[ CFM , Definition 1.1] Let $K$ be a Hopf algebra and $A$ be a right $K$ -comodule algebra with structure $\rho:A\rightarrow A\otimes K$ . Let $B=A^{coK}$ . We say that $B\subset A$ is a (right) $K$ -Galois extension if the map $\beta:A\otimes_{B}A\to A\otimes K$ given by

$$\beta(a\otimes b)=(a\otimes 1)\rho(b)$$

is surjective.

The subriemannian distance, also called Carnot-Carathéodory distance, between two points $x$ and $y$ on a subriemannian manifold is given by

$$d(x,y)=\inf l(\gamma),$$

where the infimum is taken over all horizontal curves that connect $x$ to $y$ . The distance is infinite if there is no such curve. The subriemannian ball of radius $\epsilon$ centered at $x\in M$ is denoted by

$$B(\epsilon,x)=\{y\in M:d(x,y)<\epsilon\}.$$

Let $D$ be commutative ring. ^{2.5 2.5} 2.5 This section is written on the base of the section [ 7 ]- LABEL:8433-5163-English-section:_Algebra_over_Ring . $A$ is an algebra over ring $D$ or $D$ -algebra , if $A$ is $D$ -module and we defined product ^{2.6 2.6} 2.6 I follow the definition given in [ 15 ], p. 1, [ 11 ], p. 4. The statement which is true for any $D$ -module, is true also for $D$ -algebra. in $A$

(2.2.1)

$$ab=f\circ(a,b)$$

where $f$ is bilinear map

$$f:A\times A\rightarrow A$$

If $A$ is free $D$ -module, then $A$ is called free algebra over ring $D$ . ∎

The commutator

$${\color[rgb]{.4,0,.9}[a,b]}=ab-ba$$

measures commutativity in $D$ -algebra $A$ . $D$ -algebra $A$ is called commutative , if

$$[a,b]=0$$

∎

The associator

(2.2.4)

$${\color[rgb]{.4,0,.9}(a,b,c)}=(ab)c-a(bc)$$

measures associativity in $D$ -algebra $A$ . $D$ -algebra $A$ is called associative , if

$$(a,b,c)=0$$

∎

Let $D$ be a ring. ^{3.1 3.1} 3.1 [ 2 ], page 89. The set ${\color[rgb]{.4,0,.9}Z(D)}$ of elements $a\in D$ such that

(3.1.1)

$$ax=xa$$

for all $x\in D$ , is called center of ring $D$ . ∎

The pipe surface of radius $r$ of a parameterized curve c $(t)$ , where $t\in[0,1]$ is given by

$$\textbf{ p}(t,\theta)=\textbf{ c}(t)+r[cos(\theta)\textbf{ n}(t)+sin(\theta)\textbf{ b}(t)],$$

where $\theta\in[0,2\pi]$ and $\textbf{ n}(t)$ and $\textbf{ b}(t)$ are, respectively, the normal and bi-normal vectors at the point $\textbf{c}(t)$ , as given by the Frenet-Serret trihedron. The curve c is called a spine curve.

Let $K$ be a field, and let $V$ be a vector space over $K$ equipped with an additional binary operation from $V\times V\to V$ , denoted here by $\cdot$ (i.e. if x and y are any two elements of $V,x\cdot y$ is the product of $x$ and $y$ ). Then $V$ is an algebra over $K$ if the following identities hold for any three elements $x,y$ , and $z$ of $V$ , and all elements (”scalars”) $a$ and $b$ of $K$ :

$$(x+y)\cdot z=x\cdot z+y\cdot z$$

$$x\cdot(y+z)=x\cdot y+x\cdot z$$

$$(ax)\cdot(by)=(ab)(x\cdot y)$$

We call the algebra $V$ assiciative if

$$(x\cdot y)\cdot z=x\cdot(y\cdot z)$$

We call the algebra $V$ unital if there is an element $e\in V$ :

$$e\cdot x=x\cdot e=x$$

for all $x\in V$

Let U and V be two $Z_{2}$ -graded vector spaces. A bilinear mapping of $V\times V$ into U is called supersymmetric / skew-supersymmetric if $sg\pm g$ , i.e. if

$$b(y,x)=\pm(-1)^{ab}b(x,y)$$

for all $x\in V_{a},y\in V_{b};a,b\in Z_{2}$ .

Let U and V be two Klein graded vector spaces. A bilinear mapping of $V\times V$ into U is called supersymmetric / skew-supersymmetric if $sg\pm g$ , i.e. if

$$b(y,x)=\pm\theta(a,b)b(x,y)$$

for all $x\in V_{a},y\in V_{b};a,b\in\mathcal{K}$ .

[ 13 ] A hypergroupoid $(H,\circ)$ is called an LA-semihypergroup if, for all $x,y,z\in H,$

$$(x\circ y)\circ z=(z\circ y)\circ x.$$