In mathematics, specifically in representation theory, a semisimple representation (also called a completely reducible representation) is a linear representation of a group or an algebra that is a direct sum of simple representations (also called irreducible representations). It is an example of the general mathematical notion of semisimplicity.
Many representations that appear in applications of representation theory are semisimple or can be approximated by semisimple representations. A semisimple module over an algebra over a field is an example of a semisimple representation. Conversely, a semisimple representation of a group G over a field k is a semisimple module over the group algebra k[''G'' ].
Let V be a representation of a group G; or more generally, let V be a vector space with a set of linear endomorphisms acting on it. In general, a vector space acted on by a set of linear endomorphisms is said to be simple (or irreducible) if the only invariant subspaces for those operators are zero and the vector space itself; a semisimple representation then is a direct sum of simple representations in that sense.
The following are equivalent:
V=W ⊕ W'
The equivalence of the above conditions can be proved based on the following lemma, which is of independent interest:
Proof of the lemma: Write
V=oplusi\inVi
Vi
Vi
VJ:=oplusiVi\subsetV
J\subsetI
K\subsetJ
J\subsetI
\operatorname{ker}p\capVJ=0
V=\operatorname{ker}p ⊕ VJ
\operatorname{ker}p\capVJ=0
V=\operatorname{ker}p+VJ
\operatorname{ker}p+VJ
V
k\inI-J
Vk\not\subset\operatorname{ker}p+VJ
Vk
Vk\cap(\operatorname{ker}p+VJ)=0
J
V=\operatorname{ker}p ⊕ VJ
W\simeqV/\operatorname{ker}p\simeqVJ\toV
\square
Note that we cannot take
J
i
\ker(p)\capVi=0
X
Y ⊕ Z
X\capY=0=X\capZ
X
Y
Z
R2
A=\operatorname{Mat}2F
V=A
A
V1=l\{\begin{pmatrix}a&0\\b&0\end{pmatrix}r\}
V2=l\{\begin{pmatrix}0&c\\0&d\end{pmatrix}r\}
W=l\{\begin{pmatrix}c&c\\d&d\end{pmatrix}r\}
V1
V2
W
A
V=V1 ⊕ V2
p:V\toV/W
\operatorname{ker}p=W\ne0
V1\cap\operatorname{ker}p=0=V2\cap\operatorname{ker}p
W\simeqV1\simeqV2
V\ne\operatorname{ker}p ⊕ V1 ⊕ V2
Proof of equivalences
1. ⇒ 3.
V\toV/W
V/W
3. ⇒ 2.
We shall first observe that every nonzero subrepresentation W has a simple subrepresentation. Shrinking W to a (nonzero) cyclic subrepresentation we can assume it is finitely generated. Then it has a maximal subrepresentation U. By the condition 3.,
V=U ⊕ U'
U'
W=U ⊕ (W\capU')
(W\capU')\simeqW/U
W
W'
W'\ne0
W'
W\capW'\ne0
W'=0
2. ⇒ 1.
The implication is a direct generalization of a basic fact in linear algebra that a basis can be extracted from a spanning set of a vector space. That is we can prove the following slightly more precise statement:
V=\sumiVi
V=oplusi\inVi
I'\subsetI
W
Vi
Vi\subsetW
Vi\capW=0
W ⊕ Vi
Vi\subsetW
\square
A finite-dimensional unitary representation (i.e., a representation factoring through a unitary group) is a basic example of a semisimple representation. Such a representation is semisimple since if W is a subrepresentation, then the orthogonal complement to W is a complementary representation because if
v\inW\bot
g\inG
\langle\pi(g)v,w\rangle=\langlev,\pi(g-1)w\rangle=0
\pi(g)v\inW\bot
For example, given a continuous finite-dimensional complex representation
\pi:G\toGL(V)
\langle,\rangle
\langle\pi(g)v,\pi(g)w\rangle=\langlev,w\rangle
\pi(g)
\pi
By Weyl's theorem on complete reducibility, every finite-dimensional representation of a semisimple Lie algebra over a field of characteristic zero is semisimple.[1]
Given a linear endomorphism T of a vector space V, V is semisimple as a representation of T (i.e., T is a semisimple operator) if and only if the minimal polynomial of T is separable; i.e., a product of distinct irreducible polynomials.
Given a finite-dimensional representation V, the Jordan–Hölder theorem says there is a filtration by subrepresentations:
V=V0\supsetV1\supset … \supsetVn=0
Vi/Vi+1
\operatorname{gr}V:=
n-1 | |
oplus | |
i=0 |
Vi/Vi+1
A representation of a unipotent group is generally not semisimple. Take
G
\begin{bmatrix} 1&a\\ 0&1 \end{bmatrix}
V=R2
\begin{bmatrix} 1\\ 0 \end{bmatrix}
See also: Decomposition of a module. The decomposition of a semisimple representation into simple ones, called a semisimple decomposition, need not be unique; for example, for a trivial representation, simple representations are one-dimensional vector spaces and thus a semisimple decomposition amounts to a choice of a basis of the representation vector space. The isotypic decomposition, on the other hand, is an example of a unique decomposition.
However, for a finite-dimensional semisimple representation V over an algebraically closed field, the numbers of simple representations up to isomorphism appearing in the decomposition of V (1) are unique and (2) completely determine the representation up to isomorphism; this is a consequence of Schur's lemma in the following way. Suppose a finite-dimensional semisimple representation V over an algebraically closed field is given: by definition, it is a direct sum of simple representations. By grouping together simple representations in the decomposition that are isomorphic to each other, up to an isomorphism, one finds a decomposition (not necessarily unique):
V\simeqoplusi
⊕ mi | |
V | |
i |
Vi
mi
mi=\dim\operatorname{Hom}equiv(Vi,V)=\dim\operatorname{Hom}equiv(V,Vi)
\operatorname{Hom}equiv
mi
Vi
Vi
mi
Vi
In general, given a finite-dimensional representation
\pi:G\toGL(V)
\chiV:G\overset{\pi}{\to}GL(V)\overset{\operatorname{tr}}{\to}k
(\pi,V)
(\pi,V)
V\simeqoplusi
⊕ mi | |
V | |
i |
\operatorname{tr}(\pi(g))
\pi(g):Vi\toVi
\chiV=\sumimi
\chi | |
Vi |
\chi | |
Vi |
Vi
V
mi=\langle\chiV,
\chi | |
Vi |
\rangle
There is a decomposition of a semisimple representation that is unique, called the isotypic decomposition of the representation. By definition, given a simple representation S, the isotypic component of type S of a representation V is the sum of all subrepresentations of V that are isomorphic to S; note the component is also isomorphic to the direct sum of some choice of subrepresentations isomorphic to S (so the component is unique, while the summands are not necessary so).
Then the isotypic decomposition of a semisimple representation V is the (unique) direct sum decomposition:
V=oplusλ
\widehat{G}
Vλ
S\inλ
Let
V
x1,x2,x3
S3
V
S3
V
W
S3
2x | |
x | |
2-x |
2x | |
1 |
+
2x | |
x | |
3-x |
2x | |
3 |
2x | |
x | |
3-x |
2x | |
2 |
+
2x | |
x | |
1-x |
2x | |
1 |
W
W1=\{a(x
2x | |
2+x |
2x | |
3)+b(x |
2x | |
1+x |
2x | |
3)+c(x |
2x | |
1+x |
2x | |
2)\mid |
a+b+c=0\}
W
W2=\{a(x
2x | |
1+x |
2x | |
1)+b(x |
2x | |
2+x |
2x | |
2)+c(x |
2x | |
3+x |
2x | |
3)\mid |
a+b+c=0\}
W3=\{ax
3\mid | |
3 |
a+b+c=0\}
W1 ⊕ W2 ⊕ W3
W
V
In Fourier analysis, one decomposes a (nice) function as the limit of the Fourier series of the function. In much the same way, a representation itself may not be semisimple but it may be the completion (in a suitable sense) of a semisimple representation. The most basic case of this is the Peter–Weyl theorem, which decomposes the left (or right) regular representation of a compact group into the Hilbert-space completion of the direct sum of all simple unitary representations. As a corollary, there is a natural decomposition for
W=L2(G)
W\simeq\widehat{oplus[(\pi,
\widehat{oplus}
(\pi,V)
When the group G is a finite group, the vector space
W=C[G]
C[G]=oplus[(\pi,V ⊕ .
S1
In quantum mechanics and particle physics, the angular momentum of an object can be described by complex representations of the rotation group SO(3), all of which are semisimple. Due to connection between SO(3) and SU(2), the non-relativistic spin of an elementary particle is described by complex representations of SU(2) and the relativistic spin is described by complex representations of SL2(C), all of which are semisimple.[2] In angular momentum coupling, Clebsch–Gordan coefficients arise from the multiplicities of irreducible representations occurring in the semisimple decomposition of a tensor product of irreducible representations.[3]
. Jean-Pierre Serre . Linear Representations of Finite Groups . registration . Graduate Texts in Mathematics, 42 . . New York–Heidelberg . 1977-09-01 . 978-0-387-90190-9 . 0450380 . 0355.20006 .