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nucleus (Definition)

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

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

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



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Also defines:  center of a nonassociative algebra, nuclear
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Cross-references: commutator bracket, center, Jordan subalgebra, associator, associative, algebra
There are 14 references to this entry.

This is version 7 of nucleus, born on 2004-12-09, modified 2004-12-15.
Object id is 6548, canonical name is Nucleus.
Accessed 3176 times total.

Classification:
AMS MSC17A01 (Nonassociative rings and algebras :: General nonassociative rings :: General theory)

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