Applications of the Jordan and Lie algebras for some dynamical systems having internal symmetries (Q2761901)

The author of this very interesting paper considers the applications of the Jordan and Lie algebras to some dynamical systems having internal symmetries. For the \(g\)-based Jordan algebra introduced by the author [Int. J. Non-linear Mech. 35, 421-429 (2000)] one assigns a matrix representation, which is proved to form a \(g\)-based Lie algebra under the commutator product. Then a new dynamical system based on the composition of the \(g\)-based Jordan and Lie algebras is derived. This dynamical system possesses the internal symmetry group \(\text{DSO}_0(n,1)\) and its projection \(\text{PDSO}_0(n,1)\). The author extends those results to the \(i\)-based formulations, of which the resulting dynamical system has internal symmetry group \(\text{DSO}(n+1)\), and its projection \(\text{PDSO}(n+1)\). The above underlying algebras are all nonassociative. However, for three-dimensional problems it is possible to add an extra term in the product to construct other algebras, which include two associative algebras, namely the \(i\)-based quaternion \(\mathbb{H}\) and the \(g\)- and \(i\)-based (mixed) quaternion \(\mathbb{H}_-^+\), and one noncommutative Jordan algebra. For the second case the author formulates a dynamical system with internal symmetry group \(\text{DSO}(2,2)\) and its projection \(\text{PDSO}(2,2)\), and also derives a rotation formula in the three-dimensional Minkowski space \(\mathbf M^3\). Several physical examples are given to illustrate this new formulation.