On Unitary Groups in Ternary and Generalized Clifford Algebras

We discuss a generalization of Clifford algebras known as generalized Clifford algebras (in particular, ternary Clifford algebras). In these objects, we have a fixed higher-degree form (in particular, a ternary form) instead of a quadratic form in ordinary Clifford algebras. We present a natural realization of unitary Lie groups, which are important in physics and other applications, using only operations in generalized Clifford algebras and without using the corresponding matrix representations. Basis-free definitions of the determinant, trace, and characteristic polynomial in generalized Clifford algebras are introduced. Explicit formulas for all coefficients of the characteristic polynomial and inverse in generalized Clifford algebras are presented. The operation of Hermitian conjugation (or Hermitian transpose) in generalized Clifford algebras is introduced without using the corresponding matrix representations.