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CG. All the consequences of this theorem are also still valid.
4. If G is abelian then every irreducible representation is one-dimensional.
5. If G and H are compact, and V and W are irreducible representations of G and H
respectively, then V —" W is an irreducible representation of G × H.
10
The Groups S1 and SU2
Representations of S1
The group
S1 = U1(C) = {A " GL1(C) | AT = I}
is a compact group.
The one-dimensional representations of S1 are the maps z ’! zn for n " Z, where we identify S1
as the unit circle in C, and GL1(C) with C itself. These are all the irreducible representations,
and every finite dimensional representation is a direct sum of these.
The group SU2
The group
¯
SU2 = {A " GL2(C) | AAT = I, det A = 1}
is a compact group. In fact
a b
SU2 = " GL2(C) | a + b¯ = 1 ,
b
-¯
b
and so SU2 is isomorphic to a 3-sphere.
Conjugacy classes of SU2
The centre of SU2 is
1 0
Z(SU2) = ± .
0 1
Define
a 0
T = | a " C, a = 1 ‚" SU2,
0 a-1
the set of diagonal matrices in SU2. Then T S1 is called a maximal torus in SU2. Every
=
conjugacy class in SU2 meets T . In fact if O is a conjugacy class in SU2 then
O if O †" Z(SU2)
O )" T =
{x, x-1} if O isn t central,
where g has eigenvalues » and »-1 and
» 0
x = .
0 »-1
Thus there exists a bijection between the set of conjugacy classes in SU2 and the interval [-1, 1],
given by
1 1
g -’! tr(g) = (» + »-1).
2 2
Now let
1
Ot = {g " SU2 | tr(g) = t},
2
where -1 d" t d" 1. Then Ot is a conjugacy class in SU2, and these are all the conjugacy classes.
If t = ±1 then Ot = {±I} and if -1
=
11
Representations of SU2
(f.d.) Representations of SU2 are precisely polynomials with integer coefficients which are sym-
metric in z and z-1. The irreducible reps are the ones zn + zn-2 + zn-4 + · · · + z2-n + z-n.
12
Lie Algebras
sln = {n × n matrices A over C | tr A = 0}. It is a vector space over C.
sln is not generally closed under multiplication, but AB - BA is in it. This is the Lie bracket.
sl2 has a basis
0 1 1 0 0 0
e = , h = , f = .
0 0 0 -1 1 0
We have the relations
[h, e] = 2e, [h, f] = -2f, [e, f] = h.
Definition of a Lie algebra and their representations. Examples.
Explanation of Lie algebras
The Lie algebra corresponsing to a group is the tangent space at 1 " G and more back-
ground/motivational stuff
Relation between reps of groups and reps of their Lie algebras.
The Lie Algebra sl2
Irreducible modules: weight spaces, highest weight vectors
Complete reducibility for f.d. reps of sl2
13 [ Pobierz całość w formacie PDF ]
zanotowane.pl doc.pisz.pl pdf.pisz.pl wyciskamy.pev.pl
CG. All the consequences of this theorem are also still valid.
4. If G is abelian then every irreducible representation is one-dimensional.
5. If G and H are compact, and V and W are irreducible representations of G and H
respectively, then V —" W is an irreducible representation of G × H.
10
The Groups S1 and SU2
Representations of S1
The group
S1 = U1(C) = {A " GL1(C) | AT = I}
is a compact group.
The one-dimensional representations of S1 are the maps z ’! zn for n " Z, where we identify S1
as the unit circle in C, and GL1(C) with C itself. These are all the irreducible representations,
and every finite dimensional representation is a direct sum of these.
The group SU2
The group
¯
SU2 = {A " GL2(C) | AAT = I, det A = 1}
is a compact group. In fact
a b
SU2 = " GL2(C) | a + b¯ = 1 ,
b
-¯
b
and so SU2 is isomorphic to a 3-sphere.
Conjugacy classes of SU2
The centre of SU2 is
1 0
Z(SU2) = ± .
0 1
Define
a 0
T = | a " C, a = 1 ‚" SU2,
0 a-1
the set of diagonal matrices in SU2. Then T S1 is called a maximal torus in SU2. Every
=
conjugacy class in SU2 meets T . In fact if O is a conjugacy class in SU2 then
O if O †" Z(SU2)
O )" T =
{x, x-1} if O isn t central,
where g has eigenvalues » and »-1 and
» 0
x = .
0 »-1
Thus there exists a bijection between the set of conjugacy classes in SU2 and the interval [-1, 1],
given by
1 1
g -’! tr(g) = (» + »-1).
2 2
Now let
1
Ot = {g " SU2 | tr(g) = t},
2
where -1 d" t d" 1. Then Ot is a conjugacy class in SU2, and these are all the conjugacy classes.
If t = ±1 then Ot = {±I} and if -1
=
11
Representations of SU2
(f.d.) Representations of SU2 are precisely polynomials with integer coefficients which are sym-
metric in z and z-1. The irreducible reps are the ones zn + zn-2 + zn-4 + · · · + z2-n + z-n.
12
Lie Algebras
sln = {n × n matrices A over C | tr A = 0}. It is a vector space over C.
sln is not generally closed under multiplication, but AB - BA is in it. This is the Lie bracket.
sl2 has a basis
0 1 1 0 0 0
e = , h = , f = .
0 0 0 -1 1 0
We have the relations
[h, e] = 2e, [h, f] = -2f, [e, f] = h.
Definition of a Lie algebra and their representations. Examples.
Explanation of Lie algebras
The Lie algebra corresponsing to a group is the tangent space at 1 " G and more back-
ground/motivational stuff
Relation between reps of groups and reps of their Lie algebras.
The Lie Algebra sl2
Irreducible modules: weight spaces, highest weight vectors
Complete reducibility for f.d. reps of sl2
13 [ Pobierz całość w formacie PDF ]