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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 group]]s. 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 sum of representations|direct sum]]s of a collection of basic [[irreducible representation]]s (governed by the [[Peter–Weyl theorem]]). The second means that there will be irreducible representations in dimensions greater than 1.

SU(2) is the [[universal covering group]] of [[SO(3)]], and so its representation theory includes{{how|date=June 2013}} that of the latter. This also specifies importance of SU(2) for description of non-relativistic [[spin (physics)|spin]] in [[theoretical physics]]; see [[#Most important irreps and their applications|below]] for other physical and historical context.

As shown below, the finite dimensional irreducible representations of SU(2) are indexed by a [[integer]] or [[half-integer]] {{math|''λ'' ≥ 0}}, with [[dimension (vector space)|dimension]] {{math|2''λ'' + 1}}.

==Lie algebra representations==
The representations of the group are found by considering representations of <math>\mathfrak{su}(2)</math>, the [[Special_unitary_group#n_.3D_2|Lie algebra of SU(2)]]. In principle this is the 'infinitesimal version' of SU(2); Lie algebras consist of [[infinitesimal transformation]]s, and their Lie groups to 'integrated' transformations. In what follows, we shall consider the complex Lie algebra (i.e. the [[complexification]] of the Lie algebra), which doesn't affect the representation theory.

The Lie algebra is spanned by three elements {{mvar|e}}, {{mvar|f}} and {{mvar|h}} with the [[Lie bracket of vector fields|Lie brackets]]

:<math>[h,e]=e</math>
:<math>[h,f]=-f</math>
:<math>[e,f]=h</math>

(These elements may be expressed in terms of matrices {{math|''I''1}}, {{math|''I''2}} and {{math|''I''3}} which are [[Pauli_matrices#SU.282.29|related to the Pauli matrices]] by multiplication by a factor of {{math|−''i''}}. {{math|1=''e'' = ''I''1 + ''i'' ''I''2}}, {{math|1=''f'' = ''I''1 − ''i'' ''I''2}}, and {{math|1=''h'' = ''I''3}}.)

Since <math>\mathfrak{su}(2)</math> is [[Semisimple Lie algebra|semisimple]], the representation {{math|''ρ''(''h'')}} is always [[diagonalizable]] (for complex number scalars). Its [[eigenvalue]]s are called the [[weight (representation theory)|weights]]. Its eigenvectors can be taken as a basis for the vector space the group acts upon. The dimension of the representation can be determined by counting the number of these eigenvectors.

Suppose {{mvar|x}} is an [[eigenvector]] of weight {{mvar|α}}. Then,

:<math>h[x]=\alpha x</math>
:<math>h[e[x]]=(\alpha +1) e[x]</math>
:<math>h[f[x]]=(\alpha -1) f[x]</math>

In other words, {{mvar|e}} raises the weight by one and {{mvar|f}} reduces the weight by one. {{mvar|e}} and {{mvar|f}} are referred to as [[Ladder_operator#Angular_momentum|ladder operators]], taking us between eigenvectors or to 0. A consequence is that

:<math>h^2+ef+fe </math>

is a [[Casimir invariant]] and commutes with the generators of the algebra. By [[Schur's lemma]], its action is proportional to the identity map, for [[irreducible representation]]s. It is convenient to write the constant of proportionality as {{math|''λ''(''λ'' + 1)}}. (The expression <math>h^2+ef+fe</math> is equal to <math>I^2</math> defined as <math>I_1^2 + I_2^2 + I_3^2</math>, which [[Angular_momentum_operator#Commutation_relations_involving_vector_magnitude|is related to the magnitude of angular momentum operator]] in quantum physics.)

===Weights===

Finite-dimensional representations only have finitely many weights, and have a greatest and least weight. (They are both [[highest weight representation]]s and [[lowest weight representation]]s.)

Let {{math|''α''1}} be a weight which is greater than all the other weights. Let {{mvar|x}} be an {{mvar|h}}-eigenvector of eigenvalue <math>\alpha_1</math>. Then {{math|1=''e''(''x'') = 0}}. If the representation is irreducible, using the commutation relations we can calculate that <math>(h^2+ef+fe) x = (\alpha_1^2 + \alpha_1) x= \lambda (\lambda +1) x</math>. Since {{mvar|x}} is nonzero, <math>\alpha_1</math> is either {{mvar|λ}} or {{math|−''λ'' − 1}}.

Likewise, let {{math|''α''2}} be a weight which is lower than all the other weights. Let {{mvar|x}} be an eigenvector of {{math|''α''2}}, so {{math|1=''f''(''x'') = 0}}. If the representation is irreducible, using the commutation relations <math>(\alpha_2^2 - \alpha_2) x=\lambda (\lambda+1) x</math>, and so <math>\alpha_2</math> is either {{math|''λ'' + 1}} or {{math|−''λ''}}.

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 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 {{math|''λ'' < '' λ'' + 1}} and {{math|−''λ'' − 1 < −''λ''}}, without any loss of generality we can assume the highest weight is {{mvar|λ}} (if it's {{math|−''λ'' − 1}}, just redefine a new {{mvar|λ′}} as {{math|−''λ'' − 1}}) and the lowest weight would then have to be {{math|−''λ''}}. This means λ has to be an integer or [[half-integer]]. Every weight is a number between {{mvar|λ}} and {{math|−''λ''}} which differs from them by an integer.

Furthermore, each weight has [[Multiplicity (mathematics)|multiplicity]] one. If this were not the case, we could define a proper [[subrepresentation]] generated by an eigenvector of {{mvar|λ}} and {{mvar|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 {{mvar|λ}}, all finite dimensional irreps with {{mvar|λ}} as its highest weight are [[equivalent representation|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]].

==Properties==
All irreps for a half-integer (not integer) {{mvar|λ}} are [[faithful representation|faithful]]. All irreps for an integer {{mvar|λ}} have the [[kernel (algebra)|kernel]] {{math|±[[identity matrix|'''1''']]}} and are virtually representations of SO(3), faithful ones for {{math|''λ'' ≥ 1}}.

==Most important irreps and their applications==
As stated above, representations of SU(2) describe non-relativistic [[spin (physics)|spin]] due to double covering of the [[rotation]] group of [[Euclidean space|Euclidean 3-space]]. [[special relativity|Relativistic]] spin is described with [[representation theory of SL2(C)|representation theory of SL2('''C''')]], a supergroup of SU(2), which in the similar way covers [[Lorentz group|SO+(1;3)]], the relativistic version of rotation group. SU(2) symmetry also supports concepts of [[isobaric spin]] and [[weak isospin]], collectively known as ''isospin''.

{{math|1=''λ'' =}} {{sfrac|1|2}} gives the '''2''' representation, the [[fundamental representation]] of SU(2). When an element of SU(2) is written as a [[complex number|complex]] {{math|2 × 2}} [[matrix (mathematics)|matrix]], it is simply a [[matrix multiplication|multiplication]] of [[column vector|column 2-vectors]]. It is known in physics as the [[spin-½]] and, historically, as the multiplication of [[quaternion]]s (more precisely, multiplication by a [[unit vector|unit]] quaternion).

{{math|1=''λ'' = 1}} gives the '''3''' representation, the [[adjoint representation]]. It corresponds to 3-d [[rotation (mathematics)|rotations]], the standard representation of SO(3), so [[real number]]s are sufficient for it. Physicists use it for description of [[rest mass|massive]] spin-1 particles, such as [[vector meson]]s, but its importance for the spin theory is much higher because it binds spin states to [[geometry]] of the physical [[three-dimensional space|3-space]].
This representation became known simultaneously with '''2''' when [[William Rowan Hamilton]] introduced [[versor]]s, his term for elements of SU(2). Note than Hamilton did not use terminology of the [[group theory]] for historical reasons.

{{math|1=''λ'' =}} {{sfrac|3|2}} representation is used in [[particle physics]] for certain [[baryon]]s, such as [[delta baryon|Δ]].

==See also==
* [[Rotation operator (vector space)]]
* [[Rotation operator (quantum mechanics)]]
* [[Representation theory of SL2(R)|representation theory of SL2('''R''')]]
* [[Electroweak interaction]]

==References==
* Gerard 't Hooft (2007), [http://www.staff.science.uu.nl/~hooft101/lectures/lieg.html ''Lie groups in Physics''], Chapter 5 "Ladder operators"

{{DEFAULTSORT:Representation Theory Of Su(2)}}
[[Category:Representation theory of Lie groups]]
[[Category:Rotation in three dimensions]]

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