Group Algebras of Geometric Algebras.

[CarlBrannen]

Given a ring or field R and a finite group G of size N, the "group algebra over R or G[R]" is obtained by increasing the elements of R by adding a group element suffix. This increases the size of R, considered as a vector space, by a factor of N. Addition is the usual vector addition. Multiplication is done term by term with products of group elements reduced by the group multiplication.

For example, the finite group with 2 elements S_2, and the field of complex numbers C, makes a group algebra S_2[C] that is a complex vector space of two dimensions. Call an element of it (a,b) where a and b are complex numbers and a has been labeled with the identity of the finite group E, and b has been labeled with the, uh, non identity S. The 4 group multiplication rules EE=E, ES=S, SE=S and SS=E, expressed in the four basis elements of the group algebra, are:

(1,0)(1,0) = (1,0);

(1,0)(0,1) = (0,1);

(0,1)(1,0) = (0,1);

(0,1)(0,1) = (1,0).

Addition of two elements is given by

(a,b) + (c,d) = (a+c,b+d)

and multiplication is given by

(a,b)(c,d) = (ac+bd,ad+bc).

If the ring R is chosen to be the 2x2 complex matrices then G[R] is a sort of generalization of the 2x2 matrices. The 2x2 matrices can be thought of as a vector space with a basis of the 2x2 unit matrix plus the Pauli spin matrices. Then G[R] is a generalization of the Pauli spin matrices. The same method can be used to generalize any geometric algebra.

Hestenes considers the Pauli spin matrices P as indicating reflections where we keep track of the original. So if we consider a point symmetry group G, then G[P] is a generalization of the Pauli spin matrices so that they now represent the combination of a reflection and a point symmetry. This makes them attractive to consider in physics, especially physics of crystals.

The wave equation for massless fermions in physics is the Weyl equation. We get two copies of it by setting the mass to zero in the Dirac equation. It uses the 2x2 unit matrix and the Pauli spin matrices so it is natural to generalize it using G[P]. The group multiplication property couples these Weyl equations but it is possible to algebraically uncouple them.

For the case of G being the full octahedral point symmetry, with Hestenes notation 43, the uncoupled Weyl equations for 43[P] have the symmetry of the Standard Model fermions plus a dark matter particle with an internal SU(2) symmetry (similar to the internal SU(3) symmetry of the quarks). The generalized Weyl equations have 18,432 terms but can be uncoupled with assistance from the group character table. The details of the calculation have not been peer reviewed but have been parked temporarily on the internet here:

https://vixra.org/abs/2004.0354

Of course any comments welcome.