Let $S$ be a locally compact group acting on a topological space $M$ preserving an ergodic probability measure $\mu$ . Let $L$ be a topological group. An $L$ -valued measurable cocycle for the $S$ -action on $M$ is a measurable map $\alpha:S\times M\rightarrow L$ satisfying

$$\alpha(gh,x)=\alpha(g,hx)\alpha(h,x)$$

for all $g,h\in S$ and almost-every $x\in M$ .

The algebra $\mathfrak{t}_{n}$ is the quotient of the free Lie algebra with $n(n-1)/2$ generators $t^{i,j}=t^{j,i}$ by the following relations

$$[t^{i,j},t^{k,l}]=0$$

(8)

if all indices $i,j,k$ , and $l$ are distinct, and

$$[t^{i,j}+t^{i,k},t^{j,k}]=0$$

(9)

for all triples of distinct indices $i,j$ , and $k$ .

The Grothendieck–Teichmüller Lie algebra is the Lie algebra spanned by the elements $\psi\in\mathfrak{lie}_{2}$ satisfying the following relations:

$$\psi(x,y)=-\psi(y,x),$$

(10)

$$\psi(x,y)+\psi(y,z)+\psi(z,x)=0,$$

(11)

where $z=-x-y$ ,

$$\psi(t^{1,2},t^{2,34})+\psi(t^{12,3},t^{3,4})=\psi(t^{2,3},t^{3,4})+\psi(t^{1,23},t^{23,4})+\psi(t^{1,2},t^{2,3}),$$

(12)

where the latter takes place in $\mathfrak{t}_{4}$ and $t^{i,jk}=t^{i,j}+t^{i,k}$ .

An $\omega$ -Lie algebra is called multiplicative if there is a linear form $\lambda:L\to K$ such that ( 4 ) holds, i.e.,

$$[[x,y],z]+[[z,x],y]+[[y,z],x]=\lambda([x,y])z+\lambda([z,x])y+\lambda([y,z])x$$

for any $x,y,z\in L$ .

A super-anticommutative superalgebra $L=L_{0}\oplus L_{1}$ is called an extended $\omega$ -Lie superalgebra if there is a super-skew-symmetric bilinear map $\omega:L\times L\to Cent(L)$ such that

(27)

$$\displaystyle(-1)^{\deg x\deg z}[x,[y,z]]+(-1)^{\deg z\deg y}[z,[x,y]]+(-1)^{\deg y\deg x}[y,[z,x]]\\ \displaystyle+(-1)^{\deg x\deg z}\omega(y,z)x+(-1)^{\deg z\deg y}\omega(x,y)z+(-1)^{\deg y\deg x}\omega(z,x)y=0$$

holds for any homogeneous elements $x,y,z\in L$ .

The system

$$\dot{x}(t)=f(x(t))+g(x(t))v(t-\tau)\;,$$

(13)

with $\tau\geq 0$ is semi-globally practically stabilizable by quantized feedback if for any $\varepsilon<R<0$ there exist a law $z(x)$ , a real number $u_{0}>0$ and an integer $j\geq 1$ such that the solution of

$$\dot{x}(t)=f(x(t))+g(x(t))\Psi(z(x(t-\tau)))\;,$$

(14)

starting from ${\cal R}=\{\varphi\in C^{1}([-2\tau,0],\mathbb{R}^{n}):||\varphi||_{c}\leq R\}$ enters $B_{\varepsilon}$ , the closed ball of radius $\varepsilon$ , at some finite time $t_{s}\geq 0$ , and remains in that set for all $t\geq t_{s}$ .

Let ${\mathfrak{g}}$ be a finite dimensional Lie algebra over $k$ , and put the standard (nonnegative) filtration on the enveloping algebra $U({\mathfrak{g}})$ , so that $U({\mathfrak{g}})_{0}=k$ and $U({\mathfrak{g}})_{1}=k+{\mathfrak{g}}$ , while $U({\mathfrak{g}})_{n}=U({\mathfrak{g}})_{1}^{n}$ for $n>1$ . The associated graded algebra is commutative, and is naturally identified with the symmetric algebra $S({\mathfrak{g}})$ of the vector space ${\mathfrak{g}}$ . In particular, we use the same symbol to denote an element of ${\mathfrak{g}}$ and its coset in $\operatorname{gr}_{1}U({\mathfrak{g}})=S({\mathfrak{g}})_{1}$ . Then $S({\mathfrak{g}})$ is the semiclassical limit of $U({\mathfrak{g}})$ , equipped with the Poisson bracket satisfying

$$\{e,f\}=[e,f]$$

for all $e,f\in{\mathfrak{g}}$ , where $[e,f]$ denotes the Lie product in ${\mathfrak{g}}$ . The above formula determines $\{-,-\}$ uniquely, since ${\mathfrak{g}}$ generates $S({\mathfrak{g}})$ .

Now view the dual space ${\mathfrak{g}}^{*}$ as an algebraic variety, namely the affine space ${\mathbb{A}}^{\dim{\mathfrak{g}}}$ . The coordinate ring ${\Cal{O}}({\mathfrak{g}}^{*})$ is a polynomial algebra over $k$ in $\dim{\mathfrak{g}}$ indeterminates, as is $S({\mathfrak{g}})$ . There is a canonical isomorphism

$$\theta:S({\mathfrak{g}})@>{\ \cong\ }>>{\Cal{O}}({\mathfrak{g}}^{*})$$

$2.6$

which sends each $e\in{\mathfrak{g}}$ to the polynomial function on ${\mathfrak{g}}^{*}$ given by evaluation at $e$ , that is, $\theta(e)(\alpha)=\alpha(e)$ for $\alpha\in{\mathfrak{g}}^{*}$ . (This isomorphism is often treated as an identification of the algebras $S({\mathfrak{g}})$ and ${\Cal{O}}({\mathfrak{g}}^{*})$ .) Via $\theta$ , the Poisson bracket on $S({\mathfrak{g}})$ obtained from the semiclassical limit process above carries over to a Poisson bracket on ${\Cal{O}}({\mathfrak{g}}^{*})$ , known as the Kirillov-Kostant-Souriau Poisson bracket .

If $\{e_{1},\dots,e_{n}\}$ is a basis for ${\mathfrak{g}}$ , then $S({\mathfrak{g}})=k[e_{1},\dots,e_{n}]$ and $\theta$ sends the $e_{i}$ to indeterminates $x_{i}$ such that ${\Cal{O}}({\mathfrak{g}}^{*})=k[x_{1},\dots,x_{n}]$ . An explicit description of the KKS Poisson bracket on ${\Cal{O}}({\mathfrak{g}}^{*})$ can be obtained in terms of the structure constants of ${\mathfrak{g}}$ , as follows. These constants are scalars $c^{l}_{ij}\in k$ such that $[e_{i},e_{j}]=\sum_{l}c^{l}_{ij}e_{l}$ for all $i$ , $j$ . Since $\{e_{i},e_{j}\}=[e_{i},e_{j}]$ in $S({\mathfrak{g}})$ , an application of $\theta$ yields $\{x_{i},x_{j}\}=\sum_{l}c^{l}_{ij}x_{l}$ for all $i$ , $j$ . It follows that

$$\{p,q\}=\sum_{i,j,l}c^{l}_{ij}x_{l}\frac{\partial p}{\partial x_{i}}\frac{\partial q}{\partial x_{j}}$$

for $p,q\in{\Cal{O}}({\mathfrak{g}}^{*})$ [ 7 , Proposition 1.3.18 ] . To see this, just check that the displayed formula determines a Poisson bracket on ${\Cal{O}}({\mathfrak{g}}^{*})$ which agrees with the KKS bracket on pairs of indeterminates.

The KKS Poisson bracket on ${\Cal{O}}({\mathfrak{g}}^{*})$ can also be obtained by applying the method of §2.1 to the homogenization of $U({\mathfrak{g}})$ , that is, the $k[h]$ -algebra $A$ with generating vector space ${\mathfrak{g}}$ and relations $ef-fe=h[e,f]$ for $e,f\in{\mathfrak{g}}$ (where $[e,f]$ again denotes the Lie product in ${\mathfrak{g}}$ ). Here $A/hA\cong S({\mathfrak{g}})\cong{\Cal{O}}({\mathfrak{g}}^{*})$ and $A/(h-\lambda)A\cong U({\mathfrak{g}})$ for all $\lambda\in k^{\times}$ .

Let ${\mathfrak{g}}$ be a finite dimensional complex Lie algebra. Treating ${\mathfrak{g}}$ for a moment just as a vector space, we have the general linear group $GL({\mathfrak{g}})$ on ${\mathfrak{g}}$ , which is a complex algebraic group whose Lie algebra is the general linear Lie algebra $\mathfrak{gl}({\mathfrak{g}})$ . Any algebraic subgroup of $GL({\mathfrak{g}})$ (i.e., any Zariski closed subgroup) has a Lie algebra which is naturally contained in $\mathfrak{gl}({\mathfrak{g}})$ . The algebraic adjoint group of ${\mathfrak{g}}$ is the smallest algebraic subgroup $G\subseteq GL({\mathfrak{g}})$ whose Lie algebra contains $\operatorname{ad}{\mathfrak{g}}=\{\operatorname{ad}x\mid x\in{\mathfrak{g}}\}$ (cf. [ 2 , §12.2; 50 , Definition 24.8.1 ] ).

The natural action of $GL({\mathfrak{g}})$ on ${\mathfrak{g}}$ by linear automorphisms restricts to an action of $G$ on ${\mathfrak{g}}$ , the adjoint action . This, in turn, induces a (left) action of $G$ on ${\mathfrak{g}}^{*}$ , the coadjoint action , under which

$$(g.\alpha)(x)=\alpha(g^{-1}.x)$$

for $g\in G$ , $\alpha\in{\mathfrak{g}}^{*}$ , and $x\in{\mathfrak{g}}$ . The orbits of this action, the coadjoint orbits , are collected in the set ${\mathfrak{g}}^{*}/G$ . We equip ${\mathfrak{g}}^{*}/G$ with the quotient topology induced from the Zariski topology on ${\mathfrak{g}}^{*}$ , and thus refer to it as the space of coadjoint orbits.

A semimodule , $M$ , over a semiring $\SS$ is an abelian monoid under addition which has a neutral element, ${0}$ , and is equipped with a law

$$\begin{array}[]{ccc}\SS\times M&\to&M\\ (s,{m})&\to&s\cdot{m}\end{array}$$

called action or scalar multiplication such that for all ${m}$ and ${m}^{\prime}$ in $M$ and $r,s\in\SS$

1. (1)

   $(s\cdot r)\cdot{m}=s\cdot(r\cdot{m})$ ,

2. (2)

   $(s+r)\cdot{m}=s\cdot{m}+r\cdot{m}$ ,

3. (3)

   $s\cdot({m}+{m}^{\prime})=s\cdot{m}+s\cdot{m}^{\prime}$ ,

4. (4)

   $1\cdot{m}={m}$ ,

5. (5)

   $s\cdot{0}={0}=0\cdot{m}$ .

A map $\tau:\SS\to\SS$ is a symmetry if $\tau$ is a left and right $\SS$ -semimodule homomorphism from $\SS$ to $\SS$ of order 2, i.e.,

(4.1a)		$\displaystyle\tau(a+b)=\tau(a)+\tau(b)$
(4.1b)		$\displaystyle\tau(0)=0$
(4.1c)		$\displaystyle\tau(a\cdot b)=a\cdot\tau(b)=\tau(a)\cdot b$
(4.1d)		$\displaystyle\tau(\tau(a))=a.$

The operad $\operatorname{Lie}^{2}$ (called also the operad of two compatible brackets) is generated by two skew-symmetric operations (brackets) $\{\cdot,\cdot\}$ and $[\cdot,\cdot]$ . The relations in this operad mean that all linear combinations of these brackets satisfy the Jacobi identity. It is equivalent to the following identities in each algebra over this operad:

	$$\displaystyle\{a,\{b,c\}\}+\{b,\{c,a\}\}+\{c,\{a,b\}\}=0,$$
	$$\displaystyle+[b,\{c,a\}]+[c,\{a,b\}]+\{a,[b,c]\}+\{b[c,a]\}+\{c,[a,b]\}=0,$$
	$$\displaystyle]+[b,[c,a]]+[c,[a,b]]=0.$$

The operad $\,{}^{2}\operatorname{Com}$ of two strongly compatible commutative products is generated by two symmetric binary operations (products) $\circ$ and $\bullet$ such that in any algebra over this operad the following identities hold:

	$$\displaystyle a\circ(b\circ c)=(a\circ b)\circ c,$$
	$$\displaystyle a\circ(b\bullet c)=a\bullet(b\circ c)=b\circ(a\bullet c)=b\bullet(a\circ c)=c\circ(a\bullet b)=c\bullet(a\circ b),$$
	$$\displaystyle a\bullet(b\bullet c)=(a\bullet b)\bullet c.$$

The operad $\operatorname{LieGriess}$ is a binary quadratic operad with two skew-symmetric generators $[\cdot{,}\cdot]$ and $\{\cdot{,}\cdot\}$ that satisfy the identities

	$$\displaystyle\{a,\{b,c\}\}+\{b,\{c,a\}\}+\{c,\{a,b\}\}=0,$$
	$$\displaystyle+[b,\{c,a\}]+[c,\{a,b\}]+\{a,[b,c]\}+\{b,[c,a]\}+\{c,[a,b]\}=0,$$

The Ramanujan operad $\operatorname{Ram}$ is a binary quadratic operad with a symmetric generator $\cdot\star\cdot$ and two skew-symmetric generator $[\cdot{,}\cdot]$ and $\{\cdot{,}\cdot\}$ for which the product $\cdot\star\cdot$ generates a suboperad isomorphic to $\operatorname{Com}$ , the operations $[\cdot{,}\cdot]$ and $\{\cdot{,}\cdot\}$ generate a suboperad isomorphic to $\operatorname{LieGriess}$ , and these suboperads together are related by a distributive law [ 20 , 8 ]

	$$\displaystyle[a,b\star c]=[a,b]\star c+b\star[a,c],$$
	$$\displaystyle\{a,b\star c\}=\{a,b\}\star c+b\star\{a,c\}.$$

A $k$ -ary function $f\in\mathcal{F}_{\!\mathrm{B}}$ is pure affine if there is a constant $w\in\mathbb{Q}^{>0}\!$ , an affine function $g\in\mathcal{P}_{k}$ and a polynomial $s\in\mathcal{P}_{k}$ such that

$$f(\bar{x})\ =\ w\hskip 1.0pt(-1)^{s(\bar{x})}g(\bar{x})\,.$$

(1)

(Pavičić and Megill [ 36 ] ) An ortholattice that satisfies the following condition:

$\displaystyle a\equiv b=1\qquad\Rightarrow\qquad(a\cup c)\equiv(b\cup c)=1$

(9)

is called a weakly orthomodular ortholattice , WOML .

(Pavičić [ 29 ] ) An ortholattice that satisfies the following condition:

$\displaystyle a\equiv b=1\qquad\Rightarrow\qquad a=b,$

(10)

is called an orthomodular lattice , OML .

(Foulis [ 39 ] , Kalmbach [ 11 ] ) An ortholattice that satisfies either of the following two conditions:

	$\displaystyle a\cup(a^{\prime}\cap(a\cup b))=a\cup b$		(11)
	$\displaystyle a\,{\mathcal{C}}\,b\quad\&\quad a\,{\mathcal{C}}\,c\quad\Rightarrow\quad a\cap(b\cup c)=(a\cap b)\cup(a\cap c)$		(12)

where $a\,{\mathcal{C}}\,b\ \ {\buildrel\rm def\over{\Longleftrightarrow}}\ \ a=(a\cap b)\cup(a\cap b\,^{\prime})$ ( $a$ commutes with $b$ ), is called an orthomodular lattice , OML .

(Pavičić and Megill [ 36 ] ) An ortholattice that satisfies the following: ^{19 19} 19 This condition is known as commensurability . [ 7 , Definition (2.13), p. 32] Commensurability is a weaker form of the commutativity from Definition 7 . Actually, a metaimplication from commensurability to commutativity is yet another way to express orthomodularity. They coincide in any OML.

$\displaystyle(a\equiv b)\cup(a\equiv b^{\prime})=(a\cap b)\cup(a\cap b^{\prime})\cup(a^{\prime}\cap b)\cup(a^{\prime}\cap b^{\prime})=1$

(13)

is called a weakly distributive ortholattice , WDOL .