 (cYBE and mcYBE) respectively. We adopt some notation $X\wedge Y = X\otimes Y - Y \otimes X$ and define e.g. $$r= T_1 \wedge T_2 + T_3 \wedge T_4 +\dots \ , \quad RX = {\operatorname{Tr}}_2 ( r (\mathbb{I}\otimes X)) \ .$$

We define an inner product by the matrix trace of generators, ${\operatorname{Tr}}(T_{A} T_{B})$, and lower and raise indices with this inner product and its inverse. In this way the $r$-matrix acts as $$\label{eq:rmatdef}
R(T_{A}) \equiv R_{A}{}^{B}T_{B} \ , \quad R_{A}{}^{B} = {\operatorname{Tr}}\left( {\operatorname{Tr}}_2 ( r (\mathbb{I}\otimes T_{A})) T^{B} \right) \ .$$

Suppose we have a $\mathbb{Z}_{2}$ grading $\mathfrak{f} = \mathfrak{g}\oplus \mathfrak{k}$ for a subgroup $\mathfrak{g}$. Let $T_{A}$ be generators for $\mathfrak{f}$, $T_{\alpha}$ those of $\mathfrak{g}$ and $T_{i}$ the remaining orthogonal generators of $\mathfrak{k}$. We introduce a projector to the coset defined by $P(T_{\alpha})= 0$ and $P(T_{i})= T_{i}$ or, in matrix form, $$P(T_{A}) \equiv P_{A}{}^{B}T_{B} \ , \quad P_{A}{}^{B} = {\operatorname{Tr}}\left( P( T_{A}) T^{B} \right) \ .$$ We also define the adjoint action for $g\in F$ by $${\operatorname{Ad}}_{g} T_{A} \equiv gT_{A} g^{-1} \equiv D_{A}{}^{B}(g) T_{B} \ , \quad D_{AB} = {\operatorname{Tr}}(g T_{A} g^{-1} T_{B} ) \ .$$

Let the two-dimensional worldsheet field $g$ be a coset representative for $F/G$ with which we define the currents $$J_\pm = J_{\pm}^{A}T_{A} = g^{-1} \partial_{\pm} g \ , \quad J_{\pm}^{A}= {\operatorname{Tr}}(g^{-1} \partial_{\pm} g T^{A}) \ ,$$ where we use light-cone coordinates $\partial_\pm = \partial_0 \pm \partial_1$.

The standard (bosonic) $\sigma$-model whose target is the coset space $F/G$ is $$\label{eq:cosetPCM}
{\cal L } = {\operatorname{Tr}}(J_+ P(J_-) ) \ .$$ To define the Yang-Baxter model first we let $$R_{g} = {\operatorname{Ad}}_{g^{-1}} R {\operatorname{Ad}}_{g} \ , \quad (R_{g})_{A}{}^{B} = D(g)_{A}{}^{C}R_{C}{}^{D}D(g^{-1})_{D}{}^{B} \ ,$$ and define the operator $${\cal O} = \mathbb{I} - \eta R_{g}P \ , \quad {