# nucleus

Let $A$ be an algebra, not necessarily associative multiplicatively. The nucleus of $A$ is:

 $\mathcal{N}(A):=\{a\in A\mid[a,A,A]=[A,a,A]=[A,A,a]=0\},$

where $[\ ,,]$ is the associator bracket. In other words, the nucleus is the set of elements that multiplicatively associate with all elements of $A$. An element $a\in A$ is nuclear if $a\in\mathcal{N}(A)$.

$\mathcal{N}(A)$ is a Jordan subalgebra of $A$. To see this, let $a,b\in\mathcal{N}(A)$. Then for any $c,d\in A$,

 $\displaystyle[ab,c,d]$ $\displaystyle=$ $\displaystyle((ab)c)d-(ab)(cd)=(a(bc))d-(ab)(cd)$ (1) $\displaystyle=$ $\displaystyle a((bc)d)-(ab)(cd)=a(b(cd))-(ab)(cd)$ (2) $\displaystyle=$ $\displaystyle a(b(cd))-a(b(cd))=0$ (3)

Similarly, $[c,ab,d]=[c,d,ab]=0$ and so $ab\in\mathcal{N}(A)$.

Accompanying the concept of a nucleus is that of the center of a nonassociative algebra $A$ (which is slightly different from the definition of the center of an associative algebra):

 $\mathcal{Z}(A):=\{a\in\mathcal{N}(A)\mid[a,A]=0\},$

where $[\ ,]$ is the commutator bracket.

Hence elements in $\mathcal{Z}(A)$ commute as well as associate with all elements of $A$. Like the nucleus, the center of $A$ is also a Jordan subalgebra of $A$.

Title nucleus Nucleus 2013-03-22 14:52:19 2013-03-22 14:52:19 CWoo (3771) CWoo (3771) 10 CWoo (3771) Definition msc 17A01 center of a nonassociative algebra nuclear