## Center of an Algebra

**Point of post: **In this post we define the center of an associative unital algebra and classify the center of .

*Motivation*

Often questions come up such as “do the linear transformations and commute?” or “classify all linear transformations which commute with “. An obvious question might be, for which transformations is the answer to this latter question “All of the endomorphsim algebra.” Said differently, which transformations have the property that they commute with *every *transformation on that space? This set, the set of all such ‘universal commuters’, is called the *center of an algebra. *It turns out that for the endomorphism algebra, the center is quite ‘small’.

**Center of an Algebra**

Let be an associative unital algebra. We define the *center *of , denoted , to be the set of all elements of which commute with all the other elements of . More formally

We prove one small theorem about the center of an algebra, namely that it is a subalgebra. Indeed:

**Theorem: ***Let be an associative unital -algebra with identity . Then the center is an associative unital subalgebra of .*

**Proof: **It’s clear that since for every

and

Note next then that if , , and , then

and

so that . The conclusion follows.

*Center of *

In this section we wish to describe, in full, for every . Namely, we prove that:

**Theorem: ***Let be a field and . Then,*

* *

* *

**Proof: **Let be the matrix whose general term is . In other words, has zeros for every entry except the entry where it’s one. Note then that if then

Note though that the left hand side of the above equations is a matrix which is zero for all entries except it’s column and the right hand size has all zero entries except its row. Thus, for the two to be equal we realize that the only non-zero entry along the row and column occurs at the diagonal position . Doing this for every we may conclude that

for some (note that this is a diagonal matrix, for those [almost surely everyone reading this] familiar with this notation).

But using the notation to represent the matrix whose general entry is we see by assumption that

But, notice that the left hand side of the above is just except the and columns are interchanged and the right hand side is with the and rows interchanged. Doing this for every such combinations of gives that for some . Thus, as desired.

Thus

and since the reverse inclusion is evidently true (in fact, it’s true via the theorem in the previous section) the conclusion follows.

From this, and a previous theorem we get the following corollary:

**Corollary: ***Let be an -dimensional space, then*

* *

**References:**

1. Golan, Jonathan S. *The Linear Algebra a Beginning Graduate Student Ought to Know*. Dordrecht: Springer, 2007. Print.

2. Halmos, Paul R. *Finite-dimensional Vector Spaces,*. New York: Springer-Verlag, 1974. Print

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