A symplectic subgroup of a pseudounitary group as a subset of Clifford algebra

TL;DR

This paper demonstrates that a specific Lie algebra derived from the Clifford algebra Cl(1,3) is isomorphic to the symplectic Lie algebra sp(4,R), and introduces a symplectic group within Clifford algebra that matches Sp(4,R).

Contribution

It establishes an isomorphism between a Clifford algebra-based Lie algebra and the classical symplectic Lie algebra, and defines a corresponding symplectic group within Clifford algebra.

Findings

Cl2(1,3)+iCl1(1,3) is isomorphic to sp(4,R)

The symplectic group of Clifford algebra is isomorphic to Sp(4,R)

Provides a new perspective linking Clifford algebra and symplectic Lie groups

Abstract

Let Cl1(1,3) and Cl2(1,3) be the subsets of elements of the Clifford algebra Cl(1,3) of ranks 1 and 2 respectively. Recently it was proved that the subset Cl2(p,q)+iCl1(p,q) of the complex Clifford algebra can be considered as a Lie algebra. In this paper we prove that for p=1, q=3 the Lie algebra Cl2(p,q)+iCl1(p,q) is isomorphic to the well known matrix Lie algebra sp(4,R) of the symplectic Lie group Sp(4,R). Also we define the so called symplectic group of Clifford algebra and prove that this Lie group is isomprphic to the symplectic matrix group Sp(4,R).