centralizers in algebra
1 Abstract definitions and properties
If we regard in the of actions we can perscribe a left action and a right action . The centralizer is thus the set of elements for which the left regular action and the right regular action agree when to .
It is generally possible to have not lie in for and , and likewise, it is also possible that if that . Therefore it should not be presumed that the centralizer is central.
With further axioms on the of operation we can deduce certain natural for the set .
If , then . In particular, .
If has an identity then is non-empty. In particular, in this case is non-empty. 11An identity of is an element such that for all .
If is associative and then , we say then that is closed to the binary operation of .
If is commutative then .
If is a subset of the center of then .
Note that it is possible for be a subset closed to the opertaion without the assumption of associativity, as for example, when is commutative.
2 Centralizers in groups
In the category of groups the centralizer in a group of a subset can be redefined as:
If one regards conjugation as a group action then it follows that the centralizer is the same as the pointwise stabilizer in of , where the action is of on itself by conjugation. Because of this overlap, in some contexts the centralizers is applied to the pointwise stabilizer of a set on which a group acts, though this context no longer refers to the action of conjugation. This is espeically common when there is a need to distinguish between the pointwise stabilizer and the setwise stabilizer.
3 Centralizers in rings and algebras
For we treat rings as algebras over and now speak only of algebras, which will include nonassociative examples.
In an algebra there is in fact two binary operations on the set in question. Thus the abstract definition of the centralizer is ambiguous. However, the additive operation of rings and algebras is always commutative and so any centralizer with respect to this operation is the set . Thus it is generally accepted practice to assume that centralizers in this context always refer to the multiplicative operation. In this way we have the following properties.
A centralizer in an algebra is also called a commutant. This terminology is mostly used in algebras of operators in functional analysis.
Let be a ring (or an algebra). For every ordered pair of elements of , we can define the additive commutator of to be the element , written . With this, one may alternatively define the centralizer of a set in a ring as
Of course, in this definition, two operations (multiplication and subtraction) are needed instead of one. But the nice aspect about this definition is that one can “measure” commutativity of a ring by the additive commutation operation. For example, one can show that, in a division ring, if every element additively commutes with every additive commutator, then the ring must be a field.
4 Centralizers in Lie algebras
Suppose is a Lie aglebra over a commutative ring of characteristic not . Given a subset of , then for and from the axioms of a Lie algebra multiplication. Therefore whenever it follows that so that . This motivates the more common redefinition of the centralizer in a Lie algebra:
Despite the incongruety in characteristic 2, this new definition replaces the original definition of centralizers for Lie algebras. The centralizer of a Lie algebra is a subalgebra.
5 Centralizers in other nonassociative algebras
The centralizer need not be a subalgebra on account of the lack of associativity. There are instances of non-associative algebras where the centralizer is however a subalgebra nontheless, for example, Lie algebras as seen above. In travial fashion, if an algebra is commutative then and so the centralizer is a subalgebra but without any useful properties. There is a suitable additional constraint to add to centralizers to force them to be subalgebras and carry with them more useful in the commutative but nonassociative setting.
We write for , called the commutator in of and also write for and call it the associator in of .33This notation for associators is non-standard but the standard is likely confusing given the usual commutator notation used already. Then we can redefine the centralizer in of a subset of as
It follows that is a subalgebra of on account of the added associator condition which forces the subset to be closed to the product.
In alternative algebras, if any one of three associators is 0 then the other three are as well and so the definition reduces to . occur of other nonassociative algebras.
|Title||centralizers in algebra|
|Date of creation||2013-03-22 17:22:30|
|Last modified on||2013-03-22 17:22:30|
|Last modified by||Algeboy (12884)|