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The two-dimensional "spin 1/2" representation of the Lie algebra so(3), for example, does not correspond to an ordinary (single-valued) representation of the group SO(3). (This fact is the origin of statements to the effect that "if you rotate the wave function of an electron by 360 degrees, you get the negative of the original wave function.")
Representation theory depends upon the type of algebraic object being represented. There are several different classes of groups, associative algebras and Lie algebras, and their representation theories all have an individual flavour. Representation theory depends upon the nature of the vector space on which the algebraic object is represented.
One of the most important applications of Lie algebra representations is to the representation theory of real reductive Lie groups. The application is based on the idea that if π {\displaystyle \pi } is a Hilbert-space representation of, say, a connected real semisimple linear Lie group G , then it has two natural actions: the complexification ...
Many of the representations, both finite-dimensional and infinite-dimensional, are important in theoretical physics. Representations appear in the description of fields in classical field theory, most importantly the electromagnetic field, and of particles in relativistic quantum mechanics, as well as of both particles and quantum fields in quantum field theory and of various objects in string ...
According to Bargmann's theorem, every projective unitary representation of the Poincaré group comes from an ordinary unitary representation of its universal cover, which is a double cover. (Bargmann's theorem applies because the double cover of the Poincaré group admits no non-trivial one-dimensional central extensions .)
Representation theory is a mathematical theory that describes the situation where elements of a group (here, the flavour rotations A in the group SU(3)) are automorphisms of a vector space (here, the set of all possible quantum states that you get from flavour-rotating a proton). Therefore, by studying the representation theory of SU(3), we can ...
A map U:G → GL(V) satisfying the above relation for some vector space V is called a projective representation or a ray representation. If ω ( f , g ) = 1 , then it is called a representation . One should note that the terminology differs between mathematics and physics.
Lie groups — Many important Lie groups are compact, so the results of compact representation theory apply to them. Other techniques specific to Lie groups are used as well. Most of the groups important in physics and chemistry are Lie groups, and their representation theory is crucial to the application of group theory in those fields.