# Supercommutator of exterior multiplication operators and their adjoints

Let $\mathfrak{h}$ be a complex Hilbert space and consider Grassmann algebra $\mathcal{F}=\bigwedge\mathfrak{h}$ with its induced inner product. For $\omega\in\mathcal{F}$ we also consider the operators $\varepsilon_\omega$ and its adjoint $\varepsilon^\dagger_\omega$ on $\mathcal{F}$, where $\varepsilon_\omega(\eta):=\omega\wedge\eta$.

Physically, $\mathcal{F}$ is the fermionic Fock space of the (1-particle) Hilbert space $\mathfrak{h}$, and $\varepsilon_\varphi,\varepsilon_\varphi^\dagger$ are the creation/annihilation operators of the 1-particle state $\varphi\in\mathfrak{h}$. A straightforward calculation shows that for $\varphi,\psi\in\mathfrak{h}\subset\mathcal{F}$ the canonical anticommutation relations $\{\varepsilon^\dagger_\varphi,\varepsilon_\psi\}=\langle\psi,\varphi\rangle_\mathfrak{h}$ are satisfied.

Now, I want to compute higher commutation relations for these operators. For example, I have found the relation $\varepsilon_\omega\varepsilon^\dagger_\varphi=(-1)^k\left(\varepsilon^\dagger_\varphi\varepsilon_\omega-\varepsilon_{\varepsilon^\dagger_\varphi\omega}\right)$, valid for all $\omega\in\bigwedge^k\mathfrak{h}$ and $\varphi\in\mathfrak{h}$. It turns out that this can equivalently be written in terms of the supercommutator $$\left[\varepsilon_\omega,\varepsilon^\dagger_\varphi\right]_s=(-1)^{k+1}\varepsilon_{\varepsilon^\dagger_\varphi\omega}.$$

Now my question is: Is there a formula for $\left[\varepsilon_\omega,\varepsilon^\dagger_\eta\right]_s$ and general $\omega,\eta\in\mathcal{F}$?

• I have had a look at the material and also the Wikipedia article on the Nijenhuis bracket, but I do not really see the connection. For example, I think my operator $\varepsilon^\dagger_\omega$ is quite different from what is denoted by $i_\omega$ in the given literature. – Robert Rauch Oct 29 '15 at 8:28
• Now I see that the Theorem on p. 359 gives a formula for the supercommutator $[\epsilon_\omega^\dagger,\epsilon_\eta^\dagger]_s$ in terms of the Frölicher-Nijenhuis bracket of $\omega,\eta$, great! Do you also have an idea/reference on what can be said about $[\epsilon_\omega,\epsilon_\eta^\dagger]_s$? – Robert Rauch Feb 21 at 10:00