# Fock representations of Zamolodchikov algebras and R-matrices

September 29, 2019

A variation of the Zamolodchikov-Faddeev algebra over a finite dimensional
Hilbert space $\mathcal{H}$ and an involutive unitary $R$-Matrix $S$ is
studied. This algebra carries a natural vacuum state, and the corresponding
Fock representation spaces $\mathcal{F}_S(\mathcal{H})$ are shown to satisfy
$\mathcal{F}_{S\boxplus R}(\mathcal{H}\oplus\mathcal{K}) \cong
\mathcal{F}_S(\mathcal{H})\otimes \mathcal{F}_R(\mathcal{K})$, where $S\boxplus
R$ is the box-sum of $S$ (on $\mathcal{H}\otimes\mathcal{H}$) and $R$ (on
$\mathcal{K}\otimes\mathcal{K}$). This analysis generalises the well-known
structure of Bose/Fermi Fock spaces and a recent result of Pennig.\par It is
also discussed to which extent the Fock representation depends on the
underlying $R$-matrix, and applications to quantum field theory (scaling limits
of integrable models) are sketched.

Keywords:

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