduction techniques involving character triples for the local-global conjectures, we show that a blockwise relation between module triples is a consequence of a derived equivalence with additional properties. Moreover, we show that this relation is compatible with wreath products.

preprint2020arXiv

2-recollements of singualrity categories and Gorenstein defect categories over triangular matrix algebras

Let $T=(A,M,0,B)$ be a triangular matrix algebra with its corner algebras $A$ and $B$ Artinian and $_AM_B$ an $A$-$B$-bimodule. The 2-recollement structures for singularity categories and Gorenstein defect categories over $T$ are studied. Under mild assumptions, we provide necessary and sufficient conditions for the existences of 2-recollements of singularity categories and Gorenstein defect categories over $T$ relative to those of $A$ and $B$. Parts of our results strengthen and unify the corresponding work in [27,28,34].

preprint2020arXiv

Quadratic Split Quaternion Polynomials: Factorization and Geometry

We investigate factorizability of a quadratic split quaternion polynomial. In addition to inequality conditions for existence of such factorization, we provide lucid geometric interpretations in the projective space over the split quaternions.

preprint2020arXiv

Repetitive cluster categories of type Dn

In this paper, we show that the repetitive cluster category of type $D_n$, defined as the orbit category $\mathcal{D}^b(\mathrm{mod}K D_n)/(τ^{-1}[1])^p$, is equivalent to a category defined on a subset of tagged edges in a regular punctured polygon. This generalizes the construction of Schiffler, \cite{S}, which we recover when $p=1$.

preprint2020arXiv

Structure of group rings and the group of units of integral group rings: an invitation

During the past three decades fundamental progress has been made on constructing large torsion-free subgroups (i.e. subgroups of finite index) of the unit group $\U (\Z G)$ of the integral group ring $\Z G$ of a finite group $G$. These constructions rely on explicit constructions of units in $\Z G$ and proofs of main results make use of the description of the Wedderburn components of the rational group algebra $\Q G$. The latter relies on explicit constructions of primitive central idempotents and the rational representations of $G$. It turns out that the existence of reduced two degree representations play a crucial role. Although the unit group is far from being understood, some structure results on this group have been obtained. In this paper we give a survey of some of the fundamental results and the essential needed techniques.

preprint2020arXiv

The evolution of the spectrum of a Frobenius Lie algebra under deformation

The category of Frobenius Lie algebras is stable under deformation, and here we examine explicit infinitesimal deformations of four and six dimensional Frobenius Lie algebras with the goal of understanding if the spectrum of a Frobenius Lie algebra can evolve under deformation. It can.

preprint2020arXiv

Cohomology of BiHom-associative algebras

Bihom-associative algebras have been recently introduced in the study of group hom-categories. In this paper, we introduce a Hochschild type cohomology for bihom-associative algebras with suitable coefficients. The underlying cochain complex (with coefficients in itself) can be given the structure of an operad with a multiplication. Hence, the cohomology inherits a Gerstenhaber structure. We show that this cohomology also control corresponding formal deformations. Finally, we introduce bihom-associative algebras up to homotopy and show that some particular classes of these homotopy algebras are related to the above Hochschild cohomology.

preprint2020arXiv

Varieties of K-lattices

In this paper we deal with varieties of commutative residuated lattices that arise from a specific kind of construction: the {\em twist-product} of a lattice. Twist-products were first considered by Kalman in 1958 to deal with order involutions on plain lattices, but the extension of this concept to residuated lattices has attracted some attention lately. Here we deal mainly with varieties of such lattices, that can be obtained by applying a specific twist-product construction to varieties of integral and commutative residuated lattices.