An explicit construction of real pre-Hurwitz algebras
by
O. Suzuki, J. Lawrynowicz, K. Nôno, N. Fujimoto
Nihon University, Tokyo, Japan

A real algebra with generators {e_\alpha}_{\alpha = 1, ... , p-1} satisfying the relations: e_\alpha e_\beta+e_\beta e_\alpha=-2\delta_\alpha\betaI
(\alpha, \beta = 1, 2, ..., p-1) is called a pre-Hurwitz algebra, if it has a matrix representation such that ^t(e_\alpha)=-e_\alpha(\alpha = 1, 2, .., p-1). Here we notice that a pre-Hurwitz algebra need not satisfy the irreducibility condition. In this paper, we give a complete table of an explicit construction of generators of all pre-Hurwitz algebras and discuss their irreducibility. Also we give an application to the signature problem of the space time and make a comment on the physical meaning.

1 Pre-Hurwitz pairs and pre-Hurwitz algebras

In 1898, A.Hurwitz considered the so called "Hurwitz problem". Namely he considered the characterization problem of the real number R, the complex number C, the quaternion number H, the octonion number Ø in terms of the devision algebra. Namely a bilinear irreducible mapping f:Rⁿ×Rⁿ --> Rⁿ satisfies the following condition:

||f(x, y)||=||x||||y|| (x in Rn, y in Rn).

(1)

derives nothing but the product of R, C, H, Ø for n=1, 2, 3, 4 respectively. In 1922-23, he took a pair of euclidean spaces (R^p, Rⁿ) and considered a bilinear mapping:f:R^p×Rⁿ --> Rⁿ with the following condition:

||f(x, y)||=||x||||y|| (x in Rp, y in Rn).

(2)

This condition is called the Hurwitz condition and the mapping is called the Hurwitz mapping. The mapping f is called irreducible, if f does not preserve any non trivial proper subspace V of Rⁿ, i.e., there exists no subspace V({0}\subsetneqq V\subsetneqq Rⁿ) s.t. f:R^p×V --> V . After the normalization, we can show that the determination of the Hurwtz pairs are reduced to that of the real Hurwitz algebras.

2 Deteminations of real Hurwitz algebras

In this talk, (1)we give a complete table of pre-Hurwitz algebras at first and then (2)we will discuss the irreducibility condition. By the extension of Hurwitz algebras to real Clifford algebra with the same representation space, we have the strong restriction of the signatures of the space times. By this (3)we can discuss the signature of the space time. Although Hurwitz pairs for hermitian vector spaces are completely determined, we have not known the explicit construction of the generators of real Hurwitz algebras till now. We have used the software Mathematica fully and know the form of generators.

Date received: August 9, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahz-79.