Sub-Riemannian (2, 3, 5, 6)-Structures

Abstract

We describe all Carnot algebras with growth vector (2, 3, 5, 6), their normal forms, an invariant that separates them, and a change of basis that transforms such an algebra into a normal form. For each normal form, Casimir functions and symplectic foliations on the Lie coalgebra are computed. An invariant and normal forms of left-invariant (2, 3, 5, 6)-distributions are described. A classification, up to isometries, of all left-invariant sub-Riemannian structures on (2, 3, 5, 6)-Carnot groups is obtained.

Sub-Riemannian structures [1] are stratified in depth, i.e., with respect to the minimum order of Lie brackets required for generating a tangent space from basis vector fields. The complexity of sub-Riemannian (SR) structures grows substantially with increasing step. At present, SR structures of step at most three have been examined in detail [2–8]. Accordingly, a task of great interest is to systematically investigate SR structures of step 4. The study of the simplest of these structures, namely, a nilpotent SR structure with growth vector (2, 3, 5, 8) (see Example 2 below) was initiated in [9–11]. Below, we obtain a complete classification of nilpotent SR structures and distributions with growth vector (2, 3, 5, 6). It is shown that all such structures are quotient structures of a nilpotent SR structure with growth vector (2, 3, 5, 8).

1 SUB-RIEMANNIAN QUOTIENT STRUCTURES

In this paper, all Lie algebras are considered over the field $\mathbb{R}$.

Definition 1. A nilpotent Lie algebra $\mathfrak{g}$ is called a Carnot algebra if

(i) $\mathfrak{g}$ is graded:

$$\mathfrak{g} = {{\mathfrak{g}}^{{(1)}}} \oplus \; \cdots \; \oplus {{\mathfrak{g}}^{{(s)}}},$$

$$[{{\mathfrak{g}}^{{(i)}}},{{\mathfrak{g}}^{{(j)}}}] \subset {{\mathfrak{g}}^{{(i + j)}}},\quad {{\mathfrak{g}}^{{(k)}}} = {\text{\{ }}0{\text{\} }}\quad {\text{for}}\quad k > s,$$

(1)

(ii) $\mathfrak{g}$ is generated by the first component:

$${\text{Lie}}({{\mathfrak{g}}^{{(1)}}}) = \mathfrak{g}.$$

(2)

The corresponding connected simply connected Lie group is called a Carnot group.

Conditions (i) and (ii) are equivalent to the condition

$$[{{\mathfrak{g}}^{{(1)}}},{{\mathfrak{g}}^{{(i)}}}] = {{\mathfrak{g}}^{{(i + 1)}}},\quad {{\mathfrak{g}}^{{(k)}}} = {\text{\{ }}0{\text{\} }}\quad {\text{for}}\quad k > s.$$

Definition 2. The growth vector of a Carnot algebra $\mathfrak{g}$ is defined as

$$({{n}_{1}},\; \ldots ,\;{{n}_{s}}),\quad {{n}_{j}} = \sum\limits_{i = 1}^j {\text{dim}{{\mathfrak{g}}^{{(i)}}}} .$$

Here, $({{n}_{1}},\; \ldots ,\;{{n}_{s}})$ is the growth vector of a left-invariant distribution on the Lie group of the Lie algebra $\mathfrak{g}$ generated by the subspace ${{\mathfrak{g}}^{{(1)}}} \subset \mathfrak{g}$.

Let $M$ be a smooth manifold. A sub-Riemannian structure on M [1] is a pair $(\Delta ,g)$ consisting of a vector distribution $\Delta \subset TM$ and a scalar product g in Δ.

Let G be a Lie group and $\mathfrak{g}$ be its Lie algebra. A left-invariant SR structure on G consists of a left-invariant distribution on G and a left-invariant scalar product in the distribution. This structure is specified by a subspace $\Delta \subset \mathfrak{g}$ and a scalar product g in $\Delta$. In this case, we say that $(\Delta ,g)$ is an SR structure in the algebra $\mathfrak{g}$.

Left-invariant SR structures on Carnot groups arise as nilpotent approximations of general SR structures on smooth manifolds [1].

Definition 3. Let $(\Delta ,g)$ be an SR structure in a Lie algebra $\mathfrak{g}$, and let $\mathfrak{i} \subset \mathfrak{g}$ be an ideal such that $\Delta \cap \mathfrak{i}$ = {0}. Let $\tilde {\mathfrak{g}} = \mathfrak{g}{\text{/}}\mathfrak{i}$ be a quotient algebra and $\pi$: $\mathfrak{g} \to \tilde {\mathfrak{g}}$ be a canonical projection. Define $\tilde {\Delta } = \pi (\Delta )$ and $\tilde {g}(\pi (X)$, π(Y)) = g(X, Y) for $X,Y \in \Delta$. Then ($\tilde {\Delta }$, $\tilde {g}$) is an SR structure in $\tilde {\mathfrak{g}}$, which will be called a quotient structure of the SR structure $(\Delta ,g)$.

Example 1. Let ${{\mathfrak{g}}^{5}}$ be a free nilpotent Lie algebra of step 3 with two generators (Cartan algebra); this is the Carnot algebra with growth vector (2, 3, 5). There exists a basis ${{\mathfrak{g}}^{5}} = {\text{span}}({{X}_{1}},\; \ldots ,\;{{X}_{5}})$ in which the nonzero Lie brackets are

$$\left[ {{{X}_{1}},{{X}_{2}}} \right] = {{X}_{3}},\quad \left[ {{{X}_{1}},{{X}_{3}}} \right] = {{X}_{4}},\quad \left[ {{{X}_{2}},{{X}_{3}}} \right] = {{X}_{5}}.$$

Consider an SR structure $(\Delta ,g)$ in ${{\mathfrak{g}}^{5}}$ with an orthonormal frame $({{X}_{1}},{{X}_{2}})$ [8]. Sequentially choosing the subspaces $\mathbb{R}{{X}_{5}}$, ${\text{span}}({{X}_{4}},{{X}_{5}})$, and ${\text{span}}({{X}_{3}},{{X}_{4}},{{X}_{5}})$ as an ideal $\mathfrak{i} \subset {{\mathfrak{g}}^{5}}$, we obtain SR quotient structures in the Engel algebra (growth vector (2, 3, 4)) [7], Heisenberg algebra (growth vector (2, 3)) [2], and the two-dimensional commutative algebra ${{\mathbb{R}}^{2}}$ (growth vector (2)).

Example 2. Let ${{\mathfrak{g}}^{8}}$ be a free nilpotent Lie algebra of step 4 with two generators; this is the Carnot algebra with growth vector (2, 3, 5, 8), which will be called a nilpotent (2, 3, 5, 8)-algebra. There exists a basis ${{\mathfrak{g}}^{8}} = {\text{span}}({{X}_{1}}, \ldots ,{{X}_{8}})$ in which the nonzero Lie brackets are given by

$$\left[ {{{X}_{1}},{{X}_{2}}} \right] = {{X}_{3}},\quad \left[ {{{X}_{1}},{{X}_{3}}} \right] = {{X}_{4}},\quad \left[ {{{X}_{2}},{{X}_{3}}} \right] = {{X}_{5}},$$

(3)

$$\begin{gathered} \left[ {{{X}_{1}},{{X}_{4}}} \right] = {{X}_{6}},\quad \left[ {{{X}_{1}},{{X}_{5}}} \right] = \left[ {{{X}_{2}},{{X}_{4}}} \right] = {{X}_{7}}, \\ \left[ {{{X}_{2}},{{X}_{5}}} \right] = {{X}_{8}}. \\ \end{gathered} $$

(4)

Consider an SR structure $(\Delta ,g)$ in ${{\mathfrak{g}}^{8}}$ with an orthonormal frame $({{X}_{1}},{{X}_{2}})$ [9–11]. It is easy to see that this SR structure is unique, up to a Lie algebra automorphism, SR structure in ${{\mathfrak{g}}^{8}}$ of rank 2 satisfying the condition ${\text{Lie}}(\Delta ) = {{\mathfrak{g}}^{8}}$; we call it a nilpotent SR (2, 3, 5, 8)-structure. In what follows, we will use a dual basis in the Lie coalgebra $({{\mathfrak{g}}^{8}}){\text{*}}$:

$${{\omega }_{1}},\; \ldots ,\;{{\omega }_{8}} \in ({{\mathfrak{g}}^{8}}){\text{*}},\quad {{\omega }_{i}}({{X}_{j}}) = {{\delta }_{{ij}}},\quad i,j = 1,\; \ldots ,\;8.$$

The goal of this paper is, given a structure $(\Delta ,g)$, to describe all SR quotient structures for two-dimensional ideals $\mathfrak{i} \subset {{\mathfrak{g}}^{8}}$. These are exactly nilpotent SR structures with growth vector (2, 3, 5, 6).

It is easy to see that a two-dimensional subspace $\mathfrak{i} \subset {{\mathfrak{g}}^{8}}$ is an ideal if and only if $\mathfrak{i} \subset Z(\mathfrak{g})$ = span(X₆, ${{X}_{7}},{{X}_{8}})$. Therefore, any quotient algebra ${{\mathfrak{g}}^{8}}{\text{/}}\mathfrak{i}$ by the two-dimensional ideal $\mathfrak{i}$ is a Carnot algebra with growth vector (2, 3, 5, 6). Let us describe such algebras.

2 CARNOT ALGEBRAS WITH GROWTH VECTOR (2, 3, 5, 6)

Theorem 1. (i) In any Carnot algebra $\mathfrak{g}$ with growth vector (2, 3, 5, 6), there is a basis $\mathfrak{g} = {\text{span}}({{X}_{1}},\; \ldots ,\;{{X}_{6}})$ in which all nonzero Lie brackets have the form

$$\left[ {{{X}_{1}},{{X}_{2}}} \right] = {{X}_{3}},\quad \left[ {{{X}_{1}},{{X}_{3}}} \right] = {{X}_{4}},\quad \left[ {{{X}_{2}},{{X}_{3}}} \right] = {{X}_{5}},$$

(5)

$$\begin{gathered} \left[ {{{X}_{1}},{{X}_{4}}} \right] = \alpha {{X}_{6}},\quad \left[ {{{X}_{1}},{{X}_{5}}} \right] = \left[ {{{X}_{2}},{{X}_{4}}} \right] = \beta {{X}_{6}}, \\ \left[ {{{X}_{2}},{{X}_{5}}} \right] = \gamma {{X}_{6}}, \\ \end{gathered} $$

(6)

$$(\alpha ,\beta ,\gamma ) \in {{\mathbb{R}}^{3}}{{\backslash \{ }}0{\text{\} }}.$$

(ii) Any two Carnot algebras with growth vector (2, 3, 5, 6) that have proportional triples $(\alpha ,\beta ,\gamma )$ in (6) are isomorphic to each other. Accordingly, such algebras are denoted by $\mathfrak{g}_{{\alpha :\beta :\gamma }}^{6}$, $(\alpha :\beta :\gamma ) \in \mathbb{R}{{P}^{2}}$.

(iii) The following isomorphism of Lie algebras holds:

$$\begin{gathered} \mathfrak{g}_{{\alpha :\beta :\gamma }}^{6} \cong {{\mathfrak{g}}^{8}}{\text{/}}(\ker\omega \cap Z({{\mathfrak{g}}^{8}})), \\ \omega = \alpha {{\omega }_{6}} + \beta {{\omega }_{7}} + \gamma {{\omega }_{8}} \ne 0. \\ \end{gathered} $$

Each basis in the algebra $\mathfrak{g}_{{\alpha :\beta :\gamma }}^{6}$ with multiplication table (5), (6) is associated with a quadratic form Q(x, y) = $\alpha {{x}^{2}} + 2\beta xy + \gamma {{y}^{2}}$. Depending on the sign of its discriminant ${\mathbf{s}} = {\text{sgn}}(\alpha \gamma - {{\beta }^{2}})$ ∈ {0, ±1}, the algebra $\mathfrak{g}_{{\alpha :\beta :\gamma }}^{6}$ is called

– parabolic if s = 0,

– elliptic if s = 1,

– hyperbolic if ${\mathbf{s}} = - 1$.

Remark 1. Depending on the number s = ${\text{sgn}}(\alpha \gamma$ – β²), the topology of the sets of triples $(\alpha :\beta :\gamma ) \in \mathbb{R}{{P}^{2}}$ is as follows:

$$\begin{gathered} {{A}_{P}} = \{ (\alpha :\beta :\gamma ) \in \mathbb{R}{{P}^{2}}|{\mathbf{s}} = 0\} , \\ {{A}_{E}} = \{ (\alpha :\beta :\gamma ) \in \mathbb{R}{{P}^{2}}|{\mathbf{s}} = 1\} , \\ {{A}_{H}} = \{ (\alpha :\beta :\gamma ) \in \mathbb{R}{{P}^{2}}|{\mathbf{s}} = - 1\} . \\ \end{gathered} $$

Topologically, A_P is a circle, A_E is an open disk, and ${{A}_{H}} = \mathbb{R}{{P}^{2}}{{\backslash }}({{A}_{P}} \cup {{A}_{E}})$ is a projective plane with a cutout hole, i.e., a Möbius strip.

Below, we give several examples of Carnot algebras with growth vector (2, 3, 5, 6), together with nonzero Lie brackets in the corresponding basis. Recall that the notation ${{N}_{{6,2, * }}}$ for these algebras was used in [12, 13]. A classification of nilpotent Lie algebras of dimension ≤7 was obtained in [12], and all Carnot algebras of dimension ≤7 were described in [13].

Example 3. The parabolic algebra $\mathfrak{g}_{{1:0:0}}^{6} = {{N}_{{6,2,7}}}$ = span(X₁, ..., X₆),

$$\begin{gathered} \left[ {{{X}_{1}},{{X}_{2}}} \right] = {{X}_{3}},\quad \left[ {{{X}_{1}},{{X}_{3}}} \right] = {{X}_{4}}, \\ [{{X}_{2}},{{X}_{3}}] = {{X}_{5}},\quad \left[ {{{X}_{1}},{{X}_{4}}} \right] = {{X}_{6}}. \\ \end{gathered} $$

Example 4. The hyperbolic algebra $\mathfrak{g}_{{1:0:( - 1)}}^{6} = {{N}_{{6,2,5}}}$ = span(X₁, ..., X₆),

$$\begin{gathered} \left[ {{{X}_{1}},{{X}_{2}}} \right] = {{X}_{3}},\quad \left[ {{{X}_{1}},{{X}_{3}}} \right] = {{X}_{4}}, \\ \left[ {{{X}_{2}},{{X}_{3}}} \right] = {{X}_{5}},\quad \left[ {{{X}_{1}},{{X}_{4}}} \right] = - \left[ {{{X}_{2}},{{X}_{5}}} \right] = {{X}_{6}}. \\ \end{gathered} $$

Example 5. The elliptic algebra $\mathfrak{g}_{{1:0:1}}^{6} = {{N}_{{6,2,5a}}}$ = span(X₁, ..., X₆),

$$\begin{gathered} \left[ {{{X}_{1}},{{X}_{2}}} \right] = {{X}_{3}},\quad \left[ {{{X}_{1}},{{X}_{3}}} \right] = {{X}_{4}}, \\ \left[ {{{X}_{2}},{{X}_{3}}} \right] = {{X}_{5}},\quad \left[ {{{X}_{1}},{{X}_{4}}} \right] = \left[ {{{X}_{2}},{{X}_{5}}} \right] = {{X}_{6}}. \\ \end{gathered} $$

Theorem 2. (i) The algebras $\mathfrak{g}_{{1:0:0}}^{6}$, $\mathfrak{g}_{{1:0:( - 1)}}^{6}$, and $\mathfrak{g}_{{1:0:1}}^{6}$ are pairwise nonisomorphic. Any algebra $\mathfrak{g}_{{\alpha :\beta :\gamma }}^{6}$, (α : β : $\gamma ) \in \mathbb{R}{{P}^{2}}$, is isomorphic to one of these algebras.

(ii) The number ${\mathbf{s}} = {\text{sgn}}(\alpha \gamma - {{\beta }^{2}}) \in {\text{\{ }}0,\; \pm {\kern 1pt} 1{\text{\} }}$ is an invariant of the Carnot algebra $\mathfrak{g}$; s is called the signature of $\mathfrak{g}$.

(ii) The algebras $\mathfrak{g}_{{\alpha :\beta :\gamma }}^{6}$ are separated by the signature ${\mathbf{s}}$:

$$\begin{gathered} \mathfrak{g}_{{\alpha :\beta :\gamma }}^{6} \cong \mathfrak{g}_{{1:0:0}}^{6} \Leftrightarrow {\mathbf{s}} = 0 \\ (parabolic\;algebra), \\ \end{gathered} $$

(7)

$$\begin{gathered} \mathfrak{g}_{{\alpha :\beta :\gamma }}^{6} \cong \mathfrak{g}_{{1:0:( - 1)}}^{6} \Leftrightarrow {\mathbf{s}} = - 1 \\ (hyperbolic\;algebra), \\ \end{gathered} $$

(8)

$$\begin{gathered} \mathfrak{g}_{{\alpha :\beta :\gamma }}^{6} \cong \mathfrak{g}_{{1:0:1}}^{6} \Leftrightarrow {\mathbf{s}} = 1 \\ (elliptic\;algebra). \\ \end{gathered} $$

(9)

Thus, the signature ${\mathbf{s}} \in {\text{\{ }}0,\; \pm {\kern 1pt} 1{\text{\} }}$ is an invariant separating three classes of isomorphism of the algebras $\mathfrak{g}_{{\alpha :\beta :\gamma }}^{6}$.

Remark 2. Item (i) in Theorem 2 was first proved in [13]. We proved it independently, together with an algorithm for reducing the multiplication table in the (2, 3, 5, 6)-algebra to the normal form in Examples 3–5.

To find a basis change in the algebra $\mathfrak{g} = \mathfrak{g}_{{\alpha :\beta :\gamma }}^{6}$ that transforms the basis into one of the normal forms indicated in Examples 3–5, it suffices to reduce the quadratic form Q(x, y) = $\alpha {{x}^{2}} + 2\beta xy + \gamma {{y}^{2}}$ to a sum of squares, to apply this change to the basis $({{X}_{1}},{{X}_{2}})$ of the space ${{\mathfrak{g}}^{{(1)}}}$, and to normalize the vector X₆ generating the space ${{\mathfrak{g}}^{{(4)}}}$.

3 NILPOTENT (2, 3, 5, 6)-DISTRIBUTIONS AND SUB-RIEMANNIAN STRUCTURES

Definition 4. Let $\mathfrak{g}$ be a Carnot algebra with growth vector (2, 3, 5, 6). A nilpotent sub-Riemannian (2, 3, 5, 6)-structure is a sub-Riemannian structure $(\Delta ,g)$ in $\mathfrak{g}$ satisfying the conditions

$$\text{dim}\Delta = 2,\quad {\text{Lie}}(\Delta ) = \mathfrak{g},$$

(10)

or, equivalently, the equality $\Delta \oplus {{\mathfrak{g}}^{{(2)}}}$ $\oplus$ ${{\mathfrak{g}}^{{(3)}}} \oplus {{\mathfrak{g}}^{{(4)}}}$ = $\mathfrak{g}$. The corresponding plane $\Delta \subset \mathfrak{g}$ is called a nilpotent (2, 3, 5, 6)-distribution in $\mathfrak{g}$.

The vector (2, 3, 5, 6) is the growth vector of the left-invariant distribution on the Lie group G of the Lie algebra $\mathfrak{g}$ defined by the plane $\Delta \subset \mathfrak{g}$.

Theorem 3. Let $\mathfrak{g}$ be a Carnot algebra with growth vector (2, 3, 5, 6) of signature ${\mathbf{s}} \in {\text{\{ }}0,\; \pm {\kern 1pt} 1{\text{\} }}$. For any nilpotent (2, 3, 5, 6)-distribution $\Delta \subset \mathfrak{g}$, there exists a frame $\Delta = {\text{span}}({{X}_{1}},{{X}_{2}})$ such that

$$\begin{gathered} \left[ {{{X}_{1}},{{X}_{2}}} \right] = {{X}_{3}},\quad \left[ {{{X}_{1}},{{X}_{3}}} \right] = {{X}_{4}},\quad \left[ {{{X}_{2}},{{X}_{3}}} \right] = {{X}_{5}}, \\ \left[ {{{X}_{1}},{{X}_{4}}} \right] = {{X}_{6}},\quad \left[ {{{X}_{1}},{{X}_{5}}} \right] = \left[ {{{X}_{2}},{{X}_{4}}} \right] = 0, \\ \left[ {{{X}_{2}},{{X}_{5}}} \right] = {\mathbf{s}}{{X}_{6}}. \\ \end{gathered} $$

Theorem 4. Let $\mathfrak{g}$ be a Carnot algebra with growth vector (2, 3, 5, 6). For any nilpotent SR (2, 3, 5, 6)-structures in $\mathfrak{g}$, there exists an orthonormal frame $({{X}_{1}},{{X}_{2}})$ and a number $\nu \in [ - 1,\;1]$ for which

$$\left[ {{{X}_{1}},{{X}_{2}}} \right] = {{X}_{3}},\quad \left[ {{{X}_{1}},{{X}_{3}}} \right] = {{X}_{4}},\quad \left[ {{{X}_{2}},{{X}_{3}}} \right] = {{X}_{5}},$$

(11)

$$\begin{gathered} \left[ {{{X}_{1}},{{X}_{4}}} \right] = {{X}_{6}},\quad \left[ {{{X}_{1}},{{X}_{5}}} \right] = \left[ {{{X}_{2}},{{X}_{4}}} \right] = 0, \\ \left[ {{{X}_{2}},{{X}_{5}}} \right] = \nu {{X}_{6}}. \\ \end{gathered} $$

(12)

The number $\nu \in [ - 1,\;1]$ in (12) is called the canonical parameter of the SR structure ($\Delta ,g)$. It is easy to see that ν is equal to the ratio of the smaller (in absolute value) eigenvalue of Q to the larger eigenvalue. Moreover, ${\mathbf{s}} = {\text{sgn}}\nu$.

Theorem 5. The canonical parameter $\nu \in [ - 1,\;1]$ is an invariant of the nilpotent SR (2, 3, 5, 6)-structure.

Let $(\Delta ,g)$ be an SR structure on a manifold $M$. The corresponding SR distance d [1] transforms M into a metric space. Recall that an isometry between metric spaces $(M,d)$ and $(\tilde {M},\tilde {d})$ is a mapping $F$: $M \to \tilde {M}$ such that

$$d(x,y) = \tilde {d}(F(x),F(y)),\quad x,y \in M.$$

The metric group [15] is a Lie group with a left-invariant distance inducing a manifold topology on this group. In particular, any Carnot group is a metric group. It was proved in [15] that

any isometry between metric groups is an analytic mapping;

any isometry between connected nilpotent metric groups is an affine mapping (i.e., the composition of a left shift and an isomorphism).

The following theorem provides a classification of left-invariant SR (2, 3, 5, 6)-structures up to isometries of corresponding Carnot groups.

Theorem 6. Let $\mathfrak{g}$ and $\tilde {\mathfrak{g}}$ be nilpotent Carnot (2, 3, 5, 6)-algebras, and let $G$ and $\tilde {G}$ be corresponding Carnot groups. Let $(\Delta ,g)$ and $(\tilde {\Delta },\tilde {g})$ be nilpotent SR (2, 3, 5, 6)-structures in g and $\tilde {\mathfrak{g}}$, $\nu$ and $\tilde {\nu }$ be corresponding canonical parameters, and $d$ and $\tilde {d}$ be corresponding left-invariant SR metrics on $G$ and $\tilde {G}$.

The metric spaces $(G,d)$ and $(\tilde {G},\tilde {d})$ are isometric if and only if $\nu = \tilde {\nu }$.

Theorem 7. Let $\mathfrak{g} = {{\mathfrak{g}}^{8}}$ be the (2, 3, 5, 8)-algebra from Example 2 with a basis $({{X}_{1}}, \ldots ,{{X}_{8}})$ according to (3), (4), and let ($\Delta ,g)$ be an SR structure in $\mathfrak{g}$ with orthonormal frame $({{X}_{1}},{{X}_{2}})$. Let $\mathfrak{i} = \ker\omega \cap Z(\mathfrak{g})$ with $\omega = \alpha {{\omega }_{6}} + \beta {{\omega }_{7}} + \gamma {{\omega }_{8}} \ne 0$ be any 2-dimensional subspace of $Z(\mathfrak{g})$. Then the SR quotient structure $(\tilde {\Delta },\tilde {g})$ is one of the SR (2, 3, 5, 6)-structures in $\tilde {\mathfrak{g}} = \mathfrak{g}{\text{/}}\mathfrak{i}$:

(i) in the parabolic case s = 0, we have $\tilde {\mathfrak{g}} \cong \mathfrak{g}_{{1:0:0}}^{6}$;

(ii) in the hyperbolic case ${\mathbf{s}} = - 1$, we have $\tilde {\mathfrak{g}} \cong \mathfrak{g}_{{1:0:( - 1)}}^{6}$;

(iii) in the elliptic case s = 1, we have $\tilde {\mathfrak{g}} \cong \mathfrak{g}_{{1:0:1}}^{6}$.

Conversely, any nilpotent (2, 3, 5, 6)-structure in each of these algebras is realized as a quotient structure of a nilpotent (2, 3, 5, 8)-structure $(\Delta ,g)$.

Remark 3. An SR structure in the elliptic algebra $\mathfrak{g}_{{1:0:1}}^{6}$ with canonical parameter ν = 1 was considered in [14]. For this structure, it was proved that the vertical subsystem of the Hamiltonian system in the Pon-tryagin maximum principle [1] is Liouville integrable. This Hamiltonian system was integrated in [11].

4 CASIMIR FUNCTIONS AND SYMPLECTIC FOLIATIONS

Let us recall some basic concepts of symplectic geometry. Let $\mathfrak{g}$ be a Lie algebra and $\mathfrak{g}{\text{*}}$ be the Lie coalgebra (the dual of the space $\mathfrak{g}$). Any function $h \in {{C}^{\infty }}(\mathfrak{g}*)$ is called a Hamiltonian. For any Hamiltonians f and g, the Poisson bracket is defined as

$$\left\{ {f,g} \right\}(\lambda ) = \left\langle {\lambda ,\left[ {d{{f}_{\lambda }},d{{g}_{\lambda }}} \right]} \right\rangle ,\quad \lambda \in \mathfrak{g}{\text{*}}.$$

The Casimir function (on a submanifold $M \subset \mathfrak{g}{\text{*}}$) is any Hamiltonian f commuting with all Hamiltonians in the sense of the Poisson bracket:

$$\begin{gathered} \left\{ {f,h} \right\}(\lambda ) = 0,\quad h \in {{C}^{\infty }}(\mathfrak{g}*), \\ \lambda \in \mathfrak{g}{\text{*}}\quad ({\text{accordingly}}\;\lambda \in M). \\ \end{gathered} $$

A symplectic foliation on $\mathfrak{g}{\text{*}}$ is a partition of $\mathfrak{g}{\text{*}}$ into symplectic leaves $L(\lambda )$, which are orbits of the coadjoint action of the Lie group $G$ of the Lie algebra $\mathfrak{g}$ in $\mathfrak{g}{\text{*}}$:

$$L(\lambda ) = \{ (Ad_{x}^{ * })(\lambda )|x \in G\} ,\quad \lambda \in \mathfrak{g}{\text{*}}.$$

The rank of the Poisson bracket {⋅, ⋅} at a point $\lambda \in \mathfrak{g}{\text{*}}$ is defined as the dimension of the symplectic leaf passing through $\lambda$ and is denoted by ${\text{rank}}(\lambda )$. The symplectic leaves and Casimir functions are invariants of the Hamiltonian vector field

$$\dot {\lambda } = \mathbf{h}(\lambda ),\quad \lambda \in \mathfrak{g}{\text{*}},$$

for any Hamiltonian h, so they are important for analyzing left-invariant Hamiltonians on the Lie group of the Lie algebra $\mathfrak{g}$, in particular, for studying left-invariant optimal control problems on this Lie group.

For coalgebras $\mathfrak{g}{\text{*}}$ of all Carnot algebras $\mathfrak{g}$ with growth vector (2, 3, 5, 6), we describe their Casimir functions and symplectic foliations. In each of the algebras $\mathfrak{g} = \mathfrak{g}_{{1:0:0}}^{6}$, $\mathfrak{g}_{{1:0:( - 1)}}^{6}$, and $\mathfrak{g}_{{1:0:1}}^{6}$, we use a canonical basis $\mathfrak{g} = {\text{span}}({{X}_{1}},\; \ldots ,\;{{X}_{6}})$ according to Examples 3–5 and corresponding linear Hamiltonians h₁, ..., h₆ $\in \mathfrak{g}{\text{**}}$, ${{h}_{i}}(\lambda ) = \left\langle {\lambda ,{{X}_{i}}} \right\rangle$. In each of the cases presented below, the symplectic leaf $L(\lambda )$ is a connected component of common level surfaces of the Casimir functions on corresponding submanifolds in $\mathfrak{g}{\text{*}}$.

4.1 Parabolic Algebra $\mathfrak{g} = \mathfrak{g}_{{1:0:0}}^{6}$

(1) If ${{h}_{5}}{{h}_{6}} \ne 0$, then ${\text{rank}}(\lambda ) = 4$.

Casimir functions: ${{h}_{5}}$, ${{h}_{6}}$.

(2) If ${{h}_{6}} = 0$ and $h_{3}^{2} + h_{4}^{2} + h_{5}^{2} \ne 0$, then ${\text{rank}}(\lambda )$ = 2.

Casimir functions: ${{h}_{4}}$, ${{h}_{5}}$, $\tfrac{{h_{3}^{2}}}{2} + {{h}_{1}}{{h}_{5}} - {{h}_{2}}{{h}_{4}}$.

(3) If ${{h}_{5}} = 0$ and $h_{3}^{2} + h_{4}^{2} + h_{6}^{2} \ne 0$, then ${\text{rank}}(\lambda )$ = 2.

Casimir functions: ${{h}_{6}}$, ${{C}_{1}} = {{h}_{3}}{{h}_{6}} - \tfrac{{h_{4}^{2}}}{2}$, C₂ = ${{h}_{2}}h_{6}^{2}$ – ${{C}_{1}}{{h}_{4}} - \tfrac{{h_{4}^{3}}}{6}$.

(4) If ${{h}_{3}} = \; \cdots \; = {{h}_{6}} = 0$, then ${\text{rank}}(\lambda ) = 0$.

Casimir functions: ${{h}_{1}}$, ${{h}_{2}}$.

4.2 Hyperbolic Algebra $\mathfrak{g} = \mathfrak{g}_{{1:0:( - 1)}}^{6}$

(1) If ${{h}_{6}} \ne 0$, then ${\text{rank}}(\lambda ) = 4$.

Casimir functions: ${{h}_{6}}$, ${{h}_{3}}{{h}_{6}} + \frac{{h_{4}^{2} - h_{5}^{2}}}{2}$.

(2) If ${{h}_{6}} = 0$ and $h_{3}^{2} + h_{4}^{2} + h_{5}^{2} \ne 0$, then ${\text{rank}}(\lambda )$ = 2.

Casimir functions: ${{h}_{4}}$, ${{h}_{5}}$, $\tfrac{{h_{3}^{2}}}{2} + {{h}_{1}}{{h}_{5}} - {{h}_{2}}{{h}_{4}}$.

(3) If ${{h}_{3}} = \; \cdots \; = {{h}_{6}} = 0$, then ${\text{rank}}(\lambda ) = 0$.

Casimir functions: ${{h}_{1}}$, ${{h}_{2}}$.

4.3 Elliptic Algebra $\mathfrak{g} = \mathfrak{g}_{{1:0:1}}^{6}$

(1) If ${{h}_{6}} \ne 0$, then ${\text{rank}}(\lambda ) = 4$.

Casimir functions: ${{h}_{6}}$, ${{h}_{3}}{{h}_{6}} - \tfrac{{h_{4}^{2} + h_{5}^{2}}}{2}$.

(2) If ${{h}_{6}} = 0$ and $h_{3}^{2} + h_{4}^{2} + h_{5}^{2} \ne 0$, then ${\text{rank}}(\lambda )$ = 2.

Casimir functions: ${{h}_{4}}$, ${{h}_{5}}$, $\tfrac{{h_{3}^{2}}}{2} + {{h}_{1}}{{h}_{5}} - {{h}_{2}}{{h}_{4}}$.

(3) If ${{h}_{3}} = \; \cdots \; = {{h}_{6}} = 0$, then ${\text{rank}}(\lambda ) = 0$.

Casimir functions: ${{h}_{1}}$, ${{h}_{2}}$.

CONCLUSIONS

The following results were obtained in this paper.

We described the Carnot algebras with growth vector (2, 3, 5, 6), i.e., quotient algebras of a free nilpotent Carnot algebra ${{\mathfrak{g}}^{8}}$ of step 4 with two generators. Previously, it was known that there are three normal forms of such algebras: parabolic, hyperbolic, and elliptic. An invariant separating these algebras was found, namely, the signature ${\mathbf{s}} \in {\text{\{ }}0,\; \pm {\kern 1pt} 1{\text{\} }}$. Additionally, a change of basis transforming the multiplication table into one of three normal forms was described.

We studied the left-invariant sub-Riemannian structures with growth vector (2, 3, 5, 6), i.e., quotient structures of a (unique) sub-Riemannian structure with growth vector (2, 3, 5, 8) by the two-dimensional subspace of the center of the algebra ${{\mathfrak{g}}^{8}}$. A classification of (2, 3, 5, 6)-structures was obtained up to isometries: all of them are uniquely parametrized by a canonical parameter ν ∈ {0, ±1} such that ${\mathbf{s}} = {\text{sgn}}\nu$. A classification of left-invariant (2, 3, 5, 6)-distributions was obtained: in each of the (2, 3, 5, 6)-algebras (parabolic, hyperbolic, elliptic) there exists a unique, up to isomorphism, distribution of this type.

For each Carnot algebra $\mathfrak{g}$ with growth vector (2, 3, 5, 6), the rank of the Poisson bracket, Casimir functions, and a symplectic foliation in the Lie coalgebra $\mathfrak{g}{\text{*}}$ were calculated.