Paravector - Lie Algebras

Lie Algebras

Clifford algebras can be used to represent any classical Lie algebra. In general it is possible to identify Lie algebras of compact groups by using anti-Hermitian elements, which can be extended to non-compact groups by adding Hermitian elements.

The bivectors of an n-dimensional Euclidean space are Hermitian elements and can be used to represent the Lie algebra.

The bivectors of the three-dimensional Euclidean space form the Lie algebra, which is isomorphic to the Lie algebra. This accidental isomorphism allows to picture a geometric interpretation of the states of the two dimensional Hilbert space by using the Bloch sphere. One of those systems is the spin 1/2 particle.

The Lie algebra can be extended by adding the three unitary vectors to form a Lie algebra isomorphic to the Lie algebra, which is the double cover of the Lorentz group . This isomorphism allows the possibility to develop a formalism of special relativity based on, which is carried out in the form of the algebra of physical space.

There is only one additional accidental isomorphism between a spin Lie algebra and a Lie algebra. This is the isomorphism between and .

Another interesting isomorphism exists between and . So, the Lie algebra can be used to generate the group. Despite that this group is smaller than the group, it is seen to be enough to span the four-dimensional Hilbert space.