centralizers in algebra

1 Abstract definitions and properties

Definition 1.

Let $S$ be a set with a binary operation $\mathrm{*}$. Let $T$ be a subset of $S$. Then definethe centralizer in $S$ of $T$ as the subset

$$C_S(T)=\{s\in S:s*t=t*s,\text{ for all }t\in T\}.$$

The center of $S$ is defined as $C_S\mathit{}\mathrm{(}S\mathrm{)}$. This is commonly denoted $Z\mathit{}\mathrm{(}S\mathrm{)}$ where $Z$ is derived from the German word zentral. Subsets and elements of the centerare called central.

If we regard $*:S\times S\to S$ in the of actions we can perscribe a leftaction $s*t$ and a right action $t*s$. The centralizer is thus the set of elementsfor which the left regular action and the right regular action agree when to $T$.

It is generally possible to have $s*t$ not lie in $T$ for $s\in C_S(T)$ and$t\in T$, and likewise, it is also possible that if $s,s^{\prime }\in C_S(T)$ that $s*s^{\prime }\ne s^{\prime }*s$.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 $C_S(T)$.

Proposition 2.

1. If $A\subseteq B$, then $C_S(B)\subseteq C_S(A)$. In particular, $C_S(\mathrm{\varnothing })=S$.

2. If $S$ has an identity then $C_S(T)$ is non-empty. In particular, in this case $Z(S)$ is non-empty.

3. If $S$ is associative and $s,s^{\prime }\in C_S(T)$ then $s*s^{\prime }\in C_S(T)$,we say then that $C_S(T)$ is closed to the binary operation of $S$.

4. If $s\in C_S(T)$ and $s$ has an (strong) inverse $s^{-1}$, then$s^{-1}\in C_S(T)$.

5. If $S$ is commutative then $C_S(T)=S$.

6. If $T$ is a subset of the center of $S$ then $C_S(T)=S$.

Note that it is possible for $C_S(T)$ be a subset closed to the opertaion without theassumption of associativity, as for example, when $S$ is commutative.

2 Centralizers in groups

In the category of groups the centralizer in a group $G$ of a subset $H$ can be redefined as:

$$C_G(H)=\{g\in G:g^{-1}hg=h,\text{ for all }h\in H\}.$$

If one regards conjugation as a group action $h^g:=g^{-1}hg$ then it follows that thecentralizer is the same as the pointwise stabilizer in $G$ of $H$, where the action isof $G$ 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. Thisis espeically common when there is a need to distinguish between the pointwise stabilizerand the setwise stabilizer.

In this category, the centralizer is always a subgroup of $G$. Furthermore, if $H$ is a normalsubgroup of $G$, then so too is $C_G(H)$.

3 Centralizers in rings and algebras

For we treat rings as algebras over $\mathbb{Z}$ and now speak only ofalgebras, which will include nonassociative examples.

In an algebra $A$ there is in fact two binary operations on the set $A$ inquestion. Thus the abstract definition of the centralizer is ambiguous. However, the additiveoperation of rings and algebras is always commutative and so any centralizer with respect to thisoperation is the set $A$. Thus it is generally accepted practice to assume that centralizersin this context always refer to the multiplicative operation. In this way we have the followingproperties.

Proposition 3.

Given an algebra $A$ over a commutative unital ring $R$ and a subset $B$ of $A$, then

1. $C_A(B)$ is a submodule of $A$.

2. If $A$ is associative then $C_A(B)$ is a subalgebra.

3. $Z(A)\le C_A(B)$, in particular, if $A$ has a 1 then $1\in C_A(B)$ and so$R$ embeds in $C_A(B)$.

Remarks.

• A centralizer in an algebra is also called a commutant. This terminology is mostly used in algebras of operators in functional analysis.

• Let $R$ be a ring (or an algebra). For every ordered pair $(a,b)$ of elements of $R$, we can define the additive commutator of $(a,b)$ to be the element $ab-ba$, written $[a,b]$. With this, one may alternatively define the centralizer of a set $S\subseteq R$ in a ring $R$ as

  $$C_R(S):=\{r\in R\mid [r,s]=0\text{ for all }s\in S\}.$$

  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 $𝔤$ is a Lie aglebra over a commutative ring of characteristic not $2$.Given a subset $T$ of $𝔤$, then $[s,t]=-[t,s]$ for $s\in 𝔤$ and $t\in T$ from the axiomsof a Lie algebra multiplication. Therefore whenever $[s,t]=[t,s]$ it follows that$-[t,s]=[t,s]$ so that $[t,s]=0$. This motivates the more common redefinition ofthe centralizer in a Lie algebra:

$$C_{𝔤}(T)=\{s\in 𝔤:[s,t]=0,\text{ for all }t\in T\}.$$

Despite the incongruety in characteristic 2, this new definition replaces the originaldefinition of centralizers for Lie algebras. The centralizer of a Lie algebra is a subalgebra.

When the Lie multiplication is regarded as a commutator, so $[a,b]=ab-ba$, for exampleit it universal enveloping algebra, then $0=[a,b]=ab-ba$ is the same as $ab=ba$ and sothe centralizer of the Lie algebra coincides with the centralizer of the associativeenvelope.

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 howevera subalgebra nontheless, for example, Lie algebras as seen above. In travial fashion, ifan algebra is commutative then $C_A(B)=A$ and so the centralizer is a subalgebrabut without any useful properties. There is a suitable additional constraintto add to centralizers to force them to be subalgebras and carry with themmore useful in the commutative but nonassociative setting.

We write $[a,b]$ for $ab-ba$, called the commutator in $A$ of $a,b\in A$ and also write$/a,b,c/$ for $(ab)c-a(bc)$ and call it the associator in $A$ of $a,b,c\in A$. Then wecan redefine the centralizer in $A$ of a subset $T$ of $A$ as

$$C_A(T)=\{s\in A:[s,t]=0,/s^{\prime },s,t/=/s^{\prime },t,s/=/t,s^{\prime },s/=0\text{ for all }t\in T,s^{\prime }\in A\}.$$

It follows that $C_A(T)$ is a subalgebra of $A$ on account of the added associator conditionwhich 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 welland so the definition reduces to $/s^{\prime },s,t/=0$. occur of other nonassociativealgebras.