In [2], an exhaustive construction is achieved for the class of all 4-dimensional unital division algebras over finite fields of odd order, whose left nucleus is not minimal and whose automorphism group contains Klein's four-group.
We generalize the approach of [2] towards all division algebras of the above specified type, but now admitting arbitrary fields k of characteristic not 2 as ground fields. For these division algebras we present an exhaustive construction that depends on a quadratic field extension of k and three parameters in k, and we derive an isomorphism criterion in terms of these parameters. As an application we classify, for o an ordered field in which every positive element is a square, all division o-algebras of the mentioned type, and in the finite field case we refine the Main Theorem of [2] to a classification even of the division algebras studied there.
The category formed by the division k-algebras investigated here is a groupoid, whose structure we describe in a supplementary section in terms of a covering by group actions. In particular, we exhibit the automorphism groups for all division algebras in this groupoid.
We generalize Albert's twisted field construction, applying it to unital division algebras with a multiplicative norm. We give conditions for the resulting algebras to be division algebras.Four- and eight-dimensional real unital and non-unital division algebras with large derivation algebras are constructed out of Hamilton's quaternion and Cayley's octonion algebra.
In a paper by E. Dieterich 2017, the category $\mathscr{C}(k)$ of four-dimensional unital division algebras, whose right nucleus is non-trivial and whose automorphism group contains Klein's four group $V$, is studied over a general ground field $k$ with $\mathrm{char}\,k\neq 2$. In particular, the objects in $\mathscr{C}(k)$ are exhaustively constructed from parameters in $k^3$ and explicit isomorphism conditions for the constructed objects are found in terms of these parameters. In this paper, we specialize to the case $k=\mathbb{Q}$ and present results towards a classification of $\mathscr{C…
New families of eight-dimensional real division algebras with large derivation algebra are presented: We generalize the classical Cayley-Dickson doubling process starting with a unital algebra with involution over a field F by allowing the scalar in the doubling to be an invertible element in the algebra. The resulting unital algebras are neither power-associative nor quadratic. Starting with a quaternion division algebra D, we obtain division algebras A for all invertible scalars chosen in D outside of F. This is independent on where the scalar is placed inside the product and three pairwise …
In this paper we describe all, up to isomorphism, left unital, right unital and unital algebra structures on two-dimensional vector space over any algebraically closed field and $\mathbb{R}$. We tabulate the algebras with the units.
In hom-associative structures, the associativity condition $(xy)z=x(yz)$ is twisted to $α(x)(yz) = (xy)α(z)$, with $α$ a map in the appropriate category. In the present paper, we consider two different unitality conditions for hom-associative algebras. The first one, existence of a unit in the classical sense, is stronger than the second one, which we call weak unitality. We show associativity conditions connected to the size of the image of the twisting map for unital hom-associative algebras. Also the problem of embedding arbitrary hom-associative algebras into unital or weakly unital ones i…
We describe Morita equivalence of unital locally matrix algebras in terms of their Steinitz parametrization. Two countable dimensional unital locally matrix algebras are Morita equivalent if and only if their Steinitz numbers are rationally connected. For an arbitrary uncountable dimension $α$ and an arbitrary not locally finite Steinitz number $s$ there exist unital locally matrix algebras $A$, $B$ such that $\dim_{F}A=\dim_{F}B=α$, $\mathbf{st}(A)=\mathbf{st}(B)=s$, however, the algebras $A$, $B$ are not Morita equivalent.
We construct a unital locally matrix algebra of uncountable dimension that (1) does not admit a primary decomposition, (2) has an infinite locally finite Steinitz number. It gives negative answers to questions from \cite{BezOl} and \cite{Kurochkin}. We also show that for an arbitrary infinite Steinitz number $s$ there exists a unital locally matrix algebra $A$ having the Steinitz number $s$ and not isomorphic to a tensor product of finite dimensional matrix algebras.
We study polynomial identities of algebras with adjoined external unit. For a wide class of algebras we prove that adjoining external unit element leads to increasing of PI-exponent precisely to 1. We also show that any real number from the interval [2,3] can be realized as PI-exponent of some unital algebra.
We construct a family of unital non-associative algebras $\{T_α\vert~ 2<α\in\mathbb R\}$ such that $\underline{exp}(T_α)=2$, whereas $α\le\overline{exp}(T_α)\leα+1$. In particular, it follows that ordinary PI-exponent of codimension growth of algebra $T_α$ does not exist for any $α> 2$. This is the first example of a unital algebra whose PI-exponent does not exist.
Left unital Kantor triple systems will be shown to correspond to structurable algebras endowed with an involutive automorphism. A related result is proved for (-1,-1) Freudenthal-Kantor triple systems. Some consequences for the associated 5-graded Lie algebras and superalgebras are deduced too. In particular, left unital (-1,-1) Freudenthal-Kantor triple systems are shown to be intimately related to Lie superalgebras graded over the root system of type B(0,1).
We construct a basis of free unital generalized Poisson superalgebras and a basis of free unital superalgebras of Jordan brackets. Also, we prove the analogue of Farkas' Theorem for PI unital generalized Poisson algebras and PI unital algebras of Jordan brackets. Relations beetwen generic Poisson superalgebras and superalgebras of Jordan brackets will be studied.
An $F$-algebra $A$ with unit $1$ is said to be a locally matrix algebra if an arbitrary finite collection of elements $a_1,$ $\ldots,$ $a_s $ from $ A$ lies in a subalgebra $B$ with $1$ of the algebra $A$, that is isomorphic to a matrix algebra $M_n(F),$ $n\geq 1.$ To an arbitrary unital locally matrix algebra $A$ we assign a Steinitz number $\mathbf{n}(A)$ and study a relationship between $\mathbf{n}(A)$ and $A$.
We classify all decompositions of $M_3(\mathbb{C})$ into a direct vector-space sum of two subalgebras such that one of the subalgebras contains the identity matrix.
We introduce hom-associative Ore extensions as non-unital, non-associative Ore extensions with a hom-associative multiplication, and give some necessary and sufficient conditions when such exist. Within this framework, we construct families of hom-associative quantum planes, universal enveloping algebras of a Lie algebra, and Weyl algebras, all being hom-associative generalizations of their classical counterparts, as well as prove that the latter are simple. We also provide a way of embedding any multiplicative hom-associative algebra into a multiplicative, weakly unital hom-associative algebr…
A Rota-Baxter operator defined on the polynomial algebra is called monomial if it maps each monomial to a monomial with some coefficient. We classify monomial Rota-Baxter operators defined on the algebra of polynomials in one variable without constant term. We also describe injective monomial Rota--Baxter operators of nonzero weight on the algebra of polynomials in several variables without constant term.
We study polynomial identities of nonassociative algebras constructed by using infinite binary words and their combinatorial properties. Infinite periodic and Sturmian words were first applied for constructing examples of algebras with arbitrary real PI-exponent greater than one. Later we used these algebras for confirmation of the conjecture that PI-exponent increases precisely by one after adjoining an external unit to a given algebra. Here we prove the same result for these algebras for graded identities and graded PI-exponent, provided that the grading group is cyclic of order two.
Let $E$ be an arbitrary (countable) graph and let $R$ be a unital commutative ring. We analyze the ideal structure of the Leavitt path algebra $\lr$ introduced by Mark Tomforde. We first modify the definition of basic ideals and we then develop the ideal characterization of Mark Tomforde. We also give necessary and sufficient conditions for the primeness and the primitivity of $\lr$. Then by applying these results we determine prime graded basic ideals and left (or right) primitive graded ideals of $\lr$. In particular, we show that when $E$ satisfies Condition (K) and $R$ is a field, the set …
Let $\mathfrak{A}$ be a unital ring with a nontrivial idempotent. In this paper, it is shown that under certain conditions every multiplicative generalized Jordan $n$-derivation $Δ:\mathfrak{A}\rightarrow\mathfrak{A}$ is additive. More precisely, it is proved that $Δ$ is of the form $Δ(t)=μt+δ(t),$ where $μ\in\mathcal{Z}(\mathfrak{A})$ and $δ:\mathfrak{A}\rightarrow\mathfrak{A}$ is a Jordan $n$-derivation. The main result is then applied to some classical examples of unital rings with nontrivial idempotents such as triangular rings, matrix rings, prime rings, nest algebras, standard operator a…
The notion of f-derivations was introduced by Beidar and Fong to unify several kinds of linear maps including derivations, Lie derivations and Jordan derivations. In this paper we introduce the notion of f-biderivations as a natural "biderivation" counterpart of the notion of "f-derivations". We first show, under some conditions, that any f-derivation is a Jordan biderivation. Then, we turn to study f-biderivations of a unital algebra with an idempotent. Our second main result shows, under some conditions, that every Jordan biderivation can be written as a sum of a biderivation, an antibideriv…
We state that all Rota-Baxter operators of nonzero weight on Grassmann algebra over a field of characteristic zero are projections on a subalgebra along another one. We show the one-to-one correspondence between the solutions of associative Yang-Baxter equation and Rota-Baxter operators of weight zero on the matrix algebra $M_n(F)$ (joint with P. Kolesnikov). We prove that all Rota-Baxter operators of weight zero on a unital associative (alternative, Jordan) algebraic algebra over a field of characteristic zero are nilpotent. For an algebra $A$, we introduce its new invariant the rb-index $\ma…
On the representability of actions of unital algebras
Université catholique de Louvain（天主教鲁汶大学）；Università degli Studi di Palermo（巴勒莫大学）；Università degli Studi di Messina（墨西拿大学）；Università degli Studi di Milano Statale（米兰州立大学）
From a commutative associative algebra $A$, the infinite dimensional unital 3-Lie Poisson algebra~$\mathfrak{L}$~is constructed, which is also a canonical Nambu 3-Lie algebra, and the structure of $\mathfrak{L}$ is discussed. It is proved that: (1) there is a minimal set of generators $S$ consisting of six vectors; (2) the quotient algebra $\mathfrak{L}/\mathbb{F}L_{0, 0}^0$ is a simple 3-Lie Poisson algebra; (3) four important infinite dimensional 3-Lie algebras: 3-Virasoro-Witt algebra $\mathcal{W}_3$, $A_ω^δ$, $A_ω$ and the 3-$W_{\infty}$ algebra can be embedded in $\mathfrak{L}$.
Using Wederburn's main theorem and a result of Gerstenhaber we prove that, over a field of characteristic zero, the maximal dimension of a proper unital subalgebra in the $n \times n$ matrix algebra is $n^2 - n + 1$ and furthermore this upper bound is attained for the so-called parabolic subalgebras. We also investigate the corresponding notion of parabolic coideals for matrix coalgebras and prove that the minimal dimension of a non-zero coideal of the matrix coalgebra ${\mathcal M}^n (k)$ is $n-1$.
Pseudoalgebras, introduced in [BDK], are multi-dimensional analogues of conformal algebras, which provide an axiomatic description of the singular part of the operator product expansion. Our main interest in this paper is the pseudoalgebra Cend_n, which is the analogue of an algebra of endomorphisms of a finite module. We study its algebraic properties. In particular, we introduce the class of unital pseudoalgebras and describe their structure and representations. Also, we classify pseudoalgebras algebraically similar to Cend_n.
Let A be a unital algebra with a nontrivial idempotent e, and f=1-e. Suppose that A satisfies that exe.eAf={0}=fAe.exe implies exe=0 and eAf.fxf={0}=fxf.fAe implies fxf=0 for each x in A. We obtain the (necessary and) sufficient conditions for a Lie n-derivation ϕ on A to be of the form ϕ=d+δ+γ, where d is a derivation on A, δ is a singular Jordan derivation on A and γ is a linear mapping from A into the centre Z(A) vanishing on all (n-1)-th commutators of A. In particular, we also discuss the (necessary and) sufficient conditions for a Lie n-derivation ϕ on A to be standard, i.e., ϕ=d+γ.
Let $n\ge 2$, let $\mathcal{R}_n$ be the graph consisting of one vertex and $n$ loops and let $\mathcal{R}_{n^-}$ be its Cuntz splice. Let $L_n=L(\mathcal{R}_n)$ and $L_{n^-}=L(\mathcal{R}_{n^-})$ be the Leavitt path algebras over a unital ring $\ell$. Let $C_m$ be the cyclic group on $2\le m\le \infty$ elements. Equip $L_n$ and $L_{n^-}$ with their natural $C_m$-gradings. We show that under mild conditions on $\ell$, which are satisfied for example when $\ell$ is a field or a PID, there are no unital $C_m$-graded ring homomorphisms $L_n\to L_{n^-}$ nor in the opposite direction.
The functor Max of Mulvey assigns to each unital C*-algebra A the unital involutive quantale Max A of closed linear subspaces of A, and it has been remarked that it classifies unital C*-algebras up to *-isomorphism. In this paper we provide a proof of this and of the stronger fact that for every isomorphism u : Max A -> Max B of unital involutive quantales there is a *-isomorphism u' : A -> B such that Max u' coincides with u when restricted to the left-sided elements of Max A. But we also show that isomorphisms u : Max A -> Max B may exist for which no isomorphism v : A -> B is such that Max …
Let $ξ:C^*(E)\to C^*(F)$ be a unital $*$-homomorphism between simple purely infinite Cuntz-Krieger algebras of finite graphs. We prove that there exists a unital $*$-homomorphism $ϕ:L(E)\to L(F)$ between the corresponding Leavitt path-algebras such that $ξ$ is homotopic to the map $\hatϕ:C^*(E)\to C^*(F)$ induced by completion. We show moreover that $\hatϕ$ is a homotopy equivalence in the $C^*$-algebraic sense if and only if $ϕ$ is a homotopy equivalence in the algebraic, polynomial sense. We deduce, in particular, that any isomorphism between simple purely infinite Cuntz-Krieger algebras is …
Over a field of characteristic zero, we show that the forgetful functor from the homotopy category of commutative dg algebras to the homotopy category of dg associative algebras is faithful. In fact, the induced map of derived mapping spaces gives an injection on all homotopy groups at any basepoint. We prove similar results both for unital and non-unital algebras, and also Koszul dually for the universal enveloping algebra functor from dg Lie algebras to dg associative algebras. An important ingredient is a natural model for these derived mapping spaces as Maurer-Cartan spaces of complete fil…
