Mathematical Physics

Lie Algebras from Octonion Algebraic Orientation Covariant Adjacent Algebras

Authors: Richard D. Lockyer
Comments: 32 pages plus 8 appendices

Attempts to apply Octonion Algebra to the Standard Model have involved creating a matrix algebra representing Octonion Algebra operating on itself, an adjacent algebra. Each used only one of sixteen Octonion division algebra orientations, and favored left application. This manuscript first demonstrates how proper use of Octonion Algebra for mathematical physics must give consideration to all sixteen in an algebraic orientation covariant fashion if one is to avoid pathological outcomes when the same analysis is repeated using another orientation.Properly constructed 8x8 adjacent algebra matrices generate an algebraic orientation covariant left/right application inclusive group also closed for all Octonion algebraic orientation changes. Orientation closed pairs of conjugacy classes lead to sixteen class equivalence partitions. Eight bracket/orientation closed Lie algebras are created from them, which demonstrate the orientation of each 8x8 adjacent algebra matrix position is fixed. Seven Lie algebras have four ideals in 4x4 submatrices that pair up to occupied rows/columns directly related to the seven Quaternion subalgebra indexes or their basic quad index set. From these, fourteen unique skew-Hermitian Dirac gamma matrix bases for su(4) are created, and within each six skew-Hermitian Pauli matrix su(2) bases. 28 unique su(2) bases span the 8x8 matrix format with a bracket/orientation closed Lie algebra. Skew-Hermitian bases for adjacent algebra su(n: 2-8) Lie algebras can be placed in any nxn submatrix position using linear combinations of su(2) Lie algebras. Two distinct representations of su(2) ⨁ su(3) are presented that combine into the direct sum su(2) ⨁ su(3) ⨁ su(3).
Category: Mathematical Physics