Complete classification of pseudo H-type algebras: II

Abstract

We classify a class of 2-step nilpotent Lie algebras related to the representations of the Clifford algebras in the following way. Let $J:{{\mathrm{\mathrm {Cl}}}}({\mathbb {R}}^{r,s})\rightarrow {{\mathrm{End}}}(U)$ be a representation of the Clifford algebra ${{\mathrm{\mathrm {Cl}}}}({\mathbb {R}}^{r,s})$ generated by the pseudo Euclidean vector space ${\mathbb {R}}^{r,s}$. Assume that the Clifford module U is endowed with a bilinear symmetric non-degenerate real form $\langle \cdot \,,\cdot \rangle _U$ making the linear map $J_z$ skew symmetric for any $z\in {\mathbb {R}}^{r,s}$. The Lie algebras and the Clifford algebras are related by $\langle J_zv,w\rangle _U=\langle z,[v,w]\rangle _{{\mathbb {R}}^{r,s}}$, $z\in {\mathbb {R}}^{r,s}$, $v,w\in U$. We detect the isomorphic and non-isomorphic Lie algebras according to the dimension of U and the range of the non-negative integers r, s.

Appendix

We give the collections $PI_{r,s}$ and $CO_{r,s}$ for basic cases (2.8) grouped in four tables. The dimensions of $E_{r,s}$ and signature of the scalar product restricted to $E_{r,s}$ are listed. First we mention trivial cases.

$$\begin{aligned}&PI_{1,0}=PI_{0,1}=PI_{2,0}=PI_{1,1}=PI_{0,2}=PI_{2,1}=PI_{0,3}=\emptyset ,\\&PI_{3,0}=PI_{1,2}=\{P=J_{z_1}J_{z_2}J_{z_3}\},\quad CO_{3,0}=CO_{1,2}=\emptyset . \end{aligned}$$

For the cases (r, s) of $r-s\equiv 3 \,(\text {mod}~4)$ and s even, there is no complementary operator which commutes with all the involutions in $PI_{r,s}$ except the last involution which is of the form $\mathcal {P}_3$ or $\mathcal {P}_4$ and anti-commutes with the last involution. In these cases the operator $J_{\Omega ^{r,s}}$ is a product of involutions in $PI_{r,s}$ and it commutes with all the operators $J_{z_k}$. This is the reason for the number of complementary operators to be $p_{r,s}-1$. The last operator in $PI_{r,s}$ of the form $\mathcal {P}_3$ or $\mathcal {P}_4$ distinguishes the two different minimal admissible modules.

The signature of the admissible scalar product restricted on the space $E_{r,s}$ is sign definite in Table 6 and is neutral for signatures (r, s) in Table 7. The latter can be seen by finding an additional negative operator other than operators in $CO_{r,s}$ which commutes with all the involutions in $PI_{r,s}$ (Tables 4, 5, 6).

Table 4 Systems $PI_{r,0}$ and $CO_{r,0}$, $r=4,\ldots ,7$

Table 5 Systems $PI_{r,4}$ and $CO_{r,4}$, $r=0,1,2$

Table 6 Systems $PI_{3,s}$ and $CO_{3,s}$, $s=1,\ldots ,7$ and $PI_{7,s}$, $CO_{7,s}$, $s=1,2,3$

Table 7 Systems $PI_{r,s}$ and $CO_{r,s}$ for Proposition 2.9