# Decomposition of the group of Bogoliubov transformations

Consider the fermion Fock space $$\mathcal{F}=\bigoplus_{k\ge 0}\bigwedge^k\mathfrak{h}$$ of some finite-dimensional 1-particle Hilbert space $$\mathfrak{h}$$. The group $$\mathrm{Bog}(\mathcal{F})$$ of Bogoliubov transformations can be defined as the set of unitary maps $$U$$ on $$\mathcal{F}$$ for which there are linear maps $$u:\mathfrak{h}\to\mathfrak{h}$$ and $$v:\mathfrak{h}\to\mathfrak{h}^*$$ such that $$Ua^*(f)U^*=a^*(uf)+a(J^*(v(f)))\quad\forall f\in\mathfrak{h},$$ where $$a^*,a:\mathfrak{h}\to\mathcal{B}(\mathcal{F})$$ denote the usual fermion creation- and annihilation operators and $$J:\mathfrak{h}\to\mathfrak{h}^*$$ denotes the Riesz isomorphism. It is not hard to see that these $$u$$ and $$v$$ define a unitary map $$\Phi(U)\in U(\mathfrak{h}\oplus\mathfrak{h}^*)$$ commuting with $$\mathcal{J}$$, where $$\Phi(U):=\begin{pmatrix}u&J^*vJ^*\\ v&JuJ^*\end{pmatrix},\quad\mathcal{J}:=\begin{pmatrix}0&J^*\\J&0\end{pmatrix}.$$

Defining $$G:=\{A\in U(\mathfrak{h}\oplus\mathfrak{h}^*)\mid A\mathcal{J}=\mathcal{J}A\}$$, it turns out that $$\Phi$$ defines a short exact sequence of Lie groups $$1\to\mathbb{S}^1\to\mathrm{Bog}(\mathcal{F})\to G\to 1,$$ Now my question is: does this sequence split (or, put differently, is $$\mathrm{Bog}(\mathcal{F})\cong\mathbb{S}^1\times G$$)?

Note that, if we are working in the category of groups (as opposed to Lie groups), central extensions of the group $$G$$ by $$\mathbb{S}^1$$ are classified (upto isomorphism) by the cohomology group $$H^2(G,\mathbb{S}^1)$$. If this classification is also valid in the Lie group setting, there might be some general result showing that $$H^2(G,\mathbb{S}^1)=0$$, which would answer my question positively.

• If i give you a $v : h^* \to h$ could you give me a preimage for the matrice with coeeficients $u=0$ and $v$ ? – InfiniteLooper Mar 1 at 14:55
• Well, yes: given $v:\mathfrak{h}\to\mathfrak{h}^*$ (note the order of $\mathfrak{h}$ and $\mathfrak{h}^*$), then there is a Bogoliubov transformation $U$ on $\mathcal{F}$ with $u=0$ and the given $v$. The main difficulty here is to construct the image $U\Omega$ of the vacuum $\Omega\in\mathcal{F}$. A proof can be found e.g. in Solovejs lecture Notes on Many Particle Quantum Mechanics (Theorem 9.5). Note that in the infinite-dimensional case $V^*V$ is required to be trace class ("Shale Stinespring condition"). – Robert Rauch Mar 4 at 9:26