What is the center of SL(n,C)?

1. Nov 15, 2009

cmj1988

What is the center of SL(n,C)?

I understand that the center of a group is where all elements commute with the group G. So I figure that I should come up with a case in which matricies commute. I remember a few facts from Linear Algebra:

Fact 1: Simultaneously diagonalizeanle matricies lend itself to commutivity
Fact 2: Matrix multiplication is associative

So given an M in SL(n,C) and an A, B in GL(n,C):
D1=MAM-1
D2=MBM-1

So, AB=M-1D1MM-1D2M=M-1D1D2M

We know that diagonal matricies are commutative

M-1D2D1M

Invoking associativity

BA

I'm not sure if I actually answered the question.

2. Nov 16, 2009

HallsofIvy

Staff Emeritus
As you said at the beginning, "the center of a group is where all elements commute with the group G". "Simultaneously diagonalizable" matrices commute with each other but not, generally, with other matrices. Also, matrices A and B may be simultaneously diagonalizable, and matrices C and D may be simultaneously diagonalizable, but A and B not simultaneously diagaonalizabel with C and D. Which pair would you take as center?