Characterization of Normalizer of Lie Superalgebra and its Application to Control Theory

TL;DR

This paper explores control systems with bosonic and fermionic variables on Lie supergroups, characterizing them via the normalizer of Lie subsuperalgebras, and proposes controllability criteria with examples.

Contribution

It introduces a novel approach to control problems in supersymmetric dynamical systems using the normalizer of Lie subsuperalgebras.

Findings

Controllability criteria for linear control systems on Lie supergroups.

Characterization of control systems using the normalizer of Lie subsuperalgebras.

Application of the theory to specific examples.

Abstract

The dynamical systems having both bosonic and fermionic variables play an important role in the theory of supersymmetry. This article addresses the control problems including both bosonic and fermionic variables on Lie supergroup as the configuration space. Here, the control systems are characterized using the normalizer of Lie subsuperalgebra of left-invariant vector fields in the Lie superalgebra of all smooth vector fields of Lie supergroup. Then, the linear control system is studied in detail and its controllability criterion is proposed along with suitable examples.