Representation of a Lie group
In mathematics and theoretical physics, a representation of a Lie group is a linear action of a Lie group on a vector space. Equivalently, a representation is a smooth homomorphism of the group into the group of invertible operators on the vector space. Representations play an important role in the study of continuous symmetry. A great deal is known about such representations, a basic tool in their study being the use of the corresponding 'infinitesimal' representations of Lie algebras.
The can be understood as giving a classification of the irreducible unitary representations of the Heisenberg group.
Stone–von Neumann theorem
for representations of the Poincaré group plays a major conceptual role in quantum field theory by showing how the mass and spin of particles can be understood in group-theoretic terms.
Wigner's classification
The was worked out by V. Bargmann and serves as the prototype for the study of unitary representations of noncompact semisimple Lie groups.
representation theory of SL(2,R)
Representation theory of connected compact groups
Lie algebra representation
Projective representation
Representation theory of SU(2)
Representation theory of the Lorentz group
Representation theory of Hopf algebras
Adjoint representation of a Lie group
List of Lie group topics
Symmetry in quantum mechanics
Wigner D-matrix
; Harris, J. (1991). Representation theory. A first course. Graduate Texts in Mathematics. Vol. 129. New York: Springer-Verlag. ISBN 978-0-387-97495-8. MR 1153249.
Fulton, W.
Hall, Brian C. (2013), Quantum Theory for Mathematicians, Graduate Texts in Mathematics, vol. 267, Springer, 978-1461471158.
ISBN
Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, 978-3319134666.
ISBN
(2002), Lie Groups Beyond an Introduction, Progress in Mathematics, vol. 140 (2nd ed.), Boston: Birkhäuser.
Knapp, Anthony W.
Rossmann, Wulf (2001), Lie Groups: An Introduction Through Linear Groups, Oxford Graduate Texts in Mathematics, Oxford University Press, 978-0-19-859683-7. The 2003 reprint corrects several typographical mistakes.
ISBN
(2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 0-521-55001-7