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 ${\mathrm{\Omega }}^{\ast }(M$). For each $X\in \mathfrak{g}$ one has operators:

and

These operators satisfy the relation

$d{i}_X+i_{X}d={\mathcal{L}}_X$

where $d$ is the de Rham differential $d:{\mathrm{\Omega }}^k(M)\to {\mathrm{\Omega }}^{k+1}(M)$, 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

$\widetilde{ad}:\mathfrak{g}\to \mathfrak{spin}(\mathfrak{g})$

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 $\mathfrak{g}$, $X_a^{\ast }$ the dual basis, then

$\widetilde{ad}(X)=\frac{1}{4}\sum _a{X}_a^{\ast }[X,X_a{]}_{\mathfrak{g}}$

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

$\gamma =\frac{1}{24}\sum _{a,b}{X}_a^{\ast }{X}_b^{\ast }[X_a,X_b{]}_{\mathfrak{g}}$

then

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

$d^{Cl}\circ d^{Cl}=0$ since ${\gamma }^2$ is a scalar which can be computed to be $-\frac{1}{48}tr{\mathrm{\Omega }}_{\mathfrak{g}}$, where ${\mathrm{\Omega }}_{\mathfrak{g}}$ 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 ${\mathrm{\Lambda }}^{\ast }(\mathfrak{g})$. This gives ${\mathrm{\Lambda }}^{\ast }(\mathfrak{g})$ 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 $H^{\ast }(\mathfrak{g},\mathbf{C})$.

$Cliff(\mathfrak{g})$ 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 ${\mathrm{\Lambda }}^{\ast }({\mathfrak{g}}^{\ast })$. Corresponding to $i_X,{\mathcal{L}}_X,d$ one has both elements of ${\mathrm{\Lambda }}^{\ast }({\mathfrak{g}}^{\ast })$ and their quantizations, the operators in $Cliff(\mathfrak{g})$ constructed above.

For more details about the above, see the following references