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Abstract algebra problem

  1. Feb 17, 2012 #1
    1. The problem statement, all variables and given/known data

    Show that the center of a Clifford algebra of order 2^n is of order 1 if n is even, and 2 if n is odd

    2. Relevant equations
    the center of an algebra is the subalgebra that commutes with all elements
    Clifford algebra of 2^n is defined as being spanned by the bases
    [itex]\gamma_0,\gamma_1,...\gamma_n[/itex]
    where [itex]\gamma_0[/itex] is the unit element, as well as
    [itex]\gamma_{\{n_k\}}=\Pi_{n_k}\gamma_{n_k}[/itex]
    where [itex]1\le n_k\le n[/itex], and the mutiplication rule is
    [itex]\gamma_u\gamma_v+\gamma_v\gamma_u=2\delta_{u,v} \gamma_0[/itex]
    where [itex]\delta[/itex] is the Kronecker Delta symbol

    3. The attempt at a solution

    I was able to show that for n even, only [itex]\gamma_0[/itex] commutes with all the other bases
    and for n odd, only [itex]\gamma_0[/itex] and [itex]\gamma_{\{1,2,...,n\}}[/itex] commutes with all the other bases, but
    how can I know that the center is spanned by the bases that commutes with
    all the other bases?

    In other words, how do I know that no linear combinations of
    non-commuting bases commutes to all bases?
     
    Last edited: Feb 17, 2012
  2. jcsd
  3. Feb 20, 2012 #2
    I finally figured it out. Two things need to be realized:
    1.All the basis elements are invertible
    2.Any pair of the basis elements either commute with each other or anti-commute with each other;
    These may be verified fairly easily from the definition.

    Now if x is in the center, x must commute with all the basis elements. For any basis element γ, x may be
    written as x=x1+x2, where x1γ=γx1 and x2γ=-γx2; since xγ=γx, we have x2γ=0; since γ is invertible, x2=0;
    since x2 which depends on γ equals 0 for all γ, x must contain only components that commutes with all basis,
    i.e., γ0 and possibly γ{1,2,...,n}, qed.

    fairly straightforward, it feels good to know the answer. Anyone has a simpler proof?
     
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