The Unapologetic Mathematician

Mathematics for the interested outsider

The Center of an Algebra

Sorry I forgot to get this posted this morning.

Given an algebra A, it’s interesting to consider the “center” Z_A of A. This is the collection of algebra elements that commute with all the others. That is,

\displaystyle Z_A=\{a\in A\vert\forall b\in A, ab=ba\}

It’s straightforward to see that sums, scalar multiples, and products of central elements — elements of Z_A — are themselves central. That is, Z_A is an algebra, and it’s a commutative one to boot. This gives us a construction that starts with an associative algebra and ends with a commutative algebra, and yet it turns out that it is not a functor! I don’t really want to get into that right now, though, but I wanted to mention it in passing, since it’s one of the few examples of a natural algebraic construction that isn’t functorial.

What I do want to get into right now, is calculating the center of the matrix algebra \mathrm{Mat}_d(\mathbb{C}). The answer is reminiscent of Schur’s lemma:

Z_{\mathrm{Mat}_d(\mathbb{C})}=\{c I_d\vert c\in\mathbb{C}\}

Suppose that C is a central d\times d matrix. Then in particular it commutes with the matrix E_{i,i}, which has a 1 at the ith place along the diagonal and 0s everywhere else. That is, CE_{i,i}=E_{i,i}C. But CE_{i,i} zeroes out everything except the ith column of C, while E_{i,i}C zeroes out everything except the ith row. For these two be equal, the ith column must be all zeroes except for the one spot along the diagonal, and similarly for the ith row. And so C must be diagonal.

For i\neq j, C must also commute with E_{i,j}+E_{j,i} — the matrix with ones in the jth column of the ith row and the ith column of the jth row. That is, C(E_{i,j}+E_{j,i})=(E_{i,j}+E_{j,i})C. Multiplying on the right by E_{i,j}+E_{j,i} swaps the ith and jth columns of C, while multiplying on the left swaps the ith and jth rows. Thus we can tell that not only is C diagonal, but all the diagonal entries must be the same. And so C=c I_d for some complex c.

October 6, 2010 Posted by | Algebra, Group theory, Representation Theory | 2 Comments