This book offers a systematic study of one particular representation of infinite-dimensional and braid groups, with a focus on the intricate relationships between ergodicity, irreducibility, and linearity. It develops a unified analytical framework for understanding how von Neumann algebras, quasi-invariant measures, and probabilistic methods interact in the representation theory of non-locally-compact groups, where classical tools such as Haar measure are not available.The first part of the book investigates representations of infinite-dimensional groups, including GL0 (28,R), using operator-algebraic and measure-theoretic techniques to establish conditions for irreducibility. By connecting ergodic theory with algebraic structures, the author introduces new methods for constructing and analyzing representations that capture the complex symmetries of infinite-dimensional systems.The second part focuses on the representation theory of braid groups Bn, addressing the celebrated problem of linearity. It clarifies the relationship between the Lawrence-Krammer and reduced Burau representations and demonstrates that the Lawrence-Krammer representation can be viewed as a quantization of the symmetric square of the reduced Burau representation. This conceptual link reveals deep connections between braid group theory, quantum symmetries, and mathematical physics, offering fresh insights into one of the most dynamic areas of modern representation theory.
