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
Canonical name	Nucleus
Date of creation	2013-03-22 14:52:19
Last modified on	2013-03-22 14:52:19
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	10
Author	CWoo (3771)
Entry type	Definition
Classification	msc 17A01
Defines	center of a nonassociative algebra
Defines	nuclear