Notes on BRST IX: Clifford Algebras and Lie Algebras

Note: I’ve started putting together the material from these postings into a proper document, available here, which will be getting updated as time goes on. I’ll be making changes and additions to the text there, not on the blog postings. For most purposes, that will be what people interested in this subject will want to take a look at.

When a Lie group with Lie algebra \mathfrak g acts on a manifold M, one gets two sorts of actions of \mathfrak g on the differential forms \Omega^*(M). For each X\in \mathfrak g one has operators:

  • \mathcal L}_X: \Omega^k(M)\rightarrow\Omega^k(M), the Lie derivative along the vector field on M corresponding to X
  • and

  • i_X:\Omega^k(M)\rightarrow\Omega^{k-1}(M), contraction by the vector field on M corresponding to X
  • These operators satisfy the relation

    di_X+i_Xd={\mathcal L}_X

    where d is the de Rham differential d:\Omega^k(M)\rightarrow \Omega^{k+1}(M), and the operators d, i_X, \mathcal L_X are (super)-derivations. In general, an algebra carrying an action by operators satisfying the same relations satisfied by d, i_X, \mathcal L_X will be called a \mathfrak g-differential algebra. It will turn out that the Clifford algebra Cliff(\mathfrak g) of a semi-simple Lie algebra \mathfrak g carries not just the Clifford algebra structure, but the additional structure of a \mathfrak g-differential algebra, in this case with \mathbf Z_2, not \mathbf Z grading.

    Note that in this section the commutator symbol will be the supercommutator in the Clifford algebra (commutator or anti-commutator, depending on the \mathbf Z_2 grading). When the Lie bracket is needed, it will be denoted [\cdot,\cdot]_{\mathfrak g}.

    To get a \mathfrak g-differential algebra on Cliff(\mathfrak g) we need to construct super-derivations i_X^{Cl}, {\mathcal L}_X^{Cl}, and d^{Cl} satisfying the appropriate relations. For the first of these we don’t need the fact that this is the Clifford algebra of a Lie algebra, and can just define

    i_X^{Cl}(\cdot)=[-\frac{1}{2}X,\cdot]

    For {\mathcal L}_X^{Cl}, we need to use the fact that since the adjoint representation preserves the inner product, it gives a homomorphism

    \widetilde{ad}:\mathfrak g \rightarrow \mathfrak{spin}(\mathfrak g)

    where \mathfrak{spin}(\mathfrak g) is the Lie algebra of the group Spin(\mathfrak g) (the spin group for the inner product space \mathfrak g), which can be identified with quadratic elements of Cliff(\mathfrak g), taking the commutator as Lie bracket. Explicitly, if X_a is a basis of \mathfrak g, X_a^* the dual basis, then

    \widetilde{ad}(X)=\frac{1}{4}\sum_a X_a^*[X,X_a]_{\mathfrak g}

    and we get operators acting on Cliff(\mathfrak g)

    {\mathcal L}_X^{Cl}(\cdot)=[\widetilde{ad}(X),\cdot]

    Remarkably, an appropriate d^{Cl} can be constructed using a cubic element of Cliff(\mathfrak g). Let

    \gamma= \frac{1}{24}\sum_{a,b}X^*_aX^*_b[X_a,X_b]_{\mathfrak g}

    then

    d^{Cl}(\cdot)=[\gamma, \cdot]

    d^{Cl}\circ d^{Cl}=0 since \gamma^2 is a scalar which can be computed to be -\frac{1}{48}tr\Omega_{\mathfrak g}, where \Omega_{\mathfrak g} is the Casimir operator in the adjoint representation.

    The above constructions give Cliff(\mathfrak g) the structure of a filtered \mathfrak g-differential algebra, with associated graded algebra \Lambda^*(\mathfrak g). This gives \Lambda^*(\mathfrak g) the structure of a \mathfrak g-differential algebra, with operators i_X, \mathcal L_X, d. The cohomology of this differential algebra is just the Lie algebra cohomology H^*(\mathfrak g, \mathbf C).

    Cliff(\mathfrak g) can be thought of as an algebra of operators corresponding to the quantization of an anti-commuting phase space \mathfrak g. Classical observables are anti-commuting functions, elements of \Lambda^*(\mathfrak g^*). Corresponding to i_X, \mathcal L_X, d one has both elements of \Lambda^*(\mathfrak g^*) and their quantizations, the operators in Cliff(\mathfrak g) constructed above.

    For more details about the above, see the following references

  • A. Alekseev and E. Meinrenken, The non-commutative Weil algebra, Invent. Math 139, 135-172 (2000), or arXiv:math/9903052
  • E. Meinrenken, Clifford algebras and Lie groups, 2005 Toronto lecture notes
  • G. Landweber, Multiplets of representations and Kostant’s Dirac operator for equal rank loop groups, Duke Mathematical Journal 110, 121-160 (2001), or arXiv:math/0005057
  • B. Kostant and S. Sternberg, Symplectic reduction , BRS cohomology and infinite-dimensional Clifford algebras, Ann. Physics 176, 49-113 (1987)
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    4 Responses to Notes on BRST IX: Clifford Algebras and Lie Algebras

    1. Shantanu says:

      Peter this maybe a bit OT, But your thoughts about this paper by
      Mcelrath
      http://arxiv.org/abs/0812.2696 ?
      Thanks

    2. lewallen says:

      Thanks a lot for the pdf! It’s great and very convenient.

    3. Peter Woit says:

      Shantanu,

      Sounds pretty implausible, but I know very little about it. For some discussion with McElrath, see the comments at

      http://blogs.discovermagazine.com/cosmicvariance/2008/12/27/gravity-emergesfrom-neutrinos/

      One problem is that of universality, raised by Bee, which McElrath acknowledges he doesn’t yet have an answer for.

    4. Pingback: best of 2008 (2) : big theorems

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