Mathematics

The differential Brauer monoid of a differential commutative ring R s defined. Its elements are the isomorphism classes of differential Azumaya R algebras with operation from tensor product subject to the relation that two such algebras are equivalent if matrix algebras over them are differentially isomorphic. The Bauer monoid, which is a group, is the same thing without the differential requirement. The differential Brauer monoid is then determined from the Brauer monoids of R and its ring of constants and the submonoid whose underlying Azumaya algebras are matrix rings.

In this article, we study Dorroh extensions of algebras and Dorroh extensions of coalgebras. Their structures are described. Some properties of these extensions are presented. We also introduce the finite duals of algebras and modules which are not necessarily unital. Using these finite duals, we determine the dual relations between the two kinds of extensions.

In this paper, we study Dorroh extensions of bialgebras and Hopf algebras. Let (H,I) be both a Dorroh pair of algebras and a Dorroh pair of coalgebras. We give necessary and sufficient conditions for H ⋉ d I to be a bialgebra and a Hopf algebra, respectively. We also describe all ideals of Dorroh extensions of algebras and subcoalgebras of Dorroh extensions of coalgebras and compute these ideals and subcoalgebras for some concrete examples.

Axial algebras are a class of commutative non-associative algebras generated by idempotents, called axes, with adjoint action semi-simple and satisfying a prescribed fusion law. Axial algebras were introduced by Hall, Rehren and Shpectorov \cite{hrs,hrs1} as a broad generalisation of Majorana algebras of Ivanov, whose axioms were derived from the properties of the Griess algebra for the Monster sporadic simple group. The class of axial algebras of Monster type includes Majorana algebras for the Monster and many other sporadic simple groups, Jordan algebras for classical and some exceptional simple groups, and Matsuo algebras corresponding to 3 -transposition groups. Thus, axial algebras of Monster type unify several strands in the theory of finite simple groups. It is shown here that double axes, i.e., sums of two orthogonal axes in a Matsuo algebra, satisfy the fusion law of Monster type (2η,η) . Primitive subalgebras generated by two single or double axes are completely classified and 3 -generated primitive subalgebras are classified in one of the three cases. These classifications further lead to the general flip construction outputting a rich variety of axial algebras of Monster type. An application of the flip construction to the case of Matsuo algebras related to the symmetric groups results in three new explicit infinite series of such algebras.

This work addresses some relevant characteristics and properties of q -generalized associative algebras and q -generalized dendriform algebras such as bimodules, matched pairs. We construct for the special case of q=−1 an antiassociative algebra with a decomposition into the direct sum of the underlying vector spaces of another antiassociative algebra and its dual such that both of them are subalgebras and the natural symmetric bilinear form is invariant or the natural antisymmetric bilinear form is sympletic. The former is called a double construction of quadratic antiassociative algebra and the later is a double construction of sympletic antiassociative algebra which is interpreted in terms of antidendrifom algebras. We classify the 2-dimensional antiassociative algebras and thoroughly give some double constructions of quadratic and sympletic antiassociative algebras.

In this paper, we focus on the duo ring property via quasinilpotent elements which gives a new kind of generalizations of commutativity. We call this kind of ring qnil-duo. Firstly, some properties of quasinilpotents in a ring are provided. Then the set of quasinilpotents is applied to the duo property of rings, in this perspective, we introduce and study right (resp., left) qnil-duo rings. We show that this concept is not left-right symmetric. Among others it is proved that if the Hurwitz series ring H(R;α) is right qnil-duo, then R is right qnil-duo. Every right qnil-duo ring is abelian. A right qnil-duo exchange ring has stable range 1.

The main goal of this article is to introduce the concept of EM−G− graded rings. This concept is an extension of the notion of EM− rings. Let G be a group and R be a G− graded commutative ring. The G− gradation of R can be extended to R[x] by taking the components (R[x] ) σ = R σ [x] . We define R to be EM−G− graded ring if every homogeneous zero divisor polynomial has an annihilating content. We provide examples of EM−G− graded rings that are not EM− rings and we prove some interesting results regarding these rings.

The algebras Q n,k (E,τ) introduced by Feigin and Odesskii as generalizations of the 4-dimensional Sklyanin algebras form a family of quadratic algebras parametrized by coprime integers n>k≥1 , a complex elliptic curve E , and a point τ∈E . The main result in this paper is that Q n,k (E,τ) has the same Hilbert series as the polynomial ring on n variables when τ is not a torsion point. We also show that Q n,k (E,τ) is a Koszul algebra, hence of global dimension n when τ is not a torsion point, and, for all but countably many τ , it is Artin-Schelter regular. The proofs use the fact that the space of quadratic relations defining Q n,k (E,τ) is the image of an operator R τ (τ) that belongs to a family of operators R τ (z): C n ⊗ C n → C n ⊗ C n , z∈C , that (we will show) satisfy the quantum Yang-Baxter equation with spectral parameter.

In this article, we introduce equivariant formal deformation theory of Lie triple systems. We introduce an equivariant deformation cohomology of Lie triple systems and using this we study the equivariant formal deformation theory of Lie triple systems.

Cancellative dimer algebras on a torus have many nice algebraic and homological properties. However, these nice properties disappear for dimer algebras on higher genus surfaces. We consider a new class of quiver algebras on surfaces, called 'geodesic ghor algebras', that reduce to cancellative dimer algebras on a torus, yet continue to have nice properties on higher genus surfaces. These algebras exhibit a rich interplay between their central geometry and the topology of the surface. We show that (nontrivial) geodesic ghor algebras do in fact exist, and give explicit descriptions of their central geometry. This article serves a companion to the article 'A generalization of cancellative dimer algebras to hyperbolic surfaces', where the main statement is proven.