Let $p\in P^{*}_{{\bar{E}\/}}$ . Let $s$ be a function with $\operatorname{dom}s={1,\dotsc,n}$ such that for all $i$ $s(i)$ satisfies definition 6.2 . Then we define $(p)_{\langle s\rangle}$ as $p^{n}$ where $p^{n}$ is defined by induction as follows:

	$\displaystyle p^{0}=p,$
	$\displaystyle p^{i+1}=p^{i}_{i}\mathop{{}^{\frown}}\dotsb\mathop{{}^{\frown}}p^{i}_{1}\mathop{{}^{\frown}}(p^{i}_{0})_{\langle s(i+1)\rangle}.$

Let $A_{1}$ , $A_{2}$ be sets, and $\Cal{P}(A_{i})$ the set of all subsets of $A_{i}$ ( $i=1,2$ ). A bipartition of $(A_{1},A_{2})$ is a subset $\pi\subset\Cal{P}(A_{1})\times\Cal{P}(A_{2})$ such that the following two conditions are satisfied: i) If $S,T\in\pi$ are distinct, then $S_{i}\cap T_{i}=\emptyset$ ( $i=1,2$ ). ii) $A_{i}=\cup_{S\in\pi}S_{i}$ ( $i=1,2$ ) A bipartition $\pi$ is called special if $w(S,T)$ is defined for all $(S,T)\in\pi$ ; that is if the following holds

$$(\emptyset,S)\text{\ \ or\ \ }(S,\emptyset)\in\pi\Rightarrow\#S=1$$

$14$

Let

$$\dot{x}=\varphi(x,u),$$

$15$

be the controlled system, where $x$ is the time-dependent $m$ -dimensional complex vector and $u$ is the control parameter. Dynamical inverse problem of representation theory for the controlled system (15) is to construct a representative dynamics

$$\dot{\mathbb{X}}=F(\mathbb{X},a)$$

and the function

$$a=a(u,x)\quad\text{such that}\quad\varphi(x,u)=f(x,a(u,x)),$$

where the operator function $F$ is defined by the Weyl (symmetric) symbol $f$ as a function of $m$ non-commuting variables $X_{1},\ldots X_{m}$ .

[Lie algebra]

A Lie algebra $\mathfrak{g}$ over field $k$ is a vector space over $k$ with operation $[\cdot,\cdot]:\mathfrak{g}\times\mathfrak{g}\rightarrow\mathfrak{g}$ that satisfies the following properties:

1. 1.

   skew-symmetry:

   $$[a,b]=-[b,a]$$

2. 2.

   Jacobi identity:

   $$[[a,b],c]+[[c,a],b]+[[b,c],a]=0$$

[Multiplicative group] Let $\mathfrak{A}$ be an associative algebra with unity. Then $\mathfrak{A}^{*}$ - the set of all invertible elements of $\mathfrak{A}$ - can be regarded as a group under the operation of multiplication.

The Lie algebra of $\mathfrak{A}^{*}$ is the algebra $\mathfrak{A}$ itself with commutator

$$[a,b]=ab-ba$$