Construction of Exceptional Lie Algebra G2 and Non-associative Algebras Using Clifford Algebra

Abstract This article uses Clifford algebra of positive definite signature to derive octonions and the Lie exceptional algebra $$\textrm{G2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>G2</mml:mtext> </mml:math> from calibrations using $$\mathrm{Pin(7)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Pin</mml:mi> <mml:mo>(</mml:mo> <mml:mn>7</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . This is simpler than the usual exterior algebra derivation and uncovers a subalgebra of $$\mathrm{Spin(}7)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mi>Spin</mml:mi> <mml:mo>(</mml:mo> </mml:mrow> <mml:mn>7</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> that enables $$\textrm{G2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>G2</mml:mtext> </mml:math> and an invertible element used to classify six new power-associative algebras, which are found to be related to the symmetries of $$\textrm{G2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>G2</mml:mtext> </mml:math> in a way that breaks the symmetry of octonions. The 4-form calibration terms of $$\mathrm{Spin(7)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Spin</mml:mi> <mml:mo>(</mml:mo> <mml:mn>7</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> are related to an ideal with three idempotents and provides a direct construction of $$\textrm{G2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>G2</mml:mtext> </mml:math> for each of the 480 representations of the octonions. Clifford algebra thus provides a new construction of $$\textrm{G2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>G2</mml:mtext> </mml:math> 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.