Structure constants of pseudo $H$-type algebras in some integral bases
We present the structural constants of low dimensional pseudo H-type algebras. 1. Definition of pseudo H-type algebra Let n = z⊕ v be a nilpotent graded Lie algebra, 〈 . , . 〉r,s a non-degenerate symmetric bi-linear form on z of index (r, s) and 〈 . , . 〉 a non-degenerate symmetric bi-linear form on v that is of index (l, l) in the case s > 0 and is of index (n, 0) if s = 0. Define the map J : z → End(v) by 〈 z, [x, y] 〉r,s = 〈 Jzx, y 〉, z ∈ z, x, y ∈ v . Then the map Jz is skew symmetric with respect to the bi-linear form 〈 . , . 〉 in the following sense (1) 〈 Jzx, y 〉+ 〈x, Jzy 〉 = 0. Definition 1. A Lie algebra n = (n, [. , .], 〈 . , . 〉r,s+ 〈 . , . 〉) is called a pseudo H-type Lie algebra if J z + 〈 z , z 〉 Idv = 0 for all z ∈ z . The pseudo H-type Lie algebras are in one to one correspondence with the Clifford Cl(z, 〈 . , . 〉r,s)-module structures on v admitting a symmetric bi-linear form 〈 . , . 〉, satisfying (1), see [1, 2, 5]. We use the isomorphism of scalar vector spaces (z, 〈 . , . 〉r,s) ∼= R, (v, 〈 . , . 〉) ∼= R or (v, 〈 . , . 〉) ∼= R, and the Clifford algebras Cl(z, 〈 . , . 〉r,s) ∼= Clr,s = Cl(R). We also write nr,s for pseudo H-type Lie algebras related to the Clr,s-module of minimal possible dimension (minimal admissible module). We denote by z1, . . . , zr, zr+1, . . . , zr+s an orthonormal basis for R r,s such that