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Construction of Exceptional Lie Algebra G2 and Non-associative Algebras Using Clifford Algebra
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Abstract
This article uses Clifford algebra of positive definite signature to derive octonions and the Lie exceptional algebra
$$\textrm{G2}$$
G2
from calibrations using
$$\mathrm{Pin(7)}$$
Pin
(
7
)
. This is simpler than the usual exterior algebra derivation and uncovers a subalgebra of
$$\mathrm{Spin(}7)$$
Spin
(
7
)
that enables
$$\textrm{G2}$$
G2
and an invertible element used to classify six new power-associative algebras, which are found to be related to the symmetries of
$$\textrm{G2}$$
G2
in a way that breaks the symmetry of octonions. The 4-form calibration terms of
$$\mathrm{Spin(7)}$$
Spin
(
7
)
are related to an ideal with three idempotents and provides a direct construction of
$$\textrm{G2}$$
G2
for each of the 480 representations of the octonions. Clifford algebra thus provides a new construction of
$$\textrm{G2}$$
G2
without using the Lie bracket. A calibration in 15 dimensions is shown to generate the sedenions and to include one of the power-associative algebras, a result previously found by Cawagas.
Title: Construction of Exceptional Lie Algebra G2 and Non-associative Algebras Using Clifford Algebra
Description:
Abstract
This article uses Clifford algebra of positive definite signature to derive octonions and the Lie exceptional algebra
$$\textrm{G2}$$
G2
from calibrations using
$$\mathrm{Pin(7)}$$
Pin
(
7
)
.
This is simpler than the usual exterior algebra derivation and uncovers a subalgebra of
$$\mathrm{Spin(}7)$$
Spin
(
7
)
that enables
$$\textrm{G2}$$
G2
and an invertible element used to classify six new power-associative algebras, which are found to be related to the symmetries of
$$\textrm{G2}$$
G2
in a way that breaks the symmetry of octonions.
The 4-form calibration terms of
$$\mathrm{Spin(7)}$$
Spin
(
7
)
are related to an ideal with three idempotents and provides a direct construction of
$$\textrm{G2}$$
G2
for each of the 480 representations of the octonions.
Clifford algebra thus provides a new construction of
$$\textrm{G2}$$
G2
without using the Lie bracket.
A calibration in 15 dimensions is shown to generate the sedenions and to include one of the power-associative algebras, a result previously found by Cawagas.
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