Quantum Theory for Mathematicians pp 333-366 | Cite as

# Lie Groups, Lie Algebras, and Representations

## Abstract

An important concept in physics is that of *symmetry*, whether it be rotational symmetry for many physical systems or Lorentz symmetry in relativistic systems. In many cases, the group of symmetries of a system is a *continuous group*, that is, a group that is parameterized by one or more real parameters. More precisely, the symmetry group is often a *Lie group*, that is, a smooth manifold endowed with a group structure in such a way that operations of inversion and group multiplication are smooth. The tangent space at the identity in a Lie group has a natural “bracket” operation that makes the tangent space into a Lie algebra. The Lie algebra of a Lie group encodes many of the properties of the Lie group, and yet the Lie algebra is easier to work with because it is a linear space.

## Keywords

Universal Cover Unitary Representation Quotient Group Projective Representation Closed Normal Subgroup## References

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