Baker-Campbell-Hausdorff Formula and its Applications in Lie Theory

Mukhopadhyay, Jeet (2025) Baker-Campbell-Hausdorff Formula and its Applications in Lie Theory. Masters thesis, Indian Institute of Science Education and Research Kolkata.

Text (MS Dissertation of Jeet Mukhopadhyay (22MP003)) 22MP003_Thesis_file.pdf - Submitted Version Restricted to Repository staff only Download (557kB)

Abstract

The Baker–Campbell–Hausdorff (BCH) formula is a fundamental result in Lie theory that provides a method to express the product of matrix exponentials in terms of their Lie algebra structure. This work explores the BCH formula in detail, beginning with foundational concepts of matrix Lie groups and Lie algebras. We examine the exponential and logarithmic maps, their convergence properties, and their role in connecting Lie groups to their associated Lie algebras. Special attention is given to the Heisenberg group as an illustrative example where the BCH formula simplifies due to nilpotency. We discuss conditions for lifting Lie algebra homomorphisms to group homomorphisms, particularly in the simply connected case. The proof of the BCH formula is presented using Poincar´e’s integral form, emphasizing its dependence on the adjoint representation. This treatment aims to provide both theoretical depth and accessibility to readers familiar with basic Lie theory.

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