Representation theory of SU(2)

In the study of the representation theory of Lie groups, the study of representations of SU(2) is fundamental to the study of representations of semisimple Lie groups. It is the first case of a Lie group that is both a compact group and a non-abelian group. The first condition implies the representation theory is discrete: representations are direct sums of a collection of basic irreducible representations (governed by the Peter-Weyl theorem). The second means that irreducible representations will occur in dimensions greater than 1.

SU(2) is the universal covering group of SO(3), and so its representation theory includes that of the latter.

Lie algebra representations

Consider first representations of the Lie algebra

:mathfrak{su}(2).

In principle this is the 'infinitesimal version' of SU(2); Lie algebras consist of infinitesimal transformations, and their Lie groups to 'integrated' transformations.

Then pass to the complex Lie algebra (i.e. complexify the Lie algebra). This doesn't affect the representation theory. The Lie algebra is spanned by three elements "e", "f" and "h" with the Lie brackets

: [h,e] =e: [h,f] =-f: [e,f] =h

Since mathfrak{su}(2) is semisimple, the representation ρ(h) is always diagonalizable (for complex number scalars). Its eigenvalues are called the weights.

Suppose "x" is an eigenvector of the weight α. Then,

:h [x] =alpha x:h [e [x] =(alpha +1) e [x] :h [f [x] =(alpha -1) f [x]

In other words, "e" raises the weight by one and "f" reduces the weight by one. A consequence is that

:h2+ef+fe

is a Casimir invariant. By Schur's lemma, its action is proportional to the identity map, for irreducible representations. The constant of proportionality is conveniently written

:λ(λ+1).

Weights

A highest weight representation is a representation with a weight α which is greater than all the other weights.

If x is an eigenvector of α, e [x] =0.

If the representation is irreducible,

:(h^2+ef+fe) x = (alpha^2 + alpha) x= lambda (lambda +1) x

and so, since x is nonzero, α is either λ or -λ-1.

A lowest weight representation is a representation with a weight α which is lower than all the other weights.

If x is an eigenvector of α, f [x] =0.

If the rep is irreducible,

:(alpha^2 - alpha) x=lambda (lambda+1) x

and so, α is either λ+1 or -λ.

Finite-dimensional representations only have finitely many weights, and so are both highest and lowest weight representations. For an irreducible finite-dimensional representation, the highest weight can't be less than the lowest weight. In addition, the difference between them has to be an integer because since

:e [f [x] eq 0

implies

:f [x] eq 0

and

:f [e [x] eq 0

implies

:e [x] eq 0.

If the difference isn't an integer, there will always be a weight which is one more or one less than any given weight, contradicting the assumption of finite dimensionality.

Since λ<λ+1 and -λ-1<-λ, without any loss of generality we can assume the highest weight is λ (if it's -λ-1, just redefine a new λ' as -λ-1) and the lowest weight would then have to be -λ. This means λ has to be an integer or half-integer. Every weight is a number between λ and -λ which differs from them by an integer and has multiplicity one. This can be seen by assuming otherwise. Then, we can define a proper subrepresentation generated by an eigenvector of λ and f applied to it any number of times, contradicting the assumption of irreducibility.

This construction also shows for any given nonnegative integer multiple of half λ, all finite dimensional irreps with λ as its highest weight are equivalent (just make an identification of a highest weight eigenvector of one with one of the other).

Another approach

See under the example for Borel–Bott–Weil theorem.

ee also

* spin (physics)
* Rotation operator
* representation theory of SL2(R)


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Representation theory — This article is about the theory of representations of algebraic structures by linear transformations and matrices. For the more general notion of representations throughout mathematics, see representation (mathematics). Representation theory is… …   Wikipedia

  • representation theory — vaizdavimo teorija statusas T sritis fizika atitikmenys: angl. representation theory vok. Darstellungstheorie, f rus. теория представления, f pranc. théorie de représentation, f …   Fizikos terminų žodynas

  • Representation theory of finite groups — In mathematics, representation theory is a technique for analyzing abstract groups in terms of groups of linear transformations. See the article on group representations for an introduction. This article discusses the representation theory of… …   Wikipedia

  • Representation theory of the symmetric group — In mathematics, the representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained. This has a large area of potential applications, from… …   Wikipedia

  • Representation theory of SL2(R) — In mathematics, the main results concerning irreducible unitary representations of the Lie group SL2(R) are due to Gelfand and Naimark (1946), V. Bargmann (1947), and Harish Chandra (1952). Structure of the complexified Lie algebra We choose a… …   Wikipedia

  • Representation theory of the Lorentz group — The Lorentz group of theoretical physics has a variety of representations, corresponding to particles with integer and half integer spins in quantum field theory. These representations are normally constructed out of spinors.The group may also be …   Wikipedia

  • Representation theory of the Galilean group — In nonrelativistic quantum mechanics, an account can be given of the existence of mass and spin as follows:The spacetime symmetry group of nonrelativistic quantum mechanics is the Galilean group. In 3+1 dimensions, this is the subgroup of the… …   Wikipedia

  • Representation theory of the Poincaré group — In mathematics, the representation theory of the double cover of the Poincaré group is an example of the theory for a Lie group, in a case that is neither a compact group nor a semisimple group. It is important in relation with theoretical… …   Wikipedia

  • Representation theory of diffeomorphism groups — In mathematics, a source for the representation theory of the group of diffeomorphisms of a smooth manifold M is the initial observation that (for M connected) that group acts transitively on M .HistoryA survey paper from 1975 of the subject by… …   Wikipedia

  • Representation theory of Hopf algebras — In abstract algebra, a representation of a Hopf algebra is a representation of its underlying associative algebra. That is, a representation of a Hopf algebra H over a field K is a K vector space V with an action H × V → V usually denoted by… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.