References

Definition 2.1

Let $P$ be a smooth manifold and let $C^{\infty}(P)$ be the ring of smooth functions on it. A Leibniz bracket on $P$ is a bilinear map $[\cdot,\cdot]:C^{\infty}(P)\times C^{\infty}(P)\to C^{\infty}(P)$ that is a derivation on each entry, that is,

[fg,h]=[f,h]g+f[g,h]\text{\ \ \ and\ \ \ }[f,gh]=g[f,h]+h[f,g],

for any $f,g,h\in C^{\infty}(P)$ . We will say that the pair $(P,[\cdot,\cdot])$ is a Leibniz manifold . If the bracket $[\cdot,\cdot]$ is antisymmetric, that is, it satisfies

[f,g]=-[g,f]

for every pair of functions $f,g\in C^{\infty}(P)$ then we say that $(P,[\cdot,\cdot])$ is an almost Poisson manifold . We will usually denote the almost Poisson brackets with the symbol $\{\cdot,\cdot\}$ .

A function $f\in C^{\infty}(P)$ such that $[f,g]=0$ (respectively, $[g,f]=0$ ) for any $g\in C^{\infty}(P)$ is called a left (respectively, right ) Casimir of the Leibniz manifold $(P,[\cdot,\cdot])$ .

Definition 3

Almost-linear — Say that an expectation-valued function $f$ of possibly several expectation arguments $x,y,\cdots,z$ is almost-linear if it can be written in the form

f.x.y\cdots.z~{}~{}~{}\mathrel{\hat{=}}~{}~{}~{}w+g.x+h.y+\cdots+i.z~{},

(21)

where $w$ is an expectation and $g,h,\cdots,i$ are linear expectation-valued functions of their single arguments.

Definition 2.1

[ $t$ -Baxter operator] Let $t\in k$ . Let $(A,\ \cdot)$ be an associative algebra. A $t$ -Baxter operator is a linear map $\beta:A\xrightarrow{}A$ verifying:

\beta(x)\cdot\beta(y)=\beta(x\cdot\beta(y)+\beta(x)\cdot y+tx\cdot y).

Definition 5.3 .

Given a positive integer $\ell$ we define the class ${\cal M}_{\ell}\subset{\cal M}$ of LP- representatives to consist of functions of the form

\bar{m}=m\star m\star\ldots\star m=(m\star)^{\ell},

for some $m\in{\cal M}$ .

Definition 1.4 .

Let $(A,\delta)$ be a C ${}^{*}$ -bialgebra and $h$ be a locally finite, lower semicontinuous weight on $A$ . We say that $h$ is right invariant if for any $a\in A_{+}$ such that $h(a)<\infty$ and any $\varphi\in A^{*}_{+}$ we have

h(\varphi*a)=\varphi(1)h(a).

Definition B.1 .

Let $B$ be an operator algebra acting on a Hilbert space ${\mathcal{H}}$ and $\eta$ be a (unbounded) linear mapping from $B$ into ${\mathcal{H}}$ . We say that $\eta$ is a GNS map if the domain ${\mathcal{D}}(\eta)$ is a left ideal in $B$ and

(B.1)

\eta(ab)=a\eta(b)

for any $a\in B$ and $b\in{\mathcal{D}}(\eta)$ . The GNS map $\eta$ is called closed if it is closed with respect to strong operator topology on $B$ and norm topology on ${\mathcal{H}}$ . We say that a GNS map defined on a C ${}^{*}$ -algebra is densely defined if its domain is norm dense in the C ${}^{*}$ -algebra. Similarly a GNS map defined on a von Neumann algebra is said to be densely defined if its domain is strongly dense in the von Neumann algebra.

Definition 3 .

Let us consider a higher-order state $\rho\in\mathbb{S}_{n}(A)$ . A projector $p\in A$ is called $\rho$ -compatible if

(41)

\rho(p)+\rho(1-p)=1.

The state $\rho$ is called $A$ -compatible if the set of all $\rho$ -compatible projectors generates the whole $C^{*}$ -algebra $A$ . Finally, for a given $A$ -compatible state $\rho$ , a unital $C^{*}$ -subalgebra $B\subseteq A$ is called $\rho$ -compatible, if ( 40 ) holds for all mutually orthogonal projectors from $B$ .

Definition 3.1 .

(cf. [ DW ] ) A symmetric quiver $S$ is a quiver endowed with an involution $*$ acting on both $S_{0}$ and $S_{1}$ such that

t(\varphi^{*})=(h\varphi)^{*},h(\varphi^{*})=(t\varphi)^{*}.