Mathematics

Hom-dendriform algebras are twisted analog of dendriform algebras and are splitting of hom-associative algebras. In this paper, we define a cohomology and deformation for hom-dendriform algebras. We relate this cohomology with the Hochschild-type cohomology of hom-associative algebras. We also describe similar results for the twisted analog of dendriform coalgebras.

We study representations of hemistrict Lie 2-algebras and give a functorial construction of their cohomology. We prove that both the cohomology of an injective hemistrict Lie 2-algebra L and the cohomology of the semistrict Lie 2-algebra obtained from skew-symmetrization of L are isomorphic to the Chevalley-Eilenberg cohomology of the induced Lie algebra L Lie .

Theorem 7 in Ref. [Linear Algebra Appl., 430, 1-6, (2009)] states sufficient conditions to determine whether a pair generates the algebra of 3x3 matrices over an algebraically closed field of characteristic zero. In that case, an explicit basis for the full algebra is provided, which is composed of words of small length on such pair. However, we show that this theorem is wrong since it is based on the validity of an identity which is not true in general.

A commutative algebra is exact if its multiplication endomorphisms are trace-free and is Killing metrized if its Killing type trace-form is nondegenerate and invariant. A Killing metrized exact commutative algebra is necessarily neither unital nor associative. Such algebras can be viewed as commutative analogues of semisimple Lie algebras or, alternatively, as nonassociative generalizations of étale (associative) algebras. Some basic examples are described and there are introduced quantitative measures of nonassociativity, formally analogous to curvatures of connections, that serve to facilitate the organization and characterization of these algebras.

The Rota-Baxter algebra and the shuffle product are both algebraic structures arising from integral operators and integral equations. Free commutative Rota-Baxter algebras provide an algebraic framework for integral equations with the simple Riemann integral operator. The Zinbiel algebras form a category in which the shuffle product algebra is the free object. Motivated by algebraic structures underlying integral equations involving multiple integral operators and kernels, we study commutative matching Rota-Baxter algebras and construct the free objects making use of the shuffle product with multiple decorations. We also construct free commutative matching Rota-Baxter algebras in a relative context, to emulate the action of the integral operators on the coefficient functions in an integral equation. We finally show that free commutative \match Rota-Baxter algebras give the free matching Zinbiel algebra, generalizing the characterization of the shuffle product algebra as the free Zinbiel algebra obtained by Loday.

Let R be any associative ring with 1 , n≥3 , and let A,B be two-sided ideals of R . In our previous joint works with Roozbeh Hazrat [17,15] we have found a generating set for the mixed commutator subgroup [E(n,R,A),E(n,R,B)] . Later in [29,34] we noticed that our previous results can be drastically improved and that [E(n,R,A),E(n,R,B)] is generated by 1) the elementary conjugates z ij (ab,c)= t ij (c) t ji (ab) t ij (−c) and z ij (ba,c) , 2) the elementary commutators [ t ij (a), t ji (b)] , where 1≤i≠j≤n , a∈A , b∈B , c∈R . Later in [33,35] we noticed that for the second type of generators, it even suffices to fix one pair of indices (i,j) . Here we improve the above result in yet another completely unexpected direction and prove that [E(n,R,A),E(n,R,B)] is generated by the elementary commutators [ t ij (a), t hk (b)] alone, where 1≤i≠j≤n , 1≤h≠k≤n , a∈A , b∈B . This allows us to revise the technology of relative localisation, and, in particular, to give very short proofs for a number of recent results, such as the generation of partially relativised elementary groups E(n,A ) E(n,B) , %% normality of E(n,AB+BA) inside [E(n,R,A),E(n,R,B)] , multiple commutator formulas, commutator width, and the like.

In the present paper we find generators of the mixed commutator subgroups of relative elementary groups and obtain unrelativised versions of commutator formulas in the setting of Bak's unitary groups. It is a direct sequel of our similar results were obtained for GL(n,R) and for Chevalley groups over a commutative ring with 1, respectively. Namely, let (A,Λ) be any form ring and n≥3 . We consider Bak's hyperbolic unitary group GU(2n,A,Λ) . Further, let (I,Γ) be a form ideal of (A,Λ) . One can associate with (I,Γ) the corresponding elementary subgroup FU(2n,I,Γ) and the relative elementary subgroup EU(2n,I,Γ) of GU(2n,A,Λ) . Let (J,Δ) be another form ideal of (A,Λ) . In the present paper we prove an unexpected result that the non-obvious type of generators for [EU(2n,I,Γ),EU(2n,J,Δ)] , as constructed in our previous papers with Hazrat, are redundant and can be expressed as products of the obvious generators, the elementary conjugates Z ij (ab,c)= T ji (c) T ij (ab) T ji (−c) and Z ij (ba,c) , and the elementary commutators Y ij (a,b)=[ T ji (a), T ij (b)] , where a∈(I,Γ) , b∈(J,Δ) , c∈(A,Λ) . It follows that [FU(2n,I,Γ),FU(2n,J,Δ)]=[EU(2n,I,Γ),EU(2n,J,Δ)] . In fact, we establish much more precise generation results. In particular, even the elementary commutators Y ij (a,b) should be taken for one long root position and one short root position. Moreover, Y ij (a,b) are central modulo EU(2n,(I,Γ)∘(J,Δ)) and behave as symbols. This allows us to generalise and unify many previous results,including the multiple elementary commutator formula, and dramatically simplify their proofs.

Suppose R is a 2 , 3 -torsion free unital alternative ring having an idempotent element e 1 ( e 2 =1− e 1 ) which satisfies xR⋅ e i ={0}→x=0 (i=1,2) . In this paper, we aim to characterize the commuting maps. Let φ be a commuting map of R so it is shown that φ(x)=zx+Ξ(x) for all x∈R , where z∈Z(R) and Ξ is an additive map from R into Z(R) . As a consequence a characterization of anti-commuting maps is obtained and we provide as an application, a characterization of commuting maps on von Neumann algebras relative alternative C ∗ -algebra with no central summands of type I 1 .

Let Γ be the infinite cyclic group on a generator x. To avoid confusion when working with Z -modules which also have an additional Z -action, we consider the Z -action to be a Γ -action instead. Starting from a directed graph E , one can define a cancellative commutative monoid M Γ E with a Γ -action which agrees with the monoid structure and a natural order. The order and the action enable one to label each nonzero element as being exactly one of the following: comparable (periodic or aperiodic) or incomparable. We comprehensively pair up these element features with the graph-theoretic properties of the generators of the element. We also characterize graphs such that every element of M Γ E is comparable, periodic, graphs such that every nonzero element of M Γ E is aperiodic, incomparable, graphs such that no nonzero element of M Γ E is periodic, and graphs such that no element of M Γ E is aperiodic. The Graded Classification Conjecture can be formulated to state that M Γ E is a complete invariant of the Leavitt path algebra L K (E) of E over a field K. Our characterizations indicate that the Graded Classification Conjecture may have a positive answer since the properties of E are well reflected by the structure of M Γ E . Our work also implies that some results of [R. Hazrat, H. Li, The talented monoid of a Leavitt path algebra, J. Algebra, 547 (2020) 430-455] hold without requiring the graph to be row-finite.

In this article we give various formulates for compute the number of all coninvolutions over the group of upper triangular matrix with entries into the ring of Gaussian integers module p and the ring of Quaternions integers module p , with p an odd prime number.