On conformal pseudo-subriemannian fundamental graded Lie algebras associated with pseudo $H$-type Lie algebras

TL;DR

This paper explores the structure and prolongations of conformal pseudo-subriemannian fundamental graded Lie algebras derived from pseudo H-type Lie algebras, revealing conditions under which their prolongations coincide.

Contribution

It establishes the equivalence of prolongations for these Lie algebras under specific assumptions, advancing understanding of their structural properties.

Findings

Prolongation of conformal pseudo-subriemannian graded Lie algebra matches that of the fundamental graded Lie algebra under certain conditions.

Provides new insights into the structure of pseudo H-type Lie algebras and their associated graded Lie algebras.

Enhances the theoretical framework connecting conformal structures and graded Lie algebra prolongations.

Abstract

A pseudo $H$-type Lie algebra naturally gives rise to a conformal pseudo-subriemannian fundamental graded Lie algebras. In this paper we investigate the prolongations of the associated fundamental graded Lie algebra and the associated conformal pseudo-subriemannian fundamental graded Lie algebra. In particular, we show that the prolongation of the associated conformal pseudo-subriemannian fundamental graded Lie algebra coincides with that of the associated fundamental graded Lie algebra under some assumptions.

Equations

$$⟨J_z(x)|y⟩=⟨z|[x,y]⟩x,y\in {\mathfrak{n}}_{-1},$$

$$J_z^2=-⟨z|z⟩1_{{\mathfrak{n}}_{-1}},$$

$$⟨x|y{⟩}^{\prime }=\alpha ⟨x|y⟩(x,y\in {\mathfrak{n}}_{-1}),⟨z|w{⟩}^{\prime }=\beta ⟨z|w⟩(z,w\in {\mathfrak{n}}_{-2}),⟨{\mathfrak{n}}_{-1}|{\mathfrak{n}}_{-2}{⟩}^{\prime }=0,$$

$$f(x)=|\alpha {|}^{-1/2}x(x\in {\mathfrak{g}}_{-1}),f(z)=|\alpha {|}^{-1}z(z\in {\mathfrak{g}}_{-2});$$

$$⟨f(x)|f(y){⟩}^{(2)}=|\alpha {|}^{-1}⟨x|y{⟩}^{(2)}=\mathrm{sgn}(\alpha )⟨x|y{⟩}^{(1)}(x,y\in {\mathfrak{g}}_{-1}),$$

$$⟨f(z)|f(z^{\prime }){⟩}^{(2)}=|\alpha {|}^{-2}⟨z|z^{\prime }{⟩}^{(2)}=⟨z|z^{\prime }{⟩}^{(1)}(z,z^{\prime }\in {\mathfrak{g}}_{-2}).$$

$$(x_1,x_2)(y_1,y_2)=(x_1{y}_1+\gamma \overline{y_2}{x}_2,x_2\overline{y_1}+y_2{x}_1).$$

$${\mathfrak{n}}_{-1}={\mathbb{F}}^n,{\mathfrak{n}}_{-2}=\mathrm{Im}\mathbb{F},\mathfrak{n}={\mathfrak{n}}_{-1}\oplus {\mathfrak{n}}_{-2},$$

$$[x,y]=-2\mathrm{Im}(xS{y}^{\ast })=yS{x}^{\ast }-xS{y}^{\ast }(x,y\in {\mathfrak{n}}_{-1}),[{\mathfrak{n}}_{-1},{\mathfrak{n}}_{-2}]=[{\mathfrak{n}}_{-2},{\mathfrak{n}}_{-2}]=0;$$

$$⟨x\mid y⟩=2\mathrm{Re}(xS{y}^{\ast })(x,y\in {\mathfrak{n}}_{-1}),$$

$$⟨z\mid w⟩=\mathrm{Re}(z\overline{w})=-\mathrm{Re}(zw)(z,w\in {\mathfrak{n}}_{-2}),⟨{\mathfrak{n}}_{-1}\mid {\mathfrak{n}}_{-2}⟩=0.$$

$$\phi (x)=xP(x\in {\mathfrak{h}}^{(1)}(\mathbb{F},1_{r,s}{)}_{-1}),\phi (z)=z(z\in {\mathfrak{h}}^{(1)}(\mathbb{F},1_{r,s}{)}_{-2}).$$

$$\psi (x)=x{K}_n(x\in {\mathfrak{h}}^{(1)}(\mathbb{F},1_{r,s}{)}_{-1}),\psi (z)=-z(z\in {\mathfrak{h}}^{(1)}(\mathbb{F},1_{r,s}{)}_{-2}),$$

$${\mathfrak{g}}_{-1}=(\mathbb{F}(\gamma {)}^c{)}^n,{\mathfrak{g}}_{-2}={\mathbb{F}}^c,$$

$$[\alpha ,\beta ]={\mathrm{pr}}_2(\alpha S{\beta }^{\ast })(\alpha ,\beta \in {\mathfrak{g}}_{-1}),[{\mathfrak{g}}_{-1},{\mathfrak{g}}_{-2}]=[{\mathfrak{g}}_{-2},{\mathfrak{g}}_{-2}]=0.$$

$$[\alpha ,\beta ]={\alpha }_{2}S{}^t{\beta }_1-{\beta }_{2}S{}^t{\alpha }_1.$$

$$⟨\alpha |\beta ⟩=R(\alpha S{\beta }^{\ast })(\alpha ,\beta \in {\mathfrak{g}}_{-1}),$$

$$⟨z_1|z_2⟩=-\gamma R(\overline{z_1}{z}_2)(z_1,z_2\in {\mathfrak{g}}_{-2}),⟨{\mathfrak{g}}_{-1}|{\mathfrak{g}}_{-2}⟩=0.$$

$$⟨\alpha |\beta ⟩=R({\alpha }_{1}S{}^t\overline{{\beta }_1}-\gamma \overline{{\beta }_2}S{}^t{\alpha }_2).$$

$$⟨J_z(x)|y⟩=⟨z|[x,y]⟩(x,y\in {\mathfrak{g}}_{-1})$$

$$J_z(\alpha )=-(z\ell )\alpha ,J_z^2=\gamma \overline{z}z{1}_{{\mathfrak{g}}_{-1}}.$$

$$\kappa (\alpha )=-\overline{{\alpha }_2}-\overline{{\alpha }_1}\ell ,\kappa (z)=-\overline{z},$$

$${\mathfrak{n}}^i=\underset{p<0}{⨁}{\mathfrak{n}}_p^i,{\mathfrak{n}}_p^i={\mathfrak{n}}^i\cap {\mathfrak{g}}_p.$$

$${\mathfrak{n}}_{-1}^1=\mathbb{F}(\gamma {)}^n,{\mathfrak{n}}_{-2}^1=\mathbb{F},$$

$${\mathfrak{n}}_{-1}^2=\{{\alpha }_1+\hat{\tau }({\alpha }_1)\ell :{\alpha }_1\in ({\mathbb{F}}^c{)}^n\},{\mathfrak{n}}_{-2}^2=\sqrt{-1}\mathbb{R}\oplus \mathrm{Im}(\mathbb{F}),$$

$$[\alpha ,\beta ]=\hat{\tau }({\alpha }_1)S{}^t{\beta }_1-\hat{\tau }({\beta }_1)S{}^t{\alpha }_1,$$

$$⟨\alpha \mid \beta ⟩=R({\alpha }_{1}S{}^t\overline{{\beta }_1}-\gamma \tau ({\beta }_1)S{}^t\tau (\overline{{\alpha }_1}))=(1-\gamma )R({\alpha }_{1}S{}^t\overline{{\beta }_1}).$$

$$\phi ({\alpha }_1+{\alpha }_2\ell )={\alpha }_{1}P+{\alpha }_2{1}_{r,s}P\ell ({\alpha }_1,{\alpha }_2\in {\mathbb{F}}^n),\phi (z)=z(z\in {\mathfrak{h}}^{(2)}(\mathbb{F},1_{r+s},{\gamma }^{\prime }{)}_{-2}).$$

$$\phi ({\alpha }_1+\hat{\tau }({\alpha }_1)\ell )=\eta ({\alpha }_1)Q+\hat{\tau }(\eta ({\alpha }_1)Q)\ell ({\alpha }_1\in ({\mathbb{F}}^c{)}^n),\phi (z)=z(z\in {\mathfrak{h}}^{(3)}(\mathbb{F},1_{r+s}{)}_{-2}).$$

$$\phi ({\alpha }_1+\hat{\tau }({\alpha }_1)\ell )=\eta ({\alpha }_1)R+\hat{\tau }(\eta ({\alpha }_1)R)\ell ({\alpha }_1\in ({\mathbb{F}}^c{)}^n),\phi (z)=z(z\in {\mathfrak{h}}^{(3)}(\mathbb{F},1_{r+s}{)}_{-2}),$$

$$((\mathrm{AI}{)}_l,\{{\alpha }_1,{\alpha }_l\}),({\text{(AIIIa)}}_{l,p},\{{\alpha }_1,{\alpha }_l\}),({\text{(AIIIb)}}_l,\{{\alpha }_1,{\alpha }_l\}),({\text{(AIV)}}_l,\{{\alpha }_1,{\alpha }_l\}),({\text{(BI)}}_l,\{{\alpha }_2\}),$$

$$({\text{(CI)}}_l,\{{\alpha }_1\}),({\text{(DI)}}_l,\{{\alpha }_2\}),(\text{EI},\{{\alpha }_2\}),(\text{EII},\{{\alpha }_2\}),(\text{EIII},\{{\alpha }_2\}),(\text{EIV},\{{\alpha }_2\}),$$

$$(\text{EV},\{{\alpha }_1\}),(\text{EVI},\{{\alpha }_1\}),(\text{EVII},\{{\alpha }_1\}),(\text{EVIII},\{{\alpha }_8\}),(\text{EIX},\{{\alpha }_8\}),(\text{FI},\{{\alpha }_1\}),(\text{G},\{{\alpha }_2\}),$$

$$I(\alpha )=-\gamma J_1{J}_{{\ell }_0}(\alpha )={\ell }_0(\alpha ),I(z)={\ell }_{0}z\text{if}(\mathfrak{n},⟨\cdot |\cdot ⟩)={\mathfrak{H}}^{(2)}(\mathbb{F},S,\gamma )$$

Full text

1. Introduction

In [10] A. Kaplan introduced $H$-type Lie algebras, which belong to a special class of 2-step nilpotent Lie algebras. This class is associated with the Clifford algebra for an inner product space and an admissible module of the Clifford algebra. An $H$-type Lie algebra obtained by replacing the inner product to a general scalar product first appeared in [4]. This Lie algebra with the scalar product is called a pseudo $H$-type Lie algebra, which is exactly defined below.

Let $\mathfrak{n}$ be a finite dimensional 2-step nilpotent real Lie algebras, that is, $\mathfrak{n}$ is a finite dimensional real Lie algebra satisfying $[\mathfrak{n},\mathfrak{n}]\neq 0$ and $[\mathfrak{n},[\mathfrak{n},\mathfrak{n}]]=0$. Let $\langle\cdot\,|\,\cdot\rangle$ be a scalar product on $\mathfrak{n}$ such that the center $\mathfrak{n}_{-2}$ of $\mathfrak{n}$ is a non-degenerate subspace of $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$. Here a scalar product on $\mathfrak{n}$ means a non-degenerate symmetric bilinear form on $\mathfrak{n}$. Let $\mathfrak{n}_{-1}$ be the orthogonal complement of $\mathfrak{n}_{-2}$ with respect to $\langle\cdot\,|\,\cdot\rangle$. The pair $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$ is called a pseudo $H$-type Lie algebra if for any $z\in\mathfrak{n}_{-2}$ the endomorphism $J_{z}$ of $\mathfrak{n}_{-1}$ defined by $\langle J_{z}(x)\,|\,y\rangle=\langle z\,|\,[x,y]\rangle$ $(x,y\in\mathfrak{n}_{-1})$ satisfies the Clifford condition $J_{z}^{2}=-\langle z\,|\,z\rangle 1_{\mathfrak{n}_{-1}}$, where $1_{\mathfrak{n}_{-1}}$ is the identity transformation of $\mathfrak{n}_{-1}$. In particular, if $\langle\cdot\,|\,\cdot\rangle$ is positive definite, then $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$ is simply called an $H$-type Lie algebra.

Let $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$ be a pseudo $H$-type Lie algebra. Then $\mathfrak{n}=\mathfrak{n}_{-2}\oplus\mathfrak{n}_{-1}$ becomes a non-degenerate fundamental graded Lie algebra of the second kind, which is called associated with $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$.

Now we explain the notion of a fundamental graded Lie algebra and its prolongation briefly. A finite dimensional graded Lie algebra (GLA) $\mathfrak{m}=\bigoplus\limits_{p<0}\mathfrak{g}_{p}$ is called a fundamental graded Lie algebra (FGLA) of the $\mu$-th kind if the following conditions hold: (i) $\mathfrak{g}_{-1}\neq 0$, and $\mathfrak{m}$ is generated by $\mathfrak{g}_{-1}$; (ii) $\mathfrak{g}_{p}=0$ for all $p<-\mu$, where $\mu$ is a positive integer. Furthermore an FGLA $\mathfrak{m}=\bigoplus\limits_{p<0}\mathfrak{g}_{p}$ is called non-degenerate if for $x\in\mathfrak{g}_{-1}$, $[x,\mathfrak{g}_{-1}]=0$ implies $x=0$. For a given FGLA $\mathfrak{m}=\bigoplus\limits_{p<0}\mathfrak{g}_{p}$ there exists a GLA $\check{\mathfrak{g}}=\bigoplus\limits_{p\in\mathbb{Z}}\check{\mathfrak{g}}_{p}$ satisfying the following conditions: (P1) The negative part $\check{\mathfrak{g}}_{-}=\bigoplus\limits_{p<0}\check{\mathfrak{g}}_{p}$ of $\check{\mathfrak{g}}=\bigoplus\limits_{p\in\mathbb{Z}}\check{\mathfrak{g}}_{p}$ coincides with a given FGLA $\mathfrak{m}$ as a GLA; (P2) For $x\in\check{\mathfrak{g}}_{p}$ $(p\geqq 0)$, $[x,\mathfrak{g}_{-1}]=0$ implies $x=0$; (P3) $\check{\mathfrak{g}}=\bigoplus\limits_{p\in\mathbb{Z}}\check{\mathfrak{g}}_{p}$ is maximum among GLAs satisfying the conditions (P1) and (P2) above. The GLA $\check{\mathfrak{g}}=\bigoplus\limits_{p\in\mathbb{Z}}\check{\mathfrak{g}}_{p}$ is called the (Tanaka) prolongation of the FGLA $\mathfrak{m}$. Given the prolongation $\check{\mathfrak{g}}=\bigoplus\limits_{p\in\mathbb{Z}}\check{\mathfrak{g}}_{p}$ of an FGLA $\mathfrak{m}$, an element $E$ of $\check{\mathfrak{g}}_{0}$ is called the characteristic element of $\check{\mathfrak{g}}=\bigoplus\limits_{p\in\mathbb{Z}}\check{\mathfrak{g}}_{p}$ if $[E,x]=px$ for all $x\in\check{\mathfrak{g}}_{p}$ and $p\in\mathbb{Z}$. Also $\operatorname{ad}(\check{\mathfrak{g}}_{0})|\mathfrak{m}$ is a subalgebra of $\operatorname{Der}(\mathfrak{m})$ isomorphic to $\check{\mathfrak{g}}_{0}$; we identify it with $\check{\mathfrak{g}}_{0}$ in what follows, so that $D\in\check{\mathfrak{g}}_{0}$ is identified with $\operatorname{ad}(D)|\mathfrak{m}$. (For the details of FGLAs and a construction of the prolongation, see [15, §5]).

For a given pseudo $H$-type Lie algebra $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$ the prolongation $\check{\mathfrak{g}}=\bigoplus\limits_{p\in\mathbb{Z}}\check{\mathfrak{g}}_{p}$ of the FGLA $\mathfrak{n}$ is finite dimensional if and only if $\dim\mathfrak{n}_{-2}\geqq 3$ ([1, Theorem 2.4, and Propositions 4.4 and 4.5]). Moreover in [2, Theorem 3.1] A. Altomani and A. Santi proved that if $\dim\mathfrak{n}_{-2}\geqq 3$ and the prolongation is not trivial (i.e., $\check{\mathfrak{g}}_{1}\neq 0$), then $\check{\mathfrak{g}}=\bigoplus\limits_{p\in\mathbb{Z}}\check{\mathfrak{g}}_{p}$ is a finite dimensional SGLA (In this paper we abbreviate simple GLA to SGLA).

We next give the notion of a conformal pseudo-subriemannian FGLA and its prolongation. We say that the pair $(\mathfrak{m},[g])$ of a real FGLA $\mathfrak{m}$ of the $\mu$-th kind $(\mu\geqq 2)$ and the conformal class $[g]$ of a scalar product $g$ on $\mathfrak{g}_{-1}$ is a conformal pseudo-subriemannian FGLA (cps-FGLA). Let $\check{\mathfrak{g}}=\bigoplus\limits_{p\in\mathbb{Z}}\check{\mathfrak{g}}_{p}$ be the prolongation of $\mathfrak{m}$, and let $\mathfrak{g}_{0}$ be the subalgebra of $\check{\mathfrak{g}}_{0}$ consisting of all the elements $D$ of $\check{\mathfrak{g}}_{0}$ such that $\operatorname{ad}(D)|\mathfrak{g}_{-1}\in\mathfrak{co}(\mathfrak{g}_{-1},g)$. We define a sequence $(\mathfrak{g}_{p})_{p\geqq 1}$ inductively as follows: $l$ being a positive integer, suppose that we defined $\mathfrak{g}_{1},\dots,\mathfrak{g}_{l-1}$ as subspaces of $\check{\mathfrak{g}}_{1},\dots,\check{\mathfrak{g}}_{l-1}$ respectively, in such a way that $[\mathfrak{g}_{p},\mathfrak{g}_{r}]\subset\mathfrak{g}_{p+r}$ $(0<p<l,r<0)$. Then we define $\mathfrak{g}_{l}$ to be the subspace of $\check{\mathfrak{g}}_{l}$ consisting of all the elements $D$ of $\check{\mathfrak{g}}_{l}$ such that $[D,\mathfrak{g}_{r}]\subset\mathfrak{g}_{l+r}$ $(r<0)$. If we put $\mathfrak{g}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{g}_{p}$, then it becomes a graded subalgebra of $\check{\mathfrak{g}}=\bigoplus\limits_{p\in\mathbb{Z}}\check{\mathfrak{g}}_{p}$, which is called the prolongation of $(\mathfrak{m},\mathfrak{g}_{0})$. The prolongation of $(\mathfrak{m},\mathfrak{g}_{0})$ is also called that of the cps-FGLA $(\mathfrak{m},[g])$. The prolongation $\mathfrak{g}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{g}_{p}$ of the cps-FGLA $(\mathfrak{m},[g])$ is finite dimensional. If $\mathfrak{g}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{g}_{p}$ is semisimple, then the cps-FGLA $(\mathfrak{m},[g])$ is said to be of semisimple type. In the previous paper [18] we classified the prolongations of cps-FGLAs of semisimple type.

Let $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$ be a pseudo $H$-type Lie algebra. The pair $(\mathfrak{n},[\langle\cdot\,|\,\cdot\rangle_{-1}])$ becomes a cps-FGLA, which is called associated with $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$. Here we denote by $\langle\cdot\,|\,\cdot\rangle_{k}$ the restriction of $\langle\cdot\,|\,\cdot\rangle$ to $\mathfrak{n}_{k}$.

In [13] A. Kaplan and M. Sublis introduced the notion of a $\operatorname{div}H$-type Lie algebra (or a Lie algebra of type $\operatorname{div}H$) and classified the finite dimensional real SGLAs whose negative parts are isomorphic to some $\operatorname{div}H$-type Lie algebra. In [12] they also proved that the prolongation of the FGLA associated with an $H$-type Lie algebra is not trivial if and only if it is a $\operatorname{div}H$-type Lie algebra. In §3, inspired by the studies in [13] and [7], we give a little generalization of a $\operatorname{div}H$-type Lie algebra, which is called a pseudo $\operatorname{div}H$-type Lie algebra. More precisely, the pseudo $\operatorname{div}H$-type Lie algebras consist of three classes (pseudo $\operatorname{div}H$-type Lie algebras of the first, the second and the third classes). We determine the prolongations of the FGLAs associated with pseudo $\operatorname{div}H$-type Lie algebras by an elementary method. It is known that a pseudo $H$-type Lie algebra satisfying the $J^{2}$-condition becomes a pseudo $\operatorname{div}H$-type Lie algebra of the first class, and vice versa (cf.[14]). In §4 we prove that a pseudo $H$-type Lie algebra satisfies the $J^{2}$-condition if and only if the prolongation of the associated cps-FGLA is a finite dimensional SGLA (Theorem 4.1).

By [2, Theorem 3.1] and [11, Theorem 5.3], the prolongation $\mathfrak{g}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{g}_{p}$ of the cps-FGLA associated with a pseudo $H$-type Lie algebra $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$ is a finite dimensional SGLA of real rank one if the following conditions hold: (i) $\mathfrak{g}_{1}\neq 0$; (ii) $\langle\cdot\,|\,\cdot\rangle_{-1}$ is definite. However if $\langle\cdot\,|\,\cdot\rangle_{-1}$ is indefinite, $\mathfrak{g}$ has a more complicated form. In §5 we show that if $\mathfrak{g}_{2}\neq 0$, then $\mathfrak{g}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{g}_{p}$ is a finite dimensional SGLA and coincides with the prolongation of $\mathfrak{n}$ under the additional condition “$\dim\mathfrak{n}_{-2}\geqq 3$” (Theorem 5.3).

In [5] K. Furutani et al. investigated the prolongations of the FGLAs associated with pseudo $H$-type Lie algebras. From their results, we conjecture that if the prolongation of the FGLA associated with a pseudo $H$-type Lie algebra is not trivial, then it is of pseudo $\operatorname{div}H$-type.

2. Pseudo $H$-type Lie algebras

Following [4] we define pseudo $H$-type Lie algebras. Let $\mathfrak{n}$ be a finite dimensional 2-step nilpotent real Lie algebra equipped with a non-degenerate symmetric bilinear form $\langle\cdot\,|\,\cdot\rangle$ on $\mathfrak{n}$. The pair $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$ is called a pseudo $H$-type Lie algebra if the following conditions hold:

1. (H.1)

The restriction of $\langle\cdot\,|\,\cdot\rangle$ to the center $\mathfrak{n}_{-2}$ of $\mathfrak{n}$ is non-degenerate. 2. (H.2)

Let $\mathfrak{n}_{-1}$ be the orthogonal complement of the center $\mathfrak{n}_{-2}$ of $\mathfrak{n}$ with respect to $\langle\cdot\,|\,\cdot\rangle$. For any $z\in\mathfrak{n}_{-2}$ the endomorphism $J_{z}$ of $\mathfrak{n}_{-1}$ defined by

[TABLE]

satisfies the following condition

[TABLE]

where $1_{\mathfrak{n}_{-1}}$ is the identity transformation of $\mathfrak{n}_{-1}$.

The condition (2) is called the Clifford condition. In particular if $\langle\cdot\,|\,\cdot\rangle$ is positive definite, then $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$ is simply called an $H$-type Lie algebra. Given a pseudo $H$-type Lie algebra $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$ we can easily see that:

1. (i) 

For any $z\in\mathfrak{n}_{-2}$ the linear mapping $J_{z}$ is skew-symmetric; 2. (ii) 

$\mathfrak{n}=\mathfrak{n}_{-1}\oplus\mathfrak{n}_{-2}$ is a non-degenerate FGLA of the second kind.

The FGLA $\mathfrak{n}=\mathfrak{n}_{-1}\oplus\mathfrak{n}_{-2}$ is called associated with the pseudo $H$-type Lie algebra $(\mathfrak{n},\langle\cdot\mid\cdot\rangle)$. The pair $(\mathfrak{n}=\mathfrak{n}_{-1}\oplus\mathfrak{n}_{-2},[\langle\cdot\,|\,\cdot\rangle_{-1}])$ becomes a conformal pseudo-subriemannian FGLA (cps-FGLA), which is called associated with the pseudo $H$-type Lie algebra $(\mathfrak{n},\langle\cdot\mid\cdot\rangle)$. Given two pseudo $H$-type Lie algebras $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$ and $(\mathfrak{n}^{\prime},\langle\cdot\,|\,\cdot\rangle^{\prime})$, we say that $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$ is isomorphic to $(\mathfrak{n}^{\prime},\langle\cdot\,|\,\cdot\rangle^{\prime})$ if there exists a Lie algebra isomorphism $\varphi$ of $\mathfrak{n}$ onto $\mathfrak{n}^{\prime}$ such that $\varphi$ is an isometry of $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$ onto $(\mathfrak{n}^{\prime},\langle\cdot\,|\,\cdot\rangle^{\prime})$. Moreover we say that $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$ is equivalent to $(\mathfrak{n}^{\prime},\langle\cdot\,|\,\cdot\rangle^{\prime})$ if there exists a Lie algebra isomorphism $\varphi$ of $\mathfrak{n}$ onto $\mathfrak{n}^{\prime}$ such that: (i) $\varphi(\mathfrak{n}_{-1})=\mathfrak{n}^{\prime}_{-1}$, and $\varphi|\mathfrak{n}_{-1}$ is an isometry or an anti-isometry of $(\mathfrak{n}_{-1},\langle\cdot\,|\,\cdot\rangle_{-1})$ onto $(\mathfrak{n}^{\prime}_{-1},\langle\cdot\,|\,\cdot\rangle^{\prime}_{-1})$; (ii) $\varphi|\mathfrak{n}_{-2}$ is an isometry of $(\mathfrak{n}_{-2},\langle\cdot\,|\,\cdot\rangle_{-2})$ onto $(\mathfrak{n}^{\prime}_{-2},\langle\cdot\,|\,\cdot\rangle^{\prime}_{-2})$. If a pseudo $H$-type Lie algebra $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$ is equivalent to a pseudo $H$-type Lie algebra $(\mathfrak{n}^{\prime},\langle\cdot\,|\,\cdot\rangle^{\prime})$, then the prolongation of $(\mathfrak{n},[\langle\cdot\,|\,\cdot\rangle_{-1}])$ is isomorphic to that of $(\mathfrak{n}^{\prime},[\langle\cdot\,|\,\cdot\rangle^{\prime}_{-1}])$.

Lemma 2.1**.**

Let $(\mathfrak{n}=\mathfrak{n}_{-1}\oplus\mathfrak{n}_{-2},\langle\cdot\,|\,\cdot\rangle)$ be a pseudo $H$-type Lie algebra. We define a new scalar product $\langle\cdot\,|\,\cdot\rangle^{\prime}$ on $\mathfrak{n}$ as follows:

[TABLE]

where $\alpha,\beta$ are nonzero real numbers. The pair $(\mathfrak{n}=\mathfrak{n}_{-1}\oplus\mathfrak{n}_{-2},\langle\cdot\,|\,\cdot\rangle^{\prime})$ also becomes a pseudo $H$-type Lie algebra if and only if $\alpha^{2}=\beta$. In this case, the cps-FGLA associated with $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle^{\prime})$ is $(\mathfrak{n},[\alpha\langle\cdot\,|\,\cdot\rangle_{-1}])$.

Proof.

By (1), for $x,y\in\mathfrak{n}_{-1}$ and $z\in\mathfrak{n}_{-2}$, $\langle\alpha^{-1}\beta J_{z}(x)\,|\,y\rangle^{\prime}=\beta\langle J_{z}(x)\,|\,y\rangle=\beta\langle z\,|\,[x,y]\rangle=\langle z\,|\,[x,y]\rangle^{\prime}$. By (2), $(\alpha^{-1}\beta J_{z})^{2}=\alpha^{-2}\beta^{2}J_{z}^{2}=-\alpha^{-2}\beta^{2}\langle z\,|\,z\rangle 1_{\mathfrak{n}_{-1}}=-\alpha^{-2}\beta\langle z\,|\,z\rangle^{\prime}1_{\mathfrak{n}_{-1}}$. This proves the first statement. The last statement is clear. ∎

The proof of the following lemma is due to the proof of [6, Theorem 2].

Lemma 2.2**.**

Let $(\mathfrak{n}^{(1)},\langle\cdot\,|\,\cdot\rangle^{(1)})$ and $(\mathfrak{n}^{(2)},\langle\cdot\,|\,\cdot\rangle^{(2)})$ be pseudo $H$-type Lie algebras. Assume that there exists a GLA isomorphism $\varphi$ of $\mathfrak{n}^{(1)}$ onto $\mathfrak{n}^{(2)}$. Then there exists a GLA isomorphism $\psi$ of $\mathfrak{n}^{(1)}$ onto $\mathfrak{n}^{(2)}$ and a positive real number $\alpha$ such that: (i) $\psi|\mathfrak{n}_{-2}^{(1)}$ is an isometry or an anti-isometry; (ii) $\psi|\mathfrak{n}_{-1}^{(1)}=\alpha\varphi|\mathfrak{n}_{-1}^{(1)}$.

Remark 2.1**.**

Let $(\mathfrak{n}^{(1)},\langle\cdot\,|\,\cdot\rangle^{(1)})$ and $(\mathfrak{n}^{(2)},\langle\cdot\,|\,\cdot\rangle^{(2)})$ be $H$-type Lie algebras. If $\mathfrak{n}^{(1)}$ is isomorphic to $\mathfrak{n}^{(2)}$ as a GLA, then $(\mathfrak{n}^{(1)},\langle\cdot\,|\,\cdot\rangle^{(1)})$ is isomorphic to $(\mathfrak{n}^{(2)},\langle\cdot\,|\,\cdot\rangle^{(2)})$ as an $H$-type Lie algebra ([12, Theorem 2]).

Proposition 2.1**.**

Let $\mathfrak{g}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{g}_{p}$ be a finite dimensional real SGLA such that the negative part $\mathfrak{g}_{-}=\bigoplus_{p<0}\limits\mathfrak{g}_{p}$ is an FGLA of the second kind. Let $\langle\cdot\,|\,\cdot\rangle^{(i)}$ $(i=1,2)$ be scalar products on $\mathfrak{g}_{-}$. Assume that:

1. (i)

$(\mathfrak{g}_{-},\langle\cdot\,|\,\cdot\rangle^{(1)})$* and $(\mathfrak{g}_{-},\langle\cdot\,|\,\cdot\rangle^{(2)})$ are pseudo $H$-type Lie algebras whose associated FGLAs coincide with $\mathfrak{g}_{-}$ as a GLA.* 2. (ii)

For $i=1,2$ the prolongation of the associated csp-GLA $(\mathfrak{g}_{-},[\langle\cdot\,|\,\cdot\rangle_{-1}^{(i)}])$ coincides with $\mathfrak{g}$.

Then

1. (1) 

$[\langle\cdot\,|\,\cdot\rangle_{-1}^{(1)}]$* is equal to $[\langle\cdot\,|\,\cdot\rangle_{-1}^{(2)}]$ or $[-\langle\cdot\,|\,\cdot\rangle_{-1}^{(2)}]$;* 2. (2) 

$[\langle\cdot\,|\,\cdot\rangle_{-2}^{(1)}]=[\langle\cdot\,|\,\cdot\rangle_{-2}^{(2)}]$,

Consequently, $(\mathfrak{g}_{-},\langle\cdot\,|\,\cdot\rangle^{(1)})$ is equivalent to $(\mathfrak{g}_{-},\langle\cdot\,|\,\cdot\rangle^{(2)})$.

Proof.

Let $\varphi$ be the identity transformation of $\mathfrak{g}_{-}$. By the assumption (i) $\varphi$ is a GLA isomorphism of $\mathfrak{g}_{-}$ onto itself. By Lemma 2.2, there exists a GLA isomorphism $\psi$ of $\mathfrak{g}_{-}$ onto itself such that: (i) the restriction $\psi|\mathfrak{g}_{-2}$ to $\mathfrak{g}_{-2}$ of $\psi$ is an isometry or an anti-isometry; (ii) there exist a nonzero real number $\alpha^{\prime}$ such that $\psi|\mathfrak{g}_{-2}=\alpha^{\prime 2}\varphi|\mathfrak{g}_{-2}$ and $\psi|\mathfrak{g}_{-1}=\alpha^{\prime}\varphi|\mathfrak{g}_{-1}$. Hence $\alpha^{\prime 4}\langle\cdot\,|\,\cdot\rangle^{(2)}_{-2}=\pm\langle\cdot\,|\,\cdot\rangle^{(1)}_{-2}$. By assumptions (ii), (iii) and [18, Proposition 5.2], $\langle\cdot\,|\,\cdot\rangle^{(2)}_{-1}$ coincides with $\langle\cdot\,|\,\cdot\rangle^{(1)}_{-1}$ multiplied by a nonzero real number. By Lemma 2.1, there exists a nonzero real number $\alpha$ such that $\langle\cdot\,|\,\cdot\rangle^{(2)}_{-1}=\alpha\langle\cdot\,|\,\cdot\rangle^{(1)}_{-1}$, $\langle\cdot\,|\,\cdot\rangle^{(2)}_{-2}=\alpha^{2}\langle\cdot\,|\,\cdot\rangle^{(1)}_{-2}$. Thus assertions (i) and (ii) are proved. We define a linear mapping $f$ of $\mathfrak{g}_{-}$ into itself as follows:

[TABLE]

then $f$ is a GLA isomorphism and we see that

[TABLE]

Hence $(\mathfrak{g}_{-},\langle\cdot\,|\,\cdot\rangle^{(1)})$ is equivalent to $(\mathfrak{g}_{-},\langle\cdot\,|\,\cdot\rangle^{(2)})$. ∎

3. Pseudo $\operatorname{div}H$-type Lie algebras

In this section we introduce pseudo $\operatorname{div}H$-type Lie algebras. The pseudo $\operatorname{div}H$-type Lie algebras consist of pseudo $\operatorname{div}H$-type Lie algebras $\mathfrak{H}^{(1)}(\mathbb{F},S)$ of the first class, $\mathfrak{H}^{(2)}(\mathbb{F},S,\gamma)$ of the second class, and $\mathfrak{H}^{(3)}(\mathbb{F},S)$ of the third class, which is defined below.

3.1. Cayley algebras

Let $\mathbb{F}$ be $\mathbb{C}$, $\mathbb{C}^{\prime}$, $\mathbb{H}$, $\mathbb{H}^{\prime}$, $\mathbb{O}$ or $\mathbb{O}^{\prime}$, where $\mathbb{C}$ (resp. $\mathbb{C}^{\prime}$, $\mathbb{H}$, $\mathbb{H}^{\prime}$, $\mathbb{O}$, $\mathbb{O}^{\prime}$) is a Cayley algebra of the complex numbers (resp. the split complex numbers, the Hamilton’s quaternions, the split quaternions, the Cayley’s octonions, the split octonions). Here we consider $\mathbb{F}$ as an algebra over $\mathbb{R}$. We denote by $\mathbb{F}(\gamma)$ the Cayley extension of $\mathbb{F}$ defined by $\gamma$, where $\gamma=\pm 1$ (cf. [3, Ch.3, no.5]). Namely $\mathbb{F}(\gamma)$ is an algebra over $\mathbb{R}$ which $\mathbb{F}(\gamma)=\mathbb{F}\times\mathbb{F}$ as a module and the multiplication on $\mathbb{F}(\gamma)$ is defined by

[TABLE]

Clearly $\mathbb{F}\times\{0\}$ is a subalgebra of $\mathbb{F}(\gamma)$ isomorphic to $\mathbb{F}$; we shall identify it with $\mathbb{F}$ in what follows, so that $x\in\mathbb{F}$ is identified with $(x,0)$. Let $\ell=(0,1)$, so that $(x,y)=x+y\ell$ for $x,y\in\mathbb{F}$. Note that: (i) $\ell\alpha=\overline{\alpha}\ell$; (ii) $\alpha(\beta\ell)=(\beta\alpha)\ell$; (iii) $(\alpha\ell)\beta=(\alpha\bar{\beta})\ell$; (iv) $(\alpha\ell)(\beta\ell)=\gamma(\overline{\beta}\alpha)$; (v) $\ell^{2}=\gamma$, where $\alpha,\beta\in\mathbb{F}$. When $\mathbb{F}=\mathbb{H}$ (resp. $\mathbb{F}=\mathbb{H}^{\prime}$) we put $\mathbb{F}_{0}=\mathbb{C}$, and $\gamma_{0}=-1$ (resp. $\gamma_{0}=1)$; then $\mathbb{F}=\mathbb{F}_{0}(\gamma_{0})$. Let $\ell_{0}$ be the element of $\mathbb{F}$ corresponding to the element $(0,1)\in\mathbb{F}_{0}(\gamma_{0})=\mathbb{F}_{0}\times\mathbb{F}_{0}$. We denote by $\mathbb{F}^{c}=\mathbb{F}\oplus\sqrt{-1}\mathbb{F}$, $\mathbb{F}(\gamma)^{c}=\mathbb{F}(\gamma)\oplus\sqrt{-1}\mathbb{F}(\gamma)$ the complexifications of $\mathbb{F}$, $\mathbb{F}(\gamma)$ respectively. Let $\operatorname{pr}_{1}$ and $\operatorname{pr}_{2}$ be the projections of $\mathbb{F}(\gamma)^{c}=\mathbb{F}^{c}\times\mathbb{F}^{c}$ onto $\mathbb{F}^{c}$ defined by $\operatorname{pr}_{i}(x_{1},x_{2})=x_{i}$ $(i=1,2)$. Note that $\operatorname{pr}_{1}(\overline{\alpha})=\overline{\operatorname{pr}_{1}(\alpha)}$, $\operatorname{pr}_{2}(\overline{\alpha})=-\operatorname{pr}_{2}(\alpha)$, $\operatorname{pr}_{1}(\ell\alpha)=\gamma\overline{\operatorname{pr}_{2}(\alpha)}$, $\operatorname{pr}_{2}(\ell\alpha)=\overline{\operatorname{pr}_{1}(\alpha)}$, where $\alpha\in\mathbb{F}(\gamma)^{c}$. We define a mapping $R$ of $\mathbb{F}(\gamma)^{c}$ to $\mathbb{R}$ by $R(u+\sqrt{-1}v)=\operatorname{Re}(u)$ $(u,v\in\mathbb{F}(\gamma))$. For $z\in\mathbb{F}=\mathbb{F}\times\{0\}$ and $\alpha\in\mathbb{F}(\gamma)^{c}$ we obtain $R(z\operatorname{pr}_{1}(\alpha))=R(z\alpha)$. We extend the conjugation “$\overset{-}{\cdot}$” on $\mathbb{F}(\gamma)$ to $\mathbb{F}(\gamma)^{c}$ by $\overline{u+\sqrt{-1}v}=\overline{u}+\sqrt{-1}\overline{v}$.

3.2. Pseudo $\operatorname{div}H$-type Lie algebras of the first class

Let $\mathbb{F}$ be $\mathbb{C}$, $\mathbb{C}^{\prime}$, $\mathbb{H}$, $\mathbb{H}^{\prime}$, $\mathbb{O}$ or $\mathbb{O}^{\prime}$. Let $S$ be a real symmetric matrix of order $n$ such that $S^{2}=1_{n}$, where $1_{n}$ is the identity matrix of order $n$. We put

[TABLE]

where we assume $n=1$ in case $F=\mathbb{O}$ or $\mathbb{O}^{\prime}$. Note that $\mathbb{F}^{n}$ is the set of all the $\mathbb{F}$-valued row vectors of order $n$. We define a bracket operation on $\mathfrak{n}$ as follows:

[TABLE]

then $(\mathfrak{n},[\cdot,\cdot])$ becomes an FGLA of the second kind. Furthermore we define a symmetric bilinear form $\langle\cdot\mid\cdot\rangle$ on $\mathfrak{n}$ as follows:

[TABLE]

The linear mapping $J_{z}$ defined by (2) has the following form: $J_{z}(x)=-zx$. Thus $(\mathfrak{n},\langle\cdot\mid\cdot\rangle)$ becomes a pseudo $H$-type Lie algebra, which is denoted by $\mathfrak{H}^{(1)}(\mathbb{F},S)=(\mathfrak{h}^{(1)}(\mathbb{F},S),\langle\cdot\,|\,\cdot\rangle)$. The pseudo $H$-type Lie algebra $\mathfrak{H}^{(1)}(\mathbb{F},S)$ is called a pseudo $\operatorname{div}H$-type Lie algebra of the first class. We denote the FGLA associated with $\mathfrak{H}^{(1)}(\mathbb{F},S)$ by $\mathfrak{h}^{(1)}(\mathbb{F},S)=\bigoplus\limits_{p=-1}^{-2}\mathfrak{h}^{(1)}(\mathbb{F},S)_{p}$.

Lemma 3.1**.**

Let $(r,s)$ be the signature of $S$.

1. (1) 

$\mathfrak{H}^{(1)}(\mathbb{F},S)$* is isomorphic to $\mathfrak{H}^{(1)}(\mathbb{F},1_{r,s})$.* 2. (2) 

$\mathfrak{H}^{(1)}(\mathbb{F},1_{r,s})$* is equivalent to $\mathfrak{H}^{(1)}(\mathbb{F},1_{s,r})$.*

Proof.

(1) There exists a real orthogonal matrix $P$ such that $PSP^{-1}=1_{r,s}$, where $1_{r,s}=\begin{bmatrix}1_{r}&O\\ O&-1_{s}\\ \end{bmatrix}$. We define a linear mapping $\varphi$ of $\mathfrak{h}^{(1)}(\mathbb{F},1_{r,s})$ to $\mathfrak{h}^{(1)}(\mathbb{F},S)$ as follows:

[TABLE]

Then $\varphi$ is an isomorphism as a pseudo $H$-type Lie algebra. Hence $\mathfrak{H}^{(1)}(\mathbb{F},S)$ is isomorphic to $\mathfrak{H}^{(1)}(\mathbb{F},1_{r,s})$.

(2) We define a linear mapping $\psi$ of $\mathfrak{h}^{(1)}(\mathbb{F},1_{r,s})$ to $\mathfrak{h}^{(1)}(\mathbb{F},1_{s,r})$ as follows:

[TABLE]

where $K_{n}$ is the $n\times n$ matrix whose $(i,j)$-component is $\delta_{i,n+1-j}$. Then $\psi$ is an isomorphism as a GLA. Moreover $\psi|\mathfrak{h}^{(1)}(\mathbb{F},1_{r,s})_{-2}$ is isometry and $\psi|\mathfrak{h}^{(1)}(\mathbb{F},1_{r,s})_{-1}$ is anti-isometry. Hence $\mathfrak{H}^{(1)}(\mathbb{F},1_{r,s})$ is equivalent to $\mathfrak{H}^{(1)}(\mathbb{F},1_{s,r})$. ∎

Remark 3.1**.**

The $H$-type Lie algebra $\mathfrak{H}^{(1)}(\mathbb{F},1_{r,s})$ coincides with $\mathfrak{h}^{\prime}_{r,s}(\mathbb{F})$ in [12].

3.3. Pseudo $\operatorname{div}H$-type Lie algebras of the second and the third classes

Let $\mathbb{F}$ be $\mathbb{C}$, $\mathbb{C}^{\prime}$, $\mathbb{H}$, $\mathbb{H}^{\prime}$, $\mathbb{O}$ or $\mathbb{O}^{\prime}$. We set

[TABLE]

where we assume $n=1$ in case $\mathbb{F}=\mathbb{O}$ or $\mathbb{O}^{\prime}$. Let $S$ be a real symmetric matrix of order $n$ such that $S^{2}=1_{n}$. We define a bracket operation $[\cdot\,,\,\cdot]$ on $\mathfrak{m}=\mathfrak{g}_{-2}\oplus\mathfrak{g}_{-1}$ as follows:

[TABLE]

More explicitly, the bracket operation can be written as follows: if we put $\alpha=\alpha_{1}+\alpha_{2}\ell$ and $\beta=\beta_{1}+\beta_{2}\ell$ $(\alpha_{1},\alpha_{2},\beta_{1},\beta_{2}\in(\mathbb{F}^{c})^{n}$), then

[TABLE]

Then $\mathfrak{m}$ becomes a complex FGLA of the second kind. Moreover we define a symmetric bilinear form $\langle\cdot\,|\,\cdot\rangle$ on $\mathfrak{m}$ as follows:

[TABLE]

More explicitly, the bilinear form can be written as follows: if we put $\alpha=\alpha_{1}+\alpha_{2}\ell$ and $\beta=\beta_{1}+\beta_{2}\ell$ $(\alpha_{1},\alpha_{2},\beta_{1},\beta_{2}\in(\mathbb{F}^{c})^{n}$), then

[TABLE]

For $z\in\mathfrak{g}_{-2}$ the linear mapping $J_{z}$ of $\mathfrak{g}_{-1}$ to itself defined by

[TABLE]

satisfies

[TABLE]

We denote by the same letter $\tau$ the conjugations of $\mathbb{F}^{c}$ and $\mathbb{F}(\gamma)^{c}$ with respect to $\mathbb{F}$ and $\mathbb{F}(\gamma)$ respectively. We now extend $\tau$ to a grade-preserving involution of $\mathfrak{m}$ in a natural way, which is also denoted by the same letter. Next we define a grade-preserving involution $\kappa$ of $\mathfrak{m}$ as follows:

[TABLE]

where $\alpha=\alpha_{1}+\alpha_{2}\ell\in\mathfrak{g}_{-1}$ $(\alpha_{1},\alpha_{2}\in(\mathbb{F}^{c})^{n}$, $z\in\mathfrak{g}_{-2}$). We denote by $\mathfrak{n}^{1}$ and $\mathfrak{n}^{2}$ the sets of elements which are fixed under $\tau$ and $\kappa\circ\tau$ respectively. Then $\mathfrak{n}^{1}$ and $\mathfrak{n}^{2}$ become graded subalgebras of $\mathfrak{m}_{\mathbb{R}}$ with

[TABLE]

Explicitly the subspaces $\mathfrak{n}^{i}_{p}$ are described as follows:

[TABLE]

where $\hat{\tau}$ is a mapping of $\mathbb{F}^{c}$ to itself defined by $\hat{\tau}(x)=-\tau(\overline{x})$. We note that the bracket operation and the scalar product on $\mathfrak{n}^{2}$ can be written as follows: if we put $\alpha=\alpha_{1}+\hat{\tau}(\alpha_{1})\ell$ and $\beta=\beta_{1}+\hat{\tau}(\beta_{1})\ell$ $(\alpha_{1},\beta_{1}\in(\mathbb{F}^{c})^{n}$), then

[TABLE]

We always assume that $\gamma=-1$ when we consider $\mathfrak{n}^{2}$. Since $z\overline{z}\in\mathbb{R}$ for $z\in\mathfrak{n}^{i}_{-2}$ $(i=1,2)$, $\mathfrak{n}^{1}$ and $\mathfrak{n}^{2}$ are pseudo $H$-type Lie algebras. The pseudo $H$-type Lie algebra $(\mathfrak{n}^{1},\langle\cdot\,|\,\cdot\rangle)$ is called a pseudo $\operatorname{div}H$-type Lie algebra of the second class, which is denoted by $\mathfrak{H}^{(2)}(\mathbb{F},S,\gamma)=(\mathfrak{h}^{(2)}(\mathbb{F},S,\gamma),\langle\cdot\,|\,\cdot\rangle)$. Also in case $\mathbb{F}=\mathbb{H}$, $\mathbb{H}^{\prime}$, $\mathbb{O}$ or $\mathbb{O}^{\prime}$, the pseudo $H$-type Lie algebra $(\mathfrak{n}^{2},\langle\cdot\,|\,\cdot\rangle)$ is called a pseudo $\operatorname{div}H$-type Lie algebra of the third class, which is denoted by $\mathfrak{H}^{(3)}(\mathbb{F},S)=(\mathfrak{h}^{(3)}(\mathbb{F},S),\langle\cdot\,|\,\cdot\rangle)$. We denote the FGLA associated with $\mathfrak{H}^{(2)}(\mathbb{F},S,\gamma)$ (resp. $\mathfrak{H}^{(3)}(\mathbb{F},S)$) by $\mathfrak{h}^{(2)}(\mathbb{F},S,\gamma)=\bigoplus\limits_{p=-1}^{-2}\mathfrak{h}^{(2)}(\mathbb{F},S,\gamma)_{p}$ (resp. $\mathfrak{h}^{(3)}(\mathbb{F},S)=\bigoplus\limits_{p=-1}^{-2}\mathfrak{h}^{(3)}(\mathbb{F},S)_{p})$. Note that $\mathfrak{h}^{(2)}(\mathbb{C},S,\gamma)$ becomes a complex FGLA.

Lemma 3.2**.**

Let $(r,s)$ be the signature of $S$.

1. (1) 

$\mathfrak{H}^{(2)}(\mathbb{F},S,\gamma)$* (resp. $\mathfrak{H}^{(3)}(\mathbb{F},S)$) is isomorphic to $\mathfrak{H}^{(2)}(\mathbb{F},1_{r,s},\gamma)$ (resp. $\mathfrak{H}^{(3)}(\mathbb{F},1_{r,s})$).* 2. (2) 

$\mathfrak{h}^{(2)}(\mathbb{F},S,\gamma^{\prime})$* is isomorphic to $\mathfrak{h}^{(2)}(\mathbb{F},1_{r+s},\gamma)$ as a GLA.* 3. (3) 

$\mathfrak{H}^{(2)}(\mathbb{F},1_{r,s})$* is equivalent to $\mathfrak{H}^{(2)}(\mathbb{F},1_{s,r})$.* 4. (4) 

When $\mathbb{F}=\mathbb{H}$ or $\mathbb{H}^{\prime}$, $\mathfrak{H}^{(3)}(\mathbb{F},1_{r,s})$ is isomorphic to $\mathfrak{H}^{(3)}(\mathbb{F},1_{r+s})$. Consequently, for a fixed $\mathbb{F}$ the $\mathfrak{H}^{(3)}(\mathbb{F},S)$ are mutually isomorphic.

Proof.

As in Lemma 3.1 we can prove (1) and (3).

(2) There exists a real orthogonal matrix $P$ such that $PSP^{-1}=1_{r,s}$. We define a linear mapping of $\mathfrak{h}^{(2)}(\mathbb{F},1_{r+s},\gamma^{\prime})$ to $\mathfrak{h}^{(2)}(\mathbb{F},S,\gamma)$ as follows:

[TABLE]

Then $\varphi$ is an isomorphism as a GLA.

(4) First we assume that $\mathbb{F}=\mathbb{H}^{\prime}$. We define a linear mapping of $\mathfrak{h}^{(3)}(\mathbb{F},1_{r+s})$ to $\mathfrak{h}^{(3)}(\mathbb{F},1_{r,s})$ as follows:

[TABLE]

Here $Q=\begin{bmatrix}1_{r}&O\\ O&\ell_{0}1_{s}\\ \end{bmatrix}$ and $\eta$ is the mapping of $(\mathbb{F}^{c})^{n}$ to itself defined by $\eta(\alpha_{r},\alpha_{s})=(\alpha_{r},\overline{\alpha_{s}})$ $(\alpha_{r}\in(\mathbb{F}^{c})^{r},\alpha_{s}\in(\mathbb{F}^{c})^{s})$. Then $\varphi$ is an isomorphism of $\mathfrak{H}^{(3)}(\mathbb{F},1_{r+s})$ onto $\mathfrak{H}^{(3)}(\mathbb{F},1_{r,s})$.

Next we assume that $\mathbb{F}=\mathbb{H}$. We define a linear mapping of $\mathfrak{h}^{(3)}(\mathbb{F},1_{r+s})$ to $\mathfrak{h}^{(3)}(\mathbb{F},1_{r,s})$ as follows:

[TABLE]

where $R=\begin{bmatrix}1_{r}&O\\ O&\sqrt{-1}\ell_{0}1_{s}\\ \end{bmatrix}$. Then $\varphi$ is an isomorphism of $\mathfrak{H}^{(3)}(\mathbb{F},1_{r,s})$ onto $\mathfrak{H}^{(3)}(\mathbb{F},1_{r+s})$. ∎

Remark 3.2**.**

The $H$-type Lie algebra $\mathfrak{H}^{(2)}(\mathbb{F},1_{r+s},-1)$ coincides with $\mathfrak{h}_{r+s}(\mathbb{F})$ in [12].

3.4. Pseudo $\operatorname{div}H$-type Lie algebras with $\dim\mathfrak{n}_{-2}=1$

(cf. [1, Proposition 4.5]). Now let $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$ be a pseudo $\operatorname{div}H$-type Lie algebra with $\dim\mathfrak{n}_{-2}=1$, that is, $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$ is $\mathfrak{H}^{(1)}(\mathbb{C},S)$ or $\mathfrak{H}^{(1)}(\mathbb{C}^{\prime},S)$. Note that $\mathfrak{h}^{(1)}(\mathbb{C},S)$ is isomorphic to $\mathfrak{h}^{(1)}(\mathbb{C}^{\prime},S)$ as a GLA. Since $\dim\mathfrak{n}_{-2}=1$ and the FGLA $\mathfrak{n}$ is non-degenerate, the prolongation of $\mathfrak{n}$ is isomorphic to a real contact algebra $K(N/2,\mathbb{R})$, where $N=\dim\mathfrak{n}_{-1}$. (For the details of contact algebras, see [9]). By definition an SGLA $\mathfrak{l}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{l}_{p}$ is is said to be of contact type if the negative part is an FGLA of the second kind and $\dim\mathfrak{l}_{-2}=1$. The negative part of a finite dimensional SGLA $\mathfrak{l}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{l}_{p}$ of contact type is uniquely determined by $\dim\mathfrak{l}_{-1}$ up to isomorphism. A finite dimensional real SGLA $\mathfrak{l}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{l}_{p}$ of contact type has the negative part isomorphic to $\mathfrak{h}^{(1)}(\mathbb{C},S)$ and is one of the following types:

[TABLE]

For the description of finite dimensional SGLAs, we use the notations in [17, §3].

3.5. Pseudo $\operatorname{div}H$-type Lie algebras with $\dim\mathfrak{n}_{-2}=2$

(cf. [1, Proposition 4.4]). Now let $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$ be a pseudo $\operatorname{div}H$-type Lie algebra with $\dim\mathfrak{n}_{-2}=2$, that is, $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$ is $\mathfrak{H}^{(2)}(\mathbb{F},S,\gamma)$ $(\mathbb{F}=\mathbb{C}$ or $\mathbb{C}^{\prime})$. We define an endomorphism $I$ of $\mathfrak{n}$ as follows:

[TABLE]

then $I$ satisfies $I^{2}=\gamma_{0}1_{\mathfrak{n}}$, $[Ix,y]=I[x,y]$, and $\langle Ix\,|\,y\rangle+\langle x\,|\,Iy\rangle=0$.

(i) Firstly we assume $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)=\mathfrak{H}^{(2)}(\mathbb{C},S,\gamma)$; then $(\mathfrak{n},I)$ becomes a complex Lie algebra. The prolongation of the complex FGLA $\mathfrak{n}$ is isomorphic to a complex contact algebra $K(N/4;\mathbb{C})$, where $N=\dim\mathfrak{n}_{-1}$. Hence the prolongation of the real FGLA $\mathfrak{n}$ is isomorphic to $K(N/4;\mathbb{C})_{\mathbb{R}}$ of a complex contact algebra $K(N/4;\mathbb{C})$. The signature of $\langle\cdot\,|\,\cdot\rangle_{-2}$ is $(2,0)$ (resp. $(0,2)$). The negative part of a finite dimensional complex SGLA $\mathfrak{l}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{l}_{p}$ of contact type has the negative part isomorphic to $\mathfrak{h}^{(2)}(\mathbb{C},S,\gamma)$ and is one of the following types:

[TABLE]

(ii) Next we assume $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)=\mathfrak{H}^{(2)}(\mathbb{C}^{\prime},S^{\prime},\gamma)$. We set $\mathfrak{n}^{\pm}=\{\alpha\in\mathfrak{n}:I(\alpha)=\pm\alpha\}$ and $(\mathfrak{n}^{\pm})_{p}=\mathfrak{n}_{p}\cap\mathfrak{n}^{\pm}$; then $\mathfrak{n}^{+}$ and $\mathfrak{n}^{-}$ are ideals of $\mathfrak{n}$ such that $\mathfrak{n}=\mathfrak{n}^{+}\oplus\mathfrak{n}^{-}$, $[\mathfrak{n}^{+},\mathfrak{n}^{-}]=0$, $\langle\mathfrak{n}^{+}\,|\,\mathfrak{n}^{+}\rangle=\langle\mathfrak{n}^{-}\,|\,\mathfrak{n}^{-}\rangle=0$. Let $\check{\mathfrak{g}}^{+}=\bigoplus\limits_{p\in\mathbb{Z}}\check{\mathfrak{g}}_{p}^{+}$ and $\check{\mathfrak{g}}^{-}=\bigoplus\limits_{p\in\mathbb{Z}}\check{\mathfrak{g}}_{p}^{-}$ be the prolongation of $\mathfrak{n}^{+}$ and $\mathfrak{n}^{-}$ respectively. $\check{\mathfrak{g}}^{+}=\bigoplus\limits_{p\in\mathbb{Z}}\check{\mathfrak{g}}_{p}^{+}$ and $\check{\mathfrak{g}}^{-}=\bigoplus\limits_{p\in\mathbb{Z}}\check{\mathfrak{g}}_{p}^{-}$ are both isomorphic to a real contact algebra $K(N/4;\mathbb{R})$. Hence the prolongation $\check{\mathfrak{g}}=\bigoplus\limits_{p\in\mathbb{Z}}\check{\mathfrak{g}}_{p}$ of the FGLA $\mathfrak{n}$ is the direct sum of $\check{\mathfrak{g}}^{+}=\bigoplus\limits_{p\in\mathbb{Z}}\check{\mathfrak{g}}_{p}^{+}$ and $\check{\mathfrak{g}}^{-}=\bigoplus\limits_{p\in\mathbb{Z}}\check{\mathfrak{g}}_{p}^{-}$ and hence is isomorphic to $K(N/4;\mathbb{R})\oplus K(N/4;\mathbb{R})$. Let $\mathfrak{g}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{g}_{p}$ be the prolongation of $(\mathfrak{n},[\langle\cdot\,|\,\cdot\rangle_{-1}])$; then $\mathfrak{g}_{0}=\mathbb{R}E_{+}\oplus\mathbb{R}E_{-}\oplus\mathfrak{a}$, where $\mathfrak{a}=\{~{}D-D^{\top}:D\in\check{\mathfrak{g}}^{+}_{0},[D,\mathfrak{n}_{-2}]=0~{}\}$, where $E_{+}$ (resp. $E_{-}$) is the characteristic element of $\check{\mathfrak{g}}^{+}=\bigoplus\limits_{p\in\mathbb{Z}}\check{\mathfrak{g}}_{p}^{+}$ (resp. $\check{\mathfrak{g}}^{-}=\bigoplus\limits_{p\in\mathbb{Z}}\check{\mathfrak{g}}_{p}^{-}$) and $D^{\top}$ is the adjoint of $D$ with respect to $\langle\cdot\,|\,\cdot\rangle$. The ideal $\mathfrak{a}$ of $\check{\mathfrak{g}}_{0}$ is isomorphic to $\mathfrak{sp}(\mathfrak{n}^{+}_{-1})$. Therefore the $\mathfrak{g}_{0}$-module $\mathfrak{g}_{-1}$ is completely reducible. From these results, we can easily prove that $\mathfrak{g}_{2}=0$.

3.6. Matricial models of pseudo $\operatorname{div}H$-type Lie algebras of the first class

Let $\mathbb{F}$ be $\mathbb{C}$, $\mathbb{H}$, $\mathbb{C}^{\prime}$ or $\mathbb{H}^{\prime}$. We put $\mathfrak{l}=\mathfrak{sl}(n+2,\mathbb{F})$ $(n\geqq 1)$; then $\mathfrak{l}$ is a real semisimple Lie algebra. We define an $n\times n$ symmetric real matrix $S_{p,q}$ as follows:

[TABLE]

Here the center column and the center row of $S_{p,q}$ should be deleted when $q=0$. Then $S_{p,q}$ is a symmetric real matrix with signature $(p+q,p)$. We put $\mathfrak{g}=\{\,X\in\mathfrak{l}:X^{*}S_{p,q}+S_{p,q}X=O\,\}$; then

[TABLE]

where we set $S_{0,m}=1_{m}$. Here $M(p,q,\mathbb{F})$ denotes the set of $\mathbb{F}$-valued $p\times q$-matrices. We define subspaces $\mathfrak{g}_{p}$ of $\mathfrak{g}$ as follows:

[TABLE]

Then $\mathfrak{g}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{g}_{p}$ becomes a GLA whose negative part $\mathfrak{m}$ is an FGLA of the second kind. We define a linear mapping of $\mathfrak{h}^{(1)}(\mathbb{F},S_{p-1,q})$ into $\mathfrak{g}_{-}$ as follows:

[TABLE]

then $\varphi$ becomes a GLA isomorphism. We define a symmetric bilinear form $\langle\cdot\,|\,\cdot\rangle$ on $\mathfrak{g}_{-}$ as follows:

[TABLE]

Then $(\mathfrak{g}_{-},\langle\cdot\,|\,\cdot\rangle)$ becomes a pseudo $H$-type Lie algebra and $\varphi$ is isomorphism of $\mathfrak{H}^{(1)}(\mathbb{F},S_{p-1,q})$ onto $(\mathfrak{g}_{-},\langle\cdot\,|\,\cdot\rangle)$. Since $\operatorname{ad}(\mathfrak{g}_{0})|\mathfrak{g}_{-1}\subset\mathfrak{co}(\mathfrak{g}_{-1},g)$, $\mathfrak{g}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{g}_{p}$ is the prolongation of $(\mathfrak{g}_{-},[\langle\cdot\,|\,\cdot\rangle_{-1}])$. From these results, [1, Theorem 3.6], [7, §3] and [18], a finite dimensional real SGLA $\mathfrak{s}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{s}_{p}$ that is isomorphic to the prolongation of the cps-FGLA $(\mathfrak{n},[\langle\cdot\,|\,\cdot\rangle_{-1}])$ associated with a pseudo $\operatorname{div}H$-type Lie algebras $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$ of the first class is one of the following:

[TABLE]

In particular, if $\dim\mathfrak{s}_{-2}\geqq 3$, then $\mathfrak{s}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{s}_{p}$ is the prolongation of $\mathfrak{s}_{-}$.

3.7. Matricial Models of pseudo $\operatorname{div}H$-type Lie algebras of the second class

Let $\mathbb{F}=\mathbb{C},\mathbb{C}^{\prime},\mathbb{H},\mathbb{H}^{\prime}$. Let $\mathfrak{g}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{g}_{p}$ be a finite dimensional semisimple GLA $\mathfrak{sl}(n+2,\mathbb{F})$ with the the following gradation $(\mathfrak{g}_{p})$.

[TABLE]

Note that $\mathfrak{g}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{g}_{p}$ is an SGLA except for the case $\mathbb{F}=\mathbb{C}^{\prime}$. We consider an FGLA $\mathfrak{H}^{(2)}(\mathbb{F},S,\gamma)$. That is,

[TABLE]

where $S$ is a real symmetric matrix of order $n$ such that $S^{2}=1_{n}$. We define a linear mapping $\varphi$ of $\mathfrak{h}^{(2)}(\mathbb{F},S,\gamma)$ to $\mathfrak{g}_{-}$ as follows:

[TABLE]

Then $\varphi$ is a GLA isomorphism. Moreover we define a non-degenerate symmetric bilinear form on $\mathfrak{g}_{-}$ as follows:

[TABLE]

The negative part of $\mathfrak{g}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{g}_{p}$ equipped with this scalar product becomes a pseudo $H$-type Lie algebra which is isomorphic to $\mathfrak{H}^{(2)}(\mathbb{F},S,\gamma)$ as a pseudo $H$-type Lie algebra.

1. Case 1:

$\mathbb{F}=\mathbb{C}$. $\mathfrak{g}$ is equal to $\mathfrak{sl}(n+2,\mathbb{C})_{\mathbb{R}}$. Hence the GLA $\mathfrak{g}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{g}_{p}$ is a finite dimensional SGLA of type $(A_{l},\{\alpha_{1},\alpha_{l}\})$ $(l=n+1)$. If $\gamma=-1$ (resp. $\gamma=1$), then the signature of $\langle\cdot\,|\,\cdot\rangle_{-2}$ is $(2,0)$ (resp. $(0,2)$). 2. Case 2:

$\mathbb{F}=\mathbb{C}^{\prime}$. Since $\mathbb{C}^{\prime}$ is isomorphic to $\mathbb{R}\oplus\mathbb{R}$ as a $\mathbb{R}$-algebra, $\mathfrak{g}$ is isomorphic to $\mathfrak{sl}(n+2,\mathbb{R})\times\mathfrak{sl}(n+2,\mathbb{R})$. Hence the GLA $\mathfrak{g}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{g}_{p}$ is a semisimple GLA of type $(({\rm AI})_{l},\{\alpha_{1},\alpha_{l}\})\times(({\rm AI})_{l},\{\alpha_{1},\alpha_{l}\})$, where $l=n+1$. The signature of $\langle\cdot\,|\,\cdot\rangle_{-2}$ is $(1,1)$. 3. Case 3:

$\mathbb{F}=\mathbb{H}$. The GLA $\mathfrak{g}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{g}_{p}$ is a finite dimensional SGLA of type $(({\rm AII})_{l},\{\alpha_{2},\alpha_{l-1}\})$, where $l=2n+1$. If $\gamma=-1$ (resp. $\gamma=1$), then the signature of $\langle\cdot\,|\,\cdot\rangle_{-2}$ is $(4,0)$ (resp. $(0,4)$). 4. Case 4:

$\mathbb{F}=\mathbb{H}^{\prime}$. Since $\mathbb{H}^{\prime}$ is isomorphic to $M_{2}(\mathbb{R})$ as a $\mathbb{R}$-algebra, $\mathfrak{g}$ is isomorphic to $\mathfrak{sl}(2n+2,\mathbb{R})$. Hence the GLA $\mathfrak{g}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{g}_{p}$ is a finite dimensional SGLA of type $(({\rm AI})_{l},\{\alpha_{2},\alpha_{l-1}\})$, where $l=2n-1$. The signature of $\langle\cdot\,|\,\cdot\rangle_{-2}$ is $(2,2)$.

From these results, [1, Theorem 3.6] and [7, §3], a finite dimensional real SGLA $\mathfrak{s}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{s}_{p}$ with $\dim\mathfrak{s}_{-2}\geqq 3$ whose negative part is isomorphic to a pseudo $\operatorname{div}H$-type Lie algebra of the second class is the prolongation of $\mathfrak{s}_{-}$ and is one of the following:

[TABLE]

3.8. Matricial models of pseudo $\operatorname{div}H$-type Lie algebras of the third class

Let $\mathfrak{g}$ be the simple Lie algebra $\mathfrak{su}(p+q,p)$. We define subspaces $\mathfrak{g}_{p}$ of $\mathfrak{g}$ as follows:

[TABLE]

For convenience, we denote by $X=(x_{31},x_{32})$ and $Z=(z_{41},z_{42},z_{51})$ elements

[TABLE]

of $\mathfrak{g}_{-1}$ and $\mathfrak{g}_{-2}$ respectively. Then $\mathfrak{g}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{g}_{p}$ becomes a GLA whose negative part $\mathfrak{m}$ is an FGLA of the second kind. For $X=(x_{31},x_{32}),Y=(y_{31},y_{32})\in\mathfrak{g}_{-1}$

[TABLE]

where $S^{\prime}=S_{p-2,q}$. For $X=(x_{31},x_{32})\in\mathfrak{g}_{-1}$ we denote by $X_{31}$ the $(2p+q-4)\times 2$ submatrix $\begin{bmatrix}x_{31}&x_{32}\\ \end{bmatrix}$ of $X$. Also we use the notation $x_{3i}=\begin{bmatrix}x_{3i}^{(1)}\\ x_{3i}^{(2)}\\ x_{3i}^{(3)}\end{bmatrix}$, where $x_{3i}^{(1)}$ and $x_{3i}^{(3)}$ are $(p-2)\times 1$ matrices and $x_{3i}^{(2)}$ is a $q\times 1$ matrix. We define a non-degenerate symmetric bilinear form $\langle\cdot\,|\,\cdot\rangle$ on $\mathfrak{m}$ as follows:

[TABLE]

where $m=q/2$, $Q_{m}=\begin{bmatrix}O&K_{m}\\ \\ -K_{m}&O\\ \end{bmatrix}$ and $\zeta_{0}=\pm 1$. For $Z\in\mathfrak{g}_{-2}$ let $J_{Z}$ be the mapping of $\mathfrak{g}_{-1}$ to itself defined by

[TABLE]

Then

[TABLE]

where $P_{p,q}=\begin{bmatrix}E_{p-2}&O&O\\ O&Q_{m}&O\\ O&O&-E_{p-2}\\ \end{bmatrix}$. Furthermore we obtain that

[TABLE]

3.8.1. Case of signature $(1,3)$

We assume that $p\geqq 3$, $q=0$ and $\zeta_{0}=1$. Then $(\mathfrak{g}_{-},\langle\cdot\,|\,\cdot\rangle)$ becomes a pseudo $H$-type Lie algebra. This result is a little generalization of [5, Theorem 8]. Note that the signature of the restriction of $\langle\cdot\,|\,\cdot\rangle$ to $\mathfrak{g}_{-2}$ is $(1,3)$ and $\mathfrak{g}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{g}_{p}$ is a finite dimensional SGLA of type $({\rm(AIIIb)}_{l},\{\alpha_{2},\alpha_{l-1}\})$, where $l=2p-1$. We define a linear mapping $\varPsi$ of $\mathfrak{g}_{-}$ to $\mathfrak{H}^{(3)}(\mathbb{H}^{\prime},K_{p-2})$ as follows:

[TABLE]

where $X=(x_{31},x_{32})\in\mathfrak{g}_{-1}$ and $Z=(\alpha,\beta,\gamma)\in\mathfrak{g}_{-2}$. Here for a complex number $z=a+bi$ ($a,b\in\mathbb{R}$) we denote the real part $a$ (resp. the imaginary part $b$) of $z$ by $\Re(z)$ (resp. $\Im(z)$). $\varPsi$ is isomorphic to $\mathfrak{g}_{-}$ onto $\mathfrak{n}$ as a pseudo $H$ type Lie algebra.

3.8.2. Case of signature $(3,1)$

We assume that $p=2$, $q=2m$, $m\geqq 1$ and $\zeta_{0}=-1$. Note that the signature of the restriction of $\langle\cdot\,|\,\cdot\rangle$ to $\mathfrak{g}_{-2}$ is $(3,1)$ and $\mathfrak{g}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{g}_{p}$ is a finite dimensional SGLA of type $({\rm(AIIIa)}_{l,2},\{\alpha_{2},\alpha_{l-1}\})$, where $l=2m+3$. We define a linear mapping $\varPsi$ of $\mathfrak{g}_{-}$ to $\mathfrak{H}^{(3)}(\mathbb{H},K_{q/2})$ as follows:

[TABLE]

where $X=(x_{31},x_{32})\in\mathfrak{g}_{-1}$ and $Z=(\alpha,\beta,\gamma)\in\mathfrak{g}_{-2}$. Here we use the notation $x_{3i}=x_{3i}^{(2)}=\begin{bmatrix}x_{3i}^{1}\\ x_{3i}^{2}\end{bmatrix}$, where $x_{3i}^{1}$ and $x_{3i}^{2}$ are $m\times 1$ matrices. $\varPsi$ is isomorphic to $\mathfrak{g}_{-}$ onto $\mathfrak{H}^{(3)}(\mathbb{H},K_{q/2})$ as a pseudo $H$-type Lie algebra.

From these results, [1, Theorem 3.6] and [7, §3], a finite dimensional real SGLA $\mathfrak{s}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{s}_{p}$ whose negative part is isomorphic to a pseudo $\operatorname{div}H$-type Lie algebra of the third class is the prolongation of $\mathfrak{s}_{-}$ and is one of the following :

[TABLE]

4. Pseudo $H$-type Lie algebras satisfying the $J^{2}$-condition

In this section we first see that a pseudo $H$-type Lie algebra is isomorphic to a pseudo $H$-type Lie algebra of the first class sketchily. For the details of the proof, we refer to [14]. Let $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$ be a pseudo $H$-type Lie algebra. For any $x\in\mathfrak{n}_{-1}$ with $\langle x\,|\,x\rangle\neq 0$ we set

[TABLE]

then $\mathfrak{n}_{-1}(x)$ is a non-degenerate subspace of $\mathfrak{n}_{-1}$ with respect to $\langle\cdot\,|\,\cdot\rangle$. We say that $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$ satisfies the $J^{2}$ condition if for any $z\in\mathfrak{n}_{-2}$ and any $x\in\mathfrak{n}_{-1}$ with $\langle x\,|\,x\rangle\neq 0$, $\mathfrak{n}_{-1}(x)$ is $J_{z}$-stable. Clearly if $\dim\mathfrak{n}_{-2}=1$, then $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$ satisfies the $J^{2}$-condition. If a pseudo $H$-type Lie algebra $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$ is equivalent to a pseudo $H$-type Lie algebra $(\mathfrak{n}^{\prime},\langle\cdot\,|\,\cdot\rangle^{\prime})$ satisfying the $J^{2}$ condition, then $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$ also satisfies one.

Let $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$ be a pseudo $H$-type Lie algebra satisfying the $J^{2}$-condition. For $x\in\mathfrak{n}_{-1}$ with $\langle x\,|\,x\rangle\neq 0$ we set $\mathcal{A}_{x}=\mathbb{R}\times\mathfrak{n}_{-2}$; then $\mathcal{A}_{x}$ is a real vector space. We define a multiplicative operation $\underset{x}{*}$ on $\mathcal{A}_{x}$ as follows: for $(\lambda_{1},z_{1}),(\lambda_{2},z_{2})\in\mathcal{A}_{x}$, we put

[TABLE]

where $(\lambda_{3},z_{3})$ is defined by

[TABLE]

Then $(\mathcal{A}_{x},+,\underset{x}{*})$ is an algebra over $\mathbb{R}$. We define an endomorphism $s$ of $\mathcal{A}_{x}$ as follows:

[TABLE]

then $s$ is an anti-involution of $\mathcal{A}_{x}$ and satisfies

[TABLE]

We define $N:\mathcal{A}_{x}\to\mathbb{R}$ as follows:

[TABLE]

then $N$ is a non-degenerate quadratic form on $\mathcal{A}_{x}$ and hence $(\mathcal{A}_{x},s)$ becomes a Cayley algebra.

Furthermore we can prove that $\mathcal{A}_{x}$ becomes an alternative algebra and hence a normed algebra. By Hurwitz theorem ([8, Theorem 6.37]), $\mathcal{A}_{x}$ is isomorphic to one of $\mathbb{R},\mathbb{C},\mathbb{C}^{\prime},\mathbb{H},\mathbb{H}^{\prime},\mathbb{O},\mathbb{O}^{\prime}$ as a Cayley algebra. However since $\mathfrak{n}_{-2}\neq 0$, $\mathcal{A}_{x}$ is not isomorphic to $\mathbb{R}$. Also the Cayley algebra $\mathcal{A}_{x}$ does not depend on the choice of the element $x$.

We choose elements $x_{1},\dots,x_{r+s}$ of $\mathfrak{n}_{-1}$ satisfying the following conditions:

[TABLE]

In particular, if $\mathcal{A}_{x_{i}}$ is isomorphic to $\mathbb{O}$ or $\mathbb{O}^{\prime}$ for some $i$, then $r+s=1$. We denote by $\mathbb{F}$ the Cayley algebra $\mathcal{A}_{x_{1}}$. We define a linear mapping $\varphi$ of $\mathfrak{n}$ to $\mathfrak{h}^{(1)}(\mathbb{F},1_{r,s})=\mathbb{F}^{r+s}\oplus\operatorname{Im}\mathbb{F}$ as follows:

[TABLE]

Then $\varphi$ is an isomorphism as a pseudo $H$-type Lie algebra.

Theorem 4.1**.**

Let $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$ be a pseudo $H$-type Lie algebra. The following three conditions are mutually equivalent:

1. (i)

$(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$* satisfies the $J^{2}$-condition;* 2. (ii)

$(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$* is of the first class;* 3. (iii)

The cps-FGLA associated with $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$ is of semisimple type.

Proof.

The implication (i) $\Rightarrow$ (ii) is obtained from the above result. The implication (ii) $\Rightarrow$ (iii) follows from §3.6. Finally we prove the implication (iii) $\Rightarrow$ (i). Now we assume the condition (iii). From the classification of the prolongations of cps-FGLAs of semisimple type, the prolongation of $(\mathfrak{n},[\langle\cdot\,|\,\cdot\rangle_{-1}])$ is isomorphic to the prolongation of the cps-FGLA associated with some pseudo $H$-type Lie algebra of the first class. Thus (iii) $\Rightarrow$ (i) follows from Proposition 2.1. ∎

5. The prolongations of the FGLAs and the cps-FGLAs associated with pseudo $H$ type Lie algebras

Let $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$ be a pseudo $H$-type Lie algebra, and let $\check{\mathfrak{g}}=\bigoplus\limits_{p\in\mathbb{Z}}\check{\mathfrak{g}}_{p}$ be the prolongation of $\mathfrak{n}$. The natural inclusion $\iota$ of $\mathfrak{so}(\mathfrak{n}_{-2},\langle\cdot\,|\,\cdot\rangle_{-2})$ into $\check{\mathfrak{g}}_{0}$ is defined by

[TABLE]

where $v\wedge u$ is the skew-symmetric endomorphism $\langle v\,|\,\cdot\rangle u-\langle u\,|\,\cdot\rangle v$.

Here we quote useful results from [1] and [2].

Proposition 5.1** ([1, Theorem 2.3]).**

Let $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$ be a pseudo $H$-type Lie algebra, and let $\check{\mathfrak{g}}=\bigoplus\limits_{p\in\mathbb{Z}}\check{\mathfrak{g}}_{p}$ be the prolongation of $\mathfrak{n}$. Then

[TABLE]

where $E$ is the characteristic element of the GLA $\check{\mathfrak{g}}=\bigoplus\limits_{p\in\mathbb{Z}}\check{\mathfrak{g}}_{p}$ and $\check{\mathfrak{h}}_{0}=\{~{}x\in\check{\mathfrak{g}}_{0}:[x,\mathfrak{n}_{-2}]=0~{}\}$.

Let $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$ and $\check{\mathfrak{g}}=\bigoplus\limits_{p\in\mathbb{Z}}\check{\mathfrak{g}}_{p}$ be as in Proposition 5.1. Moreover let $\mathfrak{g}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{g}_{p}$ be the prolongation of $(\mathfrak{n},[\langle\cdot\,|\,\cdot\rangle_{-1}])$. We define subspaces $\mathfrak{h}_{0}$, $\check{\mathfrak{h}}_{0}^{a}$ and $\check{\mathfrak{h}}_{0}^{s}$ of $\check{\mathfrak{g}}_{0}$ as follows:

[TABLE]

Corollary 5.1**.**

Under the above assumptions,

[TABLE]

Proof.

Since $D^{\top}\in\check{\mathfrak{h}}_{0}$ for $D\in\check{\mathfrak{h}}_{0}$, we get $\check{\mathfrak{h}}_{0}=\check{\mathfrak{h}}_{0}^{a}\oplus\check{\mathfrak{h}}_{0}^{s}$, so $\mathfrak{h}_{0}=\check{\mathfrak{h}}_{0}^{a}$. From Proposition 5.1 the last assertion is obvious. ∎

Theorem 5.1** ([2, Theorem 3.1 and Remark 3.2]).**

Let $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$ be a pseudo $H$-type Lie algebra with $\dim\mathfrak{n}_{-2}\geqq 3$, and let $\check{\mathfrak{g}}=\bigoplus\limits_{p\in\mathbb{Z}}\check{\mathfrak{g}}_{p}$ be the prolongation of $\mathfrak{n}$. If $\check{\mathfrak{g}}_{1}\neq 0$, then $\check{\mathfrak{g}}=\bigoplus\limits_{p\in\mathbb{Z}}\check{\mathfrak{g}}_{p}$ is a finite dimensional SGLA.

Let $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$ be a pseudo $H$-type Lie algebra with $\dim\mathfrak{n}_{-2}\geqq 3$. Since a pseudo $H$-type Lie algebra is a real extended translation algebra, if the prolongation of $\mathfrak{n}$ is simple, then $\dim\mathfrak{n}_{-2}=3,4,7$ or $8$ ([1, Theorem 3.6]). Hence by Theorem 5.1 we obtain the following

Corollary 5.2**.**

Let $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$ and $\check{\mathfrak{g}}=\bigoplus\limits_{p\in\mathbb{Z}}\check{\mathfrak{g}}_{p}$ be as in Theorem 5.1. If $\dim\mathfrak{n}_{-2}\neq 3,4,7,8$, then $\check{\mathfrak{g}}_{p}=0$ for all $p\geqq 1$.

Lemma 5.1**.**

Let $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$ be a pseudo $H$-type Lie algebra, and let $\mathfrak{g}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{g}_{p}$ be the prolongation of $(\mathfrak{n},[\langle\cdot\,|\,\cdot\rangle_{-1}])$. For $p\geqq 1$, the condition “$x\in\mathfrak{g}_{p}$ and $[x,\mathfrak{g}_{-2}]=0$” implies $x=0$.

Proof.

We identify $\mathfrak{h}_{0}$ with a subspace of $\mathfrak{gl}(\mathfrak{n}_{-1})$. For a subspace $\mathfrak{a}$ of $\mathfrak{gl}(\mathfrak{n}_{-1})$ we denote by $\rho^{(k)}(\mathfrak{a})$ the $k$-th (algebraic) prolongation of $\mathfrak{a}$. By Corollary 5.1, $\mathfrak{h}_{0}\subset\mathfrak{so}(\mathfrak{n}_{-1},\langle\cdot\,|\,\cdot\rangle_{-1})$; hence $\rho^{(1)}(\mathfrak{h}_{0})\subset\rho^{(1)}(\mathfrak{so}(\mathfrak{n}_{-1},\langle\cdot\,|\,\cdot\rangle_{-1}))=0$. The lemma is proved. ∎

Theorem 5.2**.**

Let $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$ be a pseudo $H$-type Lie algebra, and let $\mathfrak{g}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{g}_{p}$ be the prolongation of $(\mathfrak{n},[\langle\cdot\,|\,\cdot\rangle_{-1}])$. If $\mathfrak{g}_{2}\neq 0$ and if the $\mathfrak{g}_{0}$-module $\mathfrak{g}_{-2}$ is irreducible, then $\mathfrak{g}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{g}_{p}$ is a finite dimensional SGLA.

Proof.

Since the prolongation of a cps-FGLA of semisimple type is simple, it suffices to prove that $\mathfrak{g}$ is semisimple. Let $\mathfrak{r}$ be the radical of $\mathfrak{g}$. Then $\mathfrak{r}$ is a graded ideal of $\mathfrak{g}$. That is, putting $\mathfrak{r}_{p}=\mathfrak{r}\cap\mathfrak{g}_{p}$, we see that $\mathfrak{r}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{r}_{p}$. Let $\mathfrak{t}$ be the nilpotent radical $[\mathfrak{g},\mathfrak{r}]$ of $\mathfrak{g}$. Assume that $\mathfrak{t}\neq 0$. Since $\mathfrak{t}$ is a nilpotent ideal of $\mathfrak{g}$, there exists $k$ such that $\mathfrak{t}^{(k)}:=\mathcal{C}^{k}(\mathfrak{t})\neq 0$ and $\mathfrak{t}^{(k+1)}:=\mathcal{C}^{k+1}(\mathfrak{t})=0$, where $(\mathcal{C}^{i}(\mathfrak{t}))_{i\geqq 0}$ is the ascending central series of $\mathfrak{t}$. Clearly $\mathfrak{t}$ and $\mathfrak{t}^{(k)}$ are graded ideals of $\mathfrak{g}$; putting $\mathfrak{t}_{p}=\mathfrak{t}\cap\mathfrak{g}_{p}$ and $\mathfrak{t}^{(k)}_{p}=\mathfrak{t}^{(k)}\cap\mathfrak{g}_{p}$, we get $\mathfrak{t}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{t}_{p}$ and $\mathfrak{t}^{(k)}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{t}^{(k)}_{p}$. Since $\mathfrak{t}^{(k)}_{-2}$ is a $\mathfrak{g}_{0}$-submodule of $\mathfrak{g}_{-2}$, $\mathfrak{t}^{(k)}_{-2}=0$ or $\mathfrak{t}^{(k)}_{-2}=\mathfrak{g}_{-2}$. If $\mathfrak{t}^{(k)}_{-2}=0$, then $[\mathfrak{t}^{(k)}_{-1},\mathfrak{g}_{-1}]\subset\mathfrak{t}^{(k)}_{-2}=0$, so by non-degeneracy, $\mathfrak{t}^{(k)}_{-1}=0$. Moreover since $[\mathfrak{t}^{(k)}_{0},\mathfrak{g}_{-1}]\subset\mathfrak{t}^{(k)}_{-1}=0$, by transitivity, $\mathfrak{t}^{(k)}_{0}=0$. Similarly we see that $\mathfrak{t}^{(k)}_{p}=0$ for all $p\geqq 0$, which is a contradiction. Next if $\mathfrak{t}^{(k)}_{-2}=\mathfrak{g}_{-2}$, then $[\mathfrak{t}_{p},\mathfrak{g}_{-2}]=[\mathfrak{t}_{p},\mathfrak{t}^{(k)}_{-2}]\subset\mathfrak{t}^{(k+1)}=0$. By Lemma 5.1 $\mathfrak{t}_{p}=0$ for all $p\geqq 2$. Since $\mathfrak{t}=[\mathfrak{g},\mathfrak{r}]\supset[E,\mathfrak{r}]\supset\bigoplus\limits_{p\neq 0}\mathfrak{r}_{p}$, we obtain $\mathfrak{r}_{p}=0$ for all $p\geqq 2$. Hence $\mathfrak{g}/\mathfrak{r}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{g}_{p}/\mathfrak{r}_{p}$ is a semisimple GLA such that $\mathfrak{g}_{-2}/\mathfrak{r}_{-2}=0$ and $\mathfrak{g}_{2}/\mathfrak{r}_{2}=\mathfrak{g}_{2}\neq 0$. By semisimplicity, we get that $\dim\mathfrak{g}_{-2}/\mathfrak{r}_{-2}=\dim\mathfrak{g}_{2}/\mathfrak{r}_{2}$, which is a contradiction. Thus we obtain that $\mathfrak{t}=0$. As above $\mathfrak{r}_{p}=0$ for $p\neq 0$ and hence $\mathfrak{r}=0$. Therefore $\mathfrak{g}$ is semisimple. ∎

Theorem 5.3**.**

Let $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$ be a pseudo $H$-type Lie algebra, and let $\mathfrak{g}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{g}_{p}$ be the prolongation of the associated cps-FGLA $(\mathfrak{n},[\langle\cdot\,|\,\cdot\rangle_{-1}])$.

1. (1) 

If $\dim\mathfrak{n}_{-2}=1$, then $\mathfrak{g}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{g}_{p}$ is one of finite dimensional SGLAs of types $({\rm(AI)}_{l},\{\alpha_{1},\alpha_{l}\})$, $({\rm(AIIIa)}_{l,p},\{\alpha_{1},\alpha_{l}\})$, $({\rm(AIIIb)}_{l},\{\alpha_{1},\alpha_{l}\})$, $({\rm(AIV)}_{l},\{\alpha_{1},\alpha_{l}\})$. 2. (2) 

If $\dim\mathfrak{n}_{-2}=2$, then $\mathfrak{g}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{g}_{p}$ is not semisimple and $\mathfrak{g}_{2}=0$. 3. (3) 

Assume that $\dim\mathfrak{n}_{-2}\geqq 3$. If $\mathfrak{g}_{2}\neq 0$, then $\mathfrak{g}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{g}_{p}$ is a finite dimensional SGLA and coincides with the prolongation of $\mathfrak{n}$. Furthermore for $\mathfrak{g}_{2}$ to be nonzero, it is necessary and sufficient that $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle)$ is a pseudo $\operatorname{div}H$-type Lie algebra of the first class.

Proof.

(1) Since $\dim\mathfrak{n}_{-2}=1$, the pseudo $H$-type Lie algebra $\mathfrak{n}$ satisfies the $J^{2}$-condition. Hence (1) follows from Theorem 4.1 and the results of 3.6.

(2) If $\mathfrak{g}$ is semisimple, then $\dim\mathfrak{g}_{-2}\neq 2$ (Theorem 4.1). Hence $\mathfrak{g}$ is not semisimple. If the $\mathfrak{g}_{0}$-module $\mathfrak{g}_{-2}$ is irreducible (resp. reducible), then, by Theorem 5.2 (resp. by the results of §3.5), we obtain $\mathfrak{g}_{2}=0$.

(3) Assume that $\dim\mathfrak{n}_{-2}\geqq 3$ and $\mathfrak{g}_{2}\neq 0$. Then $\check{\mathfrak{g}}_{1}\neq 0$. By Theorem 5.1, $\check{\mathfrak{g}}$ is a finite dimensional SGLA. Let $B$ be the Killing form of $\check{\mathfrak{g}}$. Then $B([\check{\mathfrak{h}}_{0},\check{\mathfrak{g}}_{2}],\mathfrak{g}_{-2})=B(\check{\mathfrak{g}}_{2},[\check{\mathfrak{h}}_{0},\check{\mathfrak{g}}_{-2}])=0$. By non-degeneracy of the Killing form of $\check{\mathfrak{g}}$, we get $[\check{\mathfrak{h}}_{0},\check{\mathfrak{g}}_{2}]=0$. Since $\mathfrak{so}(\mathfrak{n}_{-2},\langle\cdot\,|\,\cdot\rangle_{-2})\subset\mathfrak{g}_{0}$, by Proposition 5.1 the subspace $\mathfrak{g}_{2}$ of $\check{\mathfrak{g}}_{2}$ is $\check{\mathfrak{g}}_{0}$-stable. Since the $\check{\mathfrak{g}}_{0}$-module $\mathfrak{g}_{-2}$ is irreducible, so is $\check{\mathfrak{g}}_{2}$. Since $\mathfrak{g}_{2}\neq 0$, we obtain $\mathfrak{g}_{2}=\check{\mathfrak{g}}_{2}$. By [16, Lemma 1.6], we see that $\mathfrak{g}_{1}\supset[{\mathfrak{g}}_{-1},{\mathfrak{g}}_{2}]=[\check{\mathfrak{g}}_{-1},\check{\mathfrak{g}}_{2}]=\check{\mathfrak{g}}_{1}$ and hence $\check{\mathfrak{g}}_{1}=\mathfrak{g}_{1}$. Also by [16, Lemma 1.3] we see that $\mathfrak{g}_{0}\supset[{\mathfrak{g}}_{-1},{\mathfrak{g}}_{1}]=[\check{\mathfrak{g}}_{-1},\check{\mathfrak{g}}_{1}]=\check{\mathfrak{g}}_{0}$ and hence $\check{\mathfrak{g}}_{0}=\mathfrak{g}_{0}$. By the definitions of the prolongations, we obtain that $\check{\mathfrak{g}}_{p}=\mathfrak{g}_{p}$ for all $p\geqq 0$. The last assertion follows from Theorem 4.1. ∎

Corollary 5.3**.**

Let $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle^{(1)})$ and $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle^{(2)})$ be two pseudo $H$-type Lie algebras whose associated FGLAs coincide. Let $\mathfrak{g}^{(1)}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{g}^{(1)}_{p}$ and $\mathfrak{g}^{(2)}=\bigoplus\limits_{p\in\mathbb{Z}}\mathfrak{g}^{(2)}_{p}$ be the prolongations of $(\mathfrak{n},[\langle\cdot\,|\,\cdot\rangle^{(1)}_{-1}])$ and $(\mathfrak{n},[\langle\cdot\,|\,\cdot\rangle^{(2)}_{-1}])$ respectively. If $\dim\mathfrak{n}_{-2}\geqq 3$, $\mathfrak{g}^{(1)}_{2}\neq 0$ and $\mathfrak{g}^{(2)}_{2}\neq 0$, then $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle^{(1)})$ is equivalent to $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle^{(2)})$.

Proof.

By Theorem 5.3 (3), we obtain that the prolongation $\check{\mathfrak{g}}=\bigoplus\limits_{p\in\mathbb{Z}}\check{\mathfrak{g}}_{p}$ of $\mathfrak{n}$ is an SGLA and that $\check{\mathfrak{g}}=\mathfrak{g}^{(1)}=\mathfrak{g}^{(2)}$. By Proposition 2.1 we see that $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle^{(1)})$ is equivalent to $(\mathfrak{n},\langle\cdot\,|\,\cdot\rangle^{(2)})$. ∎