INFORMATION-THEORETIC INEQUALITIES ON UNIMODULAR LIE GROUPS.

Area of Science:

• Information Theory
• Lie Group Theory
• Mathematical Physics

Background:

• Classical information theory inequalities (de Bruijn, Fisher, Cramér, Rao, Kullback) are foundational in Euclidean space.
• Unimodular Lie groups possess invariant measures, sharing useful properties with Euclidean space, including commutative and noncommutative cases.
• Relevant unimodular Lie groups include rotation, Euclidean motion, and unitary groups, crucial in geometric mechanics and quantum computing.

Purpose of the Study:

• To extend classical information-theoretic inequalities from Euclidean space to unimodular Lie groups.
• To generalize key concepts such as entropy and Fisher information matrix to the Lie group setting.
• To derive and present new inequalities analogous to classical ones within this broader mathematical framework.

Main Methods:

• Generalizing definitions of entropy and Fisher information matrix to unimodular Lie groups.
• Replacing addition of random variables with group product and employing generalized convolution of probability densities.
• Formulating fifteen new theorems presenting these extended inequalities.

Main Results:

• Established a framework for applying information theory to unimodular Lie groups.
• Derived fifteen novel inequalities analogous to classical information theory.
• Demonstrated the applicability of these results through a robotics example concerning sensory input.

Conclusions:

• The extension of information-theoretic inequalities to unimodular Lie groups provides powerful tools for analyzing information in complex systems.
• These findings have potential applications in diverse fields including mobile robotics, satellite attitude control, medical imaging, and quantum information theory.
• The generalized framework offers a new perspective on information gathering and processing in non-Euclidean settings.