arXiv subjects
Publications and source records attributed to Yoav Segev.
In a previous paper we studied ``weakly primitive axial algebras'' with respect to more general fusion rules, for which at least one axis satisfies the fusion rules. In this continuation, a concise description is provided of the $2$-generated algebras we obtained in that paper and we show that in certain cases, the algebras have a Frobenius form, a key tool in their study.
Throughout this paper $A$ is a commutative non-associative algebra over a field $\mathbb{F}$ of characteristic not $2.$ In addition $A$ posses a Frobenius form. We obtain detailed information about the multiplication in $A$ given two axes of type half in $A.$
In earlier work we studied the structure of primitive axial algebras of Jordan type (PAJ's), not necessarily commutative, in terms of their primitive axes. In this paper we weaken primitivity and permit several pairs of (left and right) eigenvalues satisfying a more general fusion rule, bringing in interesting new examples such as the band semigroup algebras and other commutative and noncommutative examples. Also we broaden our investigation and describe 2-generated algebras in which only one of the generating axes is weakly primitive and satisfies the fusion rules, on condition that its zero-eigenspace is one dimensional. We also characterize when both axes satisfy the fusion rules (weak PAJ's), and describe precisely the 2-dimensional axial algebras. In contrast to the previous situation, there are weak PAJ's of dimension~$> 3$ generated by two axes.
We show that primitive 4-generated axial algebras of Jordan type are at most 81-dimensional.
"Fusion rules" are laws of multiplication among eigenspaces of an idempotent. This terminology is relatively new and is closely related to primitive axial algebras, introduced recently by Hall, Rehren, and Shpectorov. Axial algebras, in turn, are closely related to $3$-transposition groups and vertex operator algebras. In earlier work we studied primitive axial algebras, not necessarily commutative, and showed that they all have Jordan type. In this paper, we show that all finitely generated primitive axial algebras are direct sums of specifically described flexible finite dimensional noncommutative algebras, and commutative axial algebras generated by primitive axes of the same type. In particular,all primitive axial algebras are flexible. They also have Frobenius forms. We give a precise description of all the primitive axes of axial algebras generated by two primitive axes.
Let $R$ be a ring with ${\bf 1}$ which is not commutative. Assume that a non-zero commutator in $R$ is not a zero divisor. Assume further that either $R$ is alternative, but not associative, or $R$ is associative and any commutator $v\in R$ satisfies: $v^2$ is in the center of $R.$ We prove that $R$ has no zero divisors. Furthermore, if $\text{char}(R)\ne 2,$ then the localization of $R$ at its center is an octonion division algebra, if $R$ is alternative and a quaternion division algebra, if $R$ is associative. Our proof in both cases is essentially the same and it is elementary and rather self contained.
The notion of axial algebra is closely related to $3$-transposition groups, the Monster group and vertex operator algebras. In this work we continue our previous works and compete the proof that all algebras generated by a set of primitive axes not necessarily of the same type (see the definition in the body of the paper), are primitive axial algebras of Jordan type.
"Fusion rules" are laws of multiplication among eigenspaces of an idempotent. This terminology is relatively new and is closely related to axial algebras, introduced recently by Hall, Rehren and Shpectorov. Axial algebras, in turn, are closely related to $3$-transposition groups and Vertex operator algebras. In this paper we consider fusion rules for semisimple idempotents, following Albert in the power-associative case. We examine the notion of an axis in the non-commutative setting and show that the dimension $d$ of any algebra $A$ generated by a pair $a,b$ of (not necessarily Jordan) axes of respective types $(λ,δ)$ and $(λ',δ')$ must be at most $5$; $d$ cannot be $4.$ If $d\le 3$ we list all the possibilities for $A$ up to isomorphism. We prove a variety of additional results and mention some research questions at the end.
The Peirce decomposition of a Jordan algebra with respect to an idempotent is well known. This decomposition was taken one step further and generalized recently by Hall, Rehren and Shpectorov, withtheir introduction of {\it axial algebras}, and in particular {\it primitive axial algebras of Jordan type} (PJs for short). It turns out that these notions are closely related to $3$-transposition groups and vertex operator algebras. De Medts, Peacock, Shpectorov, and M. Van Couwenberghe generalized axial algebrasto {\it decomposition algebras} which, in particular, are not necessarily commutative. This paper deals with decomposition algebras which are non-commutative versions of PJs.
Let $R$ be an associative ring with ${\bf 1}$ which is not commutative. Assume that any non-zero commutator $v\in R$ satisfies: $v^2$ is in the center of $R$ and $v$ is not a zero-divisor. (Note that our assumptions do not include finite dimensionality.) We prove that $R$ has no zero divisors, and that if ${\rm char(R)}\ne 2,$ then the localization of $R$ at its center is a quaternion division algebra.
The purpose of this short note is to prove that if $R$ is an alternative ring whose associators are not zero-divisors, then $R$ has no zero divisors. By a result of Bruck and Kleinfeld, if, in addition, the characteristic of $R$ is not $2,$ then the central quotient of $R$ is an octonion division algebra over some field.
``Fusion rules'' are laws of multiplication among eigenspaces of an idempotent. We establish fusion rules for flexible power-associative algebras, following Albert. We define the notion of an axis in the noncommutative setting (compare with [HRS]) and accumulate information about pairs of axes. We also describe a class of noncommutative examples of flexible power-associative algebras.
In this paper we prove that if $R$ is a proper alternative ring whose additive group has no $3$-torsion and whose non-zero commutators are not zero-divisors, then $R$ has no zero-divisors. It follows from a theorem of Bruck and Kleinfeld that if, in addition, the characteristic of $R$ is not $2,$ then the central quotient of $R$ is an octonion division algebra over some field. We include other characterizations of octonion division algebras and we also deal with the case where $(R,+)$ has $3$-torsion.
Let $A$ be a commutative, non-associative algebra over a field $\mathbb{F}$ of characteristic $\ne 2$. A half-axis in $A$ is an idempotent $e\in A$ such that $e$ satisfies the Peirce multiplication rules in a Jordan algebra, and, in addition, the $1$-eigenspace of ${\rm ad}_e$ (multiplication by $e$) is one dimensional. In this paper we consider the identities $(*)$ $x^2x^2=x^4$ and $x^3x^2=xx^4.$ We show that if identities $(*)$ hold strictly in $A,$ then one gets (very) interesting identities between elements in the eigenspaces of ${\rm ad}_e$ (note that if $|\mathbb{F}|>3$ and the identities $(*)$ hold in $A,$ then they hold strictly in $A$). Furthermore we prove that if $A$ is a primitive axial algebra of Jordan type half (i.e., $A$ is generated by half-axes), and the identities $(*)$ hold strictly in $A,$ then $A$ is a Jordan algebra.
In this note we give an overview of our knowledge regarding primitive axial algebras of Jordan type half and connections between $3$-transposition groups and Matsuo algebras. We also show that primitive axial algebras of Jordan type $η$ admit a Frobenius form, for any $η$.
The fixed point building of a polarity of a Moufang quadrangle of type $F_4$ is a Moufang set, as is the fixed point building of a semi-linear automorphism of order $2$ of a Moufang octagon that stabilizes at least two panels of one type but none of the other. We show that these two classes of Moufang sets are, in fact, the same, that each member of this class can be constructed as the fixed point building of a group of order $4$ acting on a building of type $F_4$ and that the group generated by all the root groups of any one of these Moufang sets is simple.
Associative or Jordan algebras generated by two idempotents are described precisely.
The purpose of this note is to observe that a homomorphism of discrete groups $f:Γ\to G$ arises as the induced map $π_0(\mathfrak{M})\to π_0(\mathfrak{X})$ on path components of some closed normal inclusion of topological groups $\mathfrak{M}\subseteq \mathfrak{X},$ if and only if the map $f$ can be equipped with a crossed module structure. In that case an essentially unique realization $\mathfrak{M}\subseteq \mathfrak{X}$ exists by homotopically discrete topological groups.