Lie theory, the theory of Lie groups, Lie algebras, and their
applications is a fundamental part of mathematics that touches on a broad
spectrum of mathematics, including geometry (classical, differential, and algebraic),
ordinary and partial differential equations, group, ring, and algebra theory,
complex and harmonic analysis, number theory, and physics (classical, quantum,
and relativistic).
It typically relies upon an array of substantial tools such
as topology, differentiable manifolds and differential geometry, covering
spaces, advanced linear algebra, measure theory, and group theory to name a
few. However, we will considerably simplify the approach to Lie theory by
restricting our attention to the most important class of examples, namely those
Lie groups that can be concretely realized as (multiplicative) groups of
matrices.
Lie theory began in the late nineteenth century, primarily
through the work of the Norwegian mathematician Sophus Lie, who called them “continuous
groups,” in contrast to the usually finite permutation groups that had been
principally studied up to that point. An early major success of the theory was
to provide a viewpoint for a systematic understanding of the newer geometries
such as hyperbolic, elliptic, and projective, that had arisen earlier in the
century.
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