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

  1. Nov 15, 2009 #1
    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. jcsd
  3. Nov 16, 2009 #2

    HallsofIvy

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    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?

    Instead of thinking about "diagonalizable" matrices, look at diagonal matrices.
     
  4. Nov 16, 2009 #3
    So all diagonal matrices?
     
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