Fock representations of multicomponent commutation relations

Let $H$ be a separable Hilbert space and $T$ be a self-adjoint bounded linear operator on $H^{\otimes 2}$ with norm $\le1$, satisfying the Yang-Baxter equation. Bożejko and Speicher (1994) proved that the operator $T$ determines a $T$-deformed Fock space $\mathcal F(H)=\bigoplus_{n=0}^\infty\mathcal F_n(H)$. We start with reviewing and extending the known results about the structure of the $n$-particle spaces $\mathcal F_n(H)$ and the commutation relations satisfied by the corresponding creation and annihilation operators acting on $\mathcal F(H)$. We then choose $H=L^2(X\to V)$, the $L^2$-space of $V$-valued functions on $X$. Here $X:=\mathbb R^d$ and $V:=\mathbb C^m$ with $m\ge2$. Furthermore, we assume that the operator $T$ acting on $H^{\otimes 2}=L^2(X^2\to V^{\otimes 2})$ is given by $(Tf^{(2)})(x,y)=C_{x,y}f^{(2)}(y,x)$. Here, for a.a.\ $(x,y)\in X^2$, $C_{x,y}$ is a linear operator on $V^{\otimes 2}$ with norm $\le1$ that satisfies $C_{x,y}^*=C_{y,x}$ and the spectral quantum Yang-Baxter equation. The corresponding creation and annihilation operators describe a multicomponent quantum system. A special choice of the operator-valued function $C_{xy}$ in the case $d=2$ is associated with plektons. For a multicomponent system, we describe its $T$-deformed Fock space and the available commutation relations satisfied by the corresponding creation and annihilation operators. Finally, we consider several examples of multicomponent quantum systems. This is joint work with A. Daletskii, A. Kalyuzhny and D. Proskurin.