Download Lecture 1: Basic notions about Lie groups

Gauge theory
Physics: accurately describes phenomena ranging from
light to the forces which holds matter together
Mathematics: intimately linked to the geometry and
topology of manifolds
Lecture 1: Basic notions about Lie groups
José Miguel Figueroa-O’Farrill
Donaldson: not every topological 4-manifold admits a
smooth structure
Donaldson: new topological invariants =⇒ Freedman:
existence of exotic differentiable structures on R4
Seiberg+Witten: new topological invariants, often simpler
to compute than Donaldson’s
The work of Donaldson and of Seiberg+Witten is based on
the study of instantons and monopoles, respectively
Michael
and Lily Atiyah Gallery
SMSTC Gauge Theory
These are special solutions to PDEs, involving geometric
objects whose definitions require the notions of Lie group
and Lie algebra
9 September 2013
Sophus Lie
José Miguel Figueroa-O’Farrill
SMSTC Gauge Theory Lecture 1
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Nordfjordeid 1842 – Christiania 1899
Sophus Lie (1842–1899)
José Miguel Figueroa-O’Farrill
SMSTC Gauge Theory Lecture 1
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Lie groups in a nutshell
After Abel, Lie is the
Lie groups are smooth manifolds with a group structure
most famous Norwegian
whose operations (multiplication and inversion) are smooth
mathematician.
In this lecture: only finite-dimensional Lie groups
The Lie groups of interest in (orthodox) gauge theory are
Professor at Christiania (Oslo)
compact
and Leipzig. Foreign Member
A Lie group, despite generally being nonlinear, is
characterised (almost fully) by its Lie algebra: a vector
RS.
space with a non-associative product
He pioneered the field ofInwhat
this lecture: (compact, matrix) Lie groups and their Lie
we now call Lie groups, one
algebras
of the most important parts
of mathematics and physics.
He also dealt with more general situations in differential
geometry, not farSMSTC
removed
from the geometric
structures that José Miguel Figueroa-O’Farrill SMSTC Gauge Theory Lecture 1
José Miguel Figueroa-O’Farrill
Gauge Theory Lecture 1
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have emerged from new ideas in physics.
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Lie groups formally defined
Simplest (nontrivial) examples
Definition
A (finite-dimensional) Lie group consists of a pointed smooth
manifold (G, e) (e the identity element) and pointed smooth
maps
(R, 0) under addition
(multiplication) µ : G × G → G, µ(a, b) = ab
(inversion) ι : G → G, ι(a) =
(R× ' R t R, 1) under multiplication
a−1
(C ' R2 , 0) under addition
obeying the usual axioms of a group. ((G × G, (e, e)) is the given
the structure of a pointed smooth manifold in the usual way.)
(U(1) = {z ∈ C|zz̄ = 1} ' S1 , 1) under multiplication
(C× ' S1 × R, 1) under multiplication
Definition
A homomorphism ϕ : G → H of Lie groups is a smooth group
homomorphism: ϕ(ab) = ϕ(a)ϕ(b). If in addition it is a
diffeomorphism, then it is an isomorphism of Lie groups.
José Miguel Figueroa-O’Farrill
SMSTC Gauge Theory Lecture 1
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The general linear groups
José Miguel Figueroa-O’Farrill
SMSTC Gauge Theory Lecture 1
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The real special linear group
2
Mat(n, R) ' Rn the n × n real matrices
GL(n, R): invertible n × n real matrices
det : Mat(n, R) → R a polynomial (hence smooth) function
GL(n, C): invertible n × n complex matrices
GL(n, R) = det−1 (R× ) is an open submanifold of Mat(n, R)
More generally, V a finite-dimensional vector space over
F = R, C and
∴ dim GL(n, R) = n2
Since R× is not connected, neither is GL(n, R)
GL(V) = {ϕ : V → V|F-linear and invertible}
The determinant defines a Lie group homomorphism
det : GL(n, R) → R×
under composition of linear maps
Its kernel is a normal closed Lie subgroup
SL(n, R) < GL(n, R): the special real linear group
∼ Fn and
A basis for V is a vector space isomorphism V =
this induces a Lie group isomorphism
∼ GL(Fn ) = GL(n, F)
GL(V) =
José Miguel Figueroa-O’Farrill
SMSTC Gauge Theory Lecture 1
It consists of n × n real matrices with unit determinant
∴ dim SL(n, R) = n2 − 1
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José Miguel Figueroa-O’Farrill
SMSTC Gauge Theory Lecture 1
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The complex special linear group
Compact Lie groups
The general and special linear groups are not compact:
they are not bounded as subsets of RN
A large class of compact Lie groups consists of Lie
subgroups of the general/special linear groups which
preserve a positive-definite inner product on Fn
Similarly, GL(n, C) is an open submanifold of
2
Mat(n, C) ' R2n
∴ dim GL(n, C) = 2n2 as a real manifold
There are three main types: orthogonal, unitary and
quaternionic unitary
GL(n, C) is connected (follows from polar decomposition)
SL(n, C) = ker det GL(n, C) → C× , a normal closed Lie
We have already seen U(1) ' S1 we will see that
SU(2) ' S3 , but no other spheres are Lie groups.
subgroup
∴ dim SL(n, C) = 2(n2 − 1) as a real manifold
Fact (Heinz Hopf, 1930s)
Every finite-dimensional Lie group is rationally homotopy
equivalent to a product of odd-dimensional spheres.
José Miguel Figueroa-O’Farrill
SMSTC Gauge Theory Lecture 1
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The orthogonal group
José Miguel Figueroa-O’Farrill
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The unitary group
= the group of linear transformations of Rn preserving the
standard euclidean inner product.
= the group of complex linear transformations of Cn preserving
the standard hermitian inner product.
The orthogonal group
The unitary group
O(n) = a ∈ Mat(n, R)aT a = I
U(n) = a ∈ Mat(n, C)āT a = I
The special orthogonal group
The special unitary group
SO(n) = O(n) ∩ SL(n, R)
SU(n) = U(n) ∩ SL(n, C)
is the group of rotations of Rn
In small dimension
In small dimension
SO(2) ' S1
SO(3) ' S3 / ∼ = RP3
SO(4) ' (S3 × S3 )/ ∼
José Miguel Figueroa-O’Farrill
SMSTC Gauge Theory Lecture 1
U(1) ' S1
SU(2) ' S3
SMSTC Gauge Theory Lecture 1
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José Miguel Figueroa-O’Farrill
SMSTC Gauge Theory Lecture 1
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Linearising a Lie group
The adjoint action of G on Te G
For a ∈ G, let La : G → G, b 7→ ab, denote
left-multiplication by a and let Ra : G → G, b 7→ ba, denote
right-multiplication by a. They are diffeomorphisms of G.
The best linear approximation to a smooth manifold at a
point p is its tangent space at p: the vector space of
velocities at p of smooth curves passing through p
La ◦ Lb = Lab , but Ra ◦ Rb = Rba !
A Lie group G is a pointed manifold, so it has associated
with it a canonical vector space: the tangent space Te G at
the identity
Associativity: for all a, b ∈ G, La ◦ Rb = Rb ◦ La
The derivative maps (Lg )∗ : Te G → Tg G and
(Rg−1 )∗ : Tg G → Te G, whence
Adg := (Lg ◦ Rg−1 )∗ : Te G → Te G
We will see that Te G is not just a vector space, but in fact
has a non-associative multiplication making it into a “Lie
algebra”
José Miguel Figueroa-O’Farrill
SMSTC Gauge Theory Lecture 1
It follows that Adg1 g2 = Adg1 ◦ Adg2
This defines a “representation” of G on Te G called the
adjoint representation of G
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Representations of Lie groups
José Miguel Figueroa-O’Farrill
SMSTC Gauge Theory Lecture 1
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The adjoint representation
Definition
Let G be a Lie group. A representation of G on a vector space
V is a Lie group homomorphism ρ : G → GL(V). It is said to be
faithful if ρ is injective, so that ker ρ = e.
Example (The adjoint representation)
Every Lie group comes with a canonical representation, namely
the adjoint representation Ad : G → GL(Te G), with
Adg = (Lg ◦ Rg−1 )∗ . For matrix Lie groups, Adg (X) = gXg−1 .
The adjoint representation need not be faithful.
Let us prove that Adg1 g2 = Adg1 ◦ Adg2 .
Adg1 g2 = Lg1 g2 ◦ R(g1 g2 )−1
∗
= Lg1 g2 ◦ Rg−1 g−1
2
1
∗
= Lg1 ◦ Lg2 ◦ Rg−1 ◦ Rg−1
2
1
∗
= Lg1 ◦ Rg−1 ◦ Lg2 ◦ Rg−1
1
2
∗
= Lg1 ◦ Rg−1 ◦ Lg2 ◦ Rg−1
∗
1
Example (The defining representation of a matrix Lie group)
2
∗
= Adg1 ◦ Adg2
Every matrix Lie group comes with a second canonical
representation, called the defining representation, given by
the inclusion G < GL(V). It is of course faithful, by definition.
José Miguel Figueroa-O’Farrill
SMSTC Gauge Theory Lecture 1
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José Miguel Figueroa-O’Farrill
SMSTC Gauge Theory Lecture 1
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The Lie bracket
The Lie bracket for matrix groups
Let G < GL(n, F) be a Lie subgroup (i.e., G is a matrix
group)
For every X ∈ Te G there is some smooth curve
0 ( 0) = X .
cX : (−ε, ε) → G with cX (0) = e and cX
Then La and (La )∗ are just left matrix multiplication by a,
and similarly for Ra and (Ra )∗
Let X ∈ Te G and let cX be one such curve.
Therefore Adg Y = gYg−1 , where all products are matrix
products
Let Y ∈ Te G and consider the curve c : (−ε, ε) → Te G on
Te G defined by c(t) = AdcX (t) Y .
∴ c(t) := AdcX (t) Y = cX (t)YcX (t)−1
Since Te G is a vector space, the velocity of that curve at
t = 0 also belongs to Te G.
Differentiating at t = 0, c 0 (0) = XY − YX, whence
[X, Y] = XY − YX
Define the Lie bracket [X, Y] := c 0 (0)
It follows at once that [X, Y] = −[Y , X]
One checks that this is independent of which curve cX one
chooses. (By the chain rule, c 0 (0) only depends on cX0 (0).)
The Lie bracket also obeys the Jacobi identity
[X, [Y , Z]] + [Y , [Z, X]] + [Z, [X, Y]] = 0
(Just expand it out!)
José Miguel Figueroa-O’Farrill
SMSTC Gauge Theory Lecture 1
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Abstract Lie algebras
SMSTC Gauge Theory Lecture 1
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Structure constants
Let g be a real N-dimensional Lie algebra
Definition
A (real) Lie algebra is a (real) vector space g together with a
bilinear multiplication [−, −] : g × g → g, called the Lie bracket,
obeying for all X, Y , Z ∈ g,
Let (t1 , . . . , tN ) be an R-basis
The Lie bracket of the basis elements is given by
(skewsymmetry) [X, Y] = −[Y , X]
[ti , tj ] =
(Jacobi identity) [X, [Y , Z]] + [Y , [Z, X]] + [Z, [X, Y]] = 0
SMSTC Gauge Theory Lecture 1
N
X
fij k tk
k=1
Definition
A homomorphism of Lie algebras ϕ : g → h is a linear map
which preserves the Lie bracket; that is, ϕ([X, Y]) = [ϕ(X), ϕ(Y)]
for all X, Y ∈ g. If ϕ is in addition a vector space isomorphism,
then it is an isomorphism of Lie algebras.
José Miguel Figueroa-O’Farrill
José Miguel Figueroa-O’Farrill
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for some structure constants fij k ∈ R
In terms of the structure constants, the properties of the Lie
bracket are
(skewsymmetry) fij k = −fji k
(Jacobi
identity)
PN
m
`
m
`
m
`
f
=0
ij fmk + fjk fmi + fki fmj
m=1
José Miguel Figueroa-O’Farrill
SMSTC Gauge Theory Lecture 1
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The Lie algebra of a Lie group
Left- and right-invariant vector fields
For G any Lie group, Te G is a Lie algebra relative to the Lie
bracket
A vector field V on G is said to be left-invariant if for all
g ∈ G, (Lg )∗ V = V ; equivalently, if V(g) = (Lg )∗ V(e) for all
Te G is called the Lie algebra of G and is often written g
g∈G
As written, the proof above is only valid for matrix groups,
but the result is completely general
Similarly, V is right-invariant if V(g) = (Rg )∗ V(e) for all
g∈G
Fact
The passage from the Lie group to its Lie algebra is functorial: if
ϕ : G → H is a Lie group homomorphism, then
(ϕ∗ )e : Te G → Te H is a Lie algebra homomorphism.
Fact
Every (finite-dimensional) Lie algebra is the Lie algebra of some
Lie group.
José Miguel Figueroa-O’Farrill
SMSTC Gauge Theory Lecture 1
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Given X ∈ Te G, we let XL (resp. XR ) denote the left- (resp.
right-) invariant vector field on G which evaluates to X at e
If (t1 , . . . , tn ) is an R-basis for Te G, then (tL1 , . . . , tLn ) is a
global frame for G consisting of left-invariant vector fields
R
Similarly, (tR
1 , . . . , tn ) is a global frame consisting of
right-invariant vector fields
In either case, we see that G has trivial tangent bundle,
hence it is parallelisable
José Miguel Figueroa-O’Farrill
SMSTC Gauge Theory Lecture 1
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The Maurer–Cartan forms
Fact
Left- (resp. right-) invariant vector fields generate right (resp.
left) translations.
G a Lie group, Xg ∈ Tg G, then (Lg−1 )∗ Xg ∈ Te G
This defines ϑL ∈ Ω1 (G; Te G) by ϑLg = (Lg−1 )∗
This might seem counterintuitive, but remember that if
(Lg )∗ V = V then Lg commutes with the flow of V .
ϑL is left-invariant: L∗g ϑL = ϑL for all g ∈ G
Indeed, for any h ∈ G,
And what commutes with all left multiplications? Right
multiplications! (That’s associativity.)
L
(L∗g ϑL )h = ϑL
gh ◦ (Lg )∗ = (L(gh)−1 )∗ ◦ (Lg )∗ = (Lh−1 )∗ = ϑh
Fact
X 7→ XL between Te G and left-invariant vector fields is a Lie
algebra homomorphism: [XL , Y L ] = [X, Y]L
ϑL is called the left-invariant Maurer–Cartan form
Fact
There is also a right-invariant Maurer–Cartan form ϑR
defined by ϑR
g = (Rg−1 )∗
unique left-invariant form in Ω1 (G; Te G) with ϑLe = idTe G
X 7→ XR between Te G and right-invariant vector fields is a Lie
algebra antihomomorphism: [XR , Y R ] = −[X, Y]R
José Miguel Figueroa-O’Farrill
SMSTC Gauge Theory Lecture 1
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José Miguel Figueroa-O’Farrill
SMSTC Gauge Theory Lecture 1
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The structure equations
Pullback of Maurer–Cartan forms of matrix Lie groups
The Maurer–Cartan forms satisfy the structure equations
dϑL + 12 [ϑL , ϑL ] = 0
and
A Lie group is a manifold and can be parametrised locally
via partial maps g : Rn · · · → G
dϑR − 12 [ϑR , ϑR ] = 0
These equations are tensorial: applied to vector fields at a
point, they only depend on the value of the vector fields at
that point!
So let us check the first equation applied to XL , Y L :
The Maurer–Cartan forms pull back via g to one-forms on
Rn with values in Te G
Fact
For G < GL(N, F),
g∗ ϑL = g−1 dg
dϑL (XL , Y L ) = XL ϑL (Y L ) − Y L ϑL (Y L ) − ϑL ([XL , Y L ])
and
g∗ ϑR = dgg−1
= 0 − 0 − ϑL ([X, Y]L )
= −[X, Y]
Here g−1 is a matrix and dg a matrix of one-forms and the
products are matrix multiplication.
whereas 12 [ϑL , ϑL ](XL , Y L ) = [ϑL (XL ), ϑL (Y L )] = [X, Y]
José Miguel Figueroa-O’Farrill
SMSTC Gauge Theory Lecture 1
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SMSTC Gauge Theory Lecture 1
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The matrix exponential
The invariance and structure equations are now very easy to
prove.
g−1 dg is left-invariant: for all h ∈ G
Let F = R, C and let X ∈ Mat(n, F).
The matrix exponential exp X ∈ Mat(n, F) is defined as
(hg)−1 d(hg) = g−1 h−1 hdg = g−1 dg
exp X = I + X + 12 X2 +
g−1 dg obeys the structure equation:
1 3
3! X
+ ··· =
∞
X
1
k!
Xk
k=0
d(g−1 dg) = dg−1 ∧ dg
with X0 = I. This power series converges uniformly for all X.
Equivalently, exp X = c(1), where c : R → Mat(n, F) is the
unique solution to either of the initial value problems
= −g−1 dgg−1 ∧ dg
= −g−1 dg ∧ g−1 dg
=
José Miguel Figueroa-O’Farrill
c 0 (t) = Xc(t) and c(0) = I
c 0 (t) = c(t)X and c(0) = I
− 12 [g−1 dg, g−1 dg]
Indeed, c(t) = exp(tX).
And similarly for dgg−1 ...
José Miguel Figueroa-O’Farrill
SMSTC Gauge Theory Lecture 1
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José Miguel Figueroa-O’Farrill
SMSTC Gauge Theory Lecture 1
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Further properties of the matrix exponential
Proposition
If X, Y ∈ Mat(n, F) are such that XY = YX, then
exp(X) exp(Y) = exp(Y) exp(X) = exp(X + Y).
Proposition
exp : Mat(n, F) → GL(n, F) and (exp X)−1 = exp(−X)
Proof.
Proof.
c(t) := exp(tX) exp(−tX) obeys c 0 (t) = 0 and c(0) = I, whence
by uniqueness of the solution of first-order initial value
problems, c(t) = I for all t. Therefore exp(−tX) = exp(tX)−1 for
all t, and in particular for t = 1.
exp(X + Y) =
=
∞
X
1
(X + Y)k
k!
k=0
∞
∞ X
X
=
∞ X
k
X
k=0 `=0
1
X` Y k−`
`!(k−`)!
`=0 k=`
Proposition
For fixed X ∈ Mat(n, F), the map ϕX : R → GL(n, F) given by
ϕX (t) = exp(tX) is a homomorphism of Lie groups and its image
is the one-parameter subgroup of GL(n, F) generated by X.
(This will follow immediately from the following proposition.)
José Miguel Figueroa-O’Farrill
SMSTC Gauge Theory Lecture 1
29 / 40
`=0
X`
∞
∞ X
X
1
`!m!
X` Y m
`=0 m=0
∞
X
1
m!
Y m = exp(X) exp(Y)
m=0
José Miguel Figueroa-O’Farrill
SMSTC Gauge Theory Lecture 1
30 / 40
Representations of Lie algebras
If c : (−ε, ε) → GL(n, F) is a curve of invertible matrices with
c(0) = I, then c 0 (0) is also a matrix, since the space of
matrices is a vector space. (Of course, c 0 (0) need not be
invertible.)
Definition
Let g be a Lie algebra. A representation of g on a vector space
V is a Lie algebra homomorphism δ : g → gl(V).
Example (Lie group representations yield Lie algebra
representations)
Conversely, given any matrix X ∈ Mat(n, F), the matrix
exponential exp(tX), for any t ∈ R, is a curve through the
identity with velocity X
∴ The Lie algebra of GL(n, F) is Mat(n, F), but usually written
gl(n, F).
More generally, the Lie algebra of GL(V) for a
finite-dimensional F-vector space V , is End(V), usually
written gl(V). (This is proved in exactly the same way,
replacing matrix multiplication by composition of
endomorphisms.)
SMSTC Gauge Theory Lecture 1
1
`!
k ` k−`
X Y
`
This implies that exp(sX) exp(tX) = exp((s + t)X) as claimed.
The Lie algebra of the general linear group
José Miguel Figueroa-O’Farrill
=
∞
X
=
1
k!
Let G be a Lie group with Lie algebra g. If ρ : G → GL(V) is a
representation, its derivative at the identity gives a Lie algebra
representation ρ∗ : g → gl(V).
Remark
The converse is not true. Not every representation δ : g → gl(V)
of a Lie algebra lifts to a representation ρ : G → GL(V); i.e.,
δ 6= ρ∗ for any ρ. However, if G is simply connected then this is
always the case.
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José Miguel Figueroa-O’Farrill
SMSTC Gauge Theory Lecture 1
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The Lie algebra of the special linear group
The Lie algebra of the orthogonal group
Let o(n) and so(n) denote the Lie algebras of O(n) and
SO(n), respectively
When does exp X have unit determinant?
Let c : (−ε, ε) → O(n) be such that c(0) = I and c 0 (0) = X
det exp X = eTr X implies that det exp X = 1 iff Tr X = 0
Differentiating c(t)T c(t) = I at t = 0, we obtain XT + X = 0
Conversely, let c : (−ε, ε) → SL(n, F) be such that c(0) = I
and c 0 (0) = X
Conversely, if XT + X = 0, t 7→ exp(tX) is a curve in O(n):
Since det c(t) = 1, the derivative of the function t 7→ det c(t)
vanishes at t = 0
−1 0
det c(t) = det c(t)
Tr c(t) 0 c (t), whence evaluating at
d
t = 0, dt det c(t) t=0 = T rc (0) = Tr X, whence Tr X = 0
d
dt
exp(tX)T = exp(tXT ) = exp(−tX) = exp(tX)−1
∴ o(n) = X ∈ Mat(n, R)XT = −X and hence dim o(n) =
∴ The Lie algebra sl(n, F) of SL(n, F) is given by the traceless
matrices in Mat(n, F)
n
2
X ∈ o(n) =⇒ Tr X = 0 =⇒ det exp(tX) = 1
∴ so(n) = o(n)
Fact: The Lie algebra only “sees” the connected component of
the identity element
José Miguel Figueroa-O’Farrill
SMSTC Gauge Theory Lecture 1
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The Lie algebra of the unitary groups
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Let G be a Lie group and g its Lie algebra
Recall Ad : G → GL(g) is defined by Adg = (Lg ◦ Rg−1 )∗
The derivative of Ad at the identity defines the adjoint
representation ad : g → gl(g) of g, given by adX (Y) = [X, Y]
Indeed, for matrix groups
Similarly to the case of the orthogonal groups,
u(n) = X ∈ Mat(n, C)X̄T = −X
d
d
=
= XY − YX
Adexp(tX) Y exp(tX)Y exp(−tX)
dt
dt
t=0
t=0
and
and
SMSTC Gauge Theory Lecture 1
The adjoint representation of the Lie algebra
Let u(n) and su(n) denote the Lie algebras of U(n) and
SU(n), respectively
su(n) = X ∈ Mat(n, C)X̄T = −X
José Miguel Figueroa-O’Farrill
Tr X = 0
A s.b.f. B on g is invariant if adZ is skew-symmetric:
B(adZ X, Y) + B(X, adZ Y) = 0
Notice that despite consisting of complex matrices, u(n)
and su(n) are real vector spaces
dim u(n) = n2 and dim su(n) = n2 − 1
José Miguel Figueroa-O’Farrill
SMSTC Gauge Theory Lecture 1
∀X, Y , Z ∈ g
If B is also nondegenerate, g is metric: most Lie algebras
aren’t; most interesting Lie algebras are!
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José Miguel Figueroa-O’Farrill
SMSTC Gauge Theory Lecture 1
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The Killing form
Semisimple Lie algebras
The Killing form is the s.b.f. κ : g × g → R defined by
Semisimple Lie algebras are those whose Killing form is
nondegenerate.
κ(X, Y) = Tr adX ◦ adY
so(n > 2), su(n > 1),sl(n > 1, F) are semisimple.
It is invariant:
gl(n, F) and u(n) are not.
The Killing forms of so(n > 2) and su(n > 1) are
negative-definite, but that of sl(n > 1, F) is indefinite.
κ(adZ X, Y) = κ([Z, X], Y) = Tr ad[Z,X] ◦ adY
= Tr[adZ , adX ] ◦ adY
The most general Lie algebra with an invariant
positive-definite symmetric bilinear form is a direct sum of
Lie algebras of types u(1), so(n), su(n) and sp(n). (This
latter is the Lie algebra of the quaternionic unitary group.)
= − Tr adX ◦[adZ , adY ]
= − Tr adX ◦ ad[Z,Y]
= −κ(X, [Z, Y]) = −κ(X, adZ Y)
However it need not be nondegenerate!
José Miguel Figueroa-O’Farrill
SMSTC Gauge Theory Lecture 1
José Miguel Figueroa-O’Farrill
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The trace form
SMSTC Gauge Theory Lecture 1
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References
The Killing form involves a trace of endomorphisms of g
Here are some of my favourite books on Lie groups and Lie
algebras:
For matrix Lie algebras, such as so(n) and su(n), the
dimension of g goes like n2
Wulf Rossmann, Lie groups: an introduction through linear
groups, Oxford University Press, 2002.
The trace form is often easier to compute
− Tr : g × g → R
defined by
(X, Y) 7→ − Tr XY
Frank Warner, Foundations of differentiable manifolds and
Lie groups, GTM 94, Springer-Verlag, 1983.
since it only involves the trace of n × n matrices
Jim Humphreys, Introduction to Lie algebras and
representation theory, GTM 9, Springer-Verlag, 1978.
The trace form is invariant: − Tr[Z, X]Y = Tr X[Z, Y]
For so(n) or su(n) the trace form is positive-definite (hence
the sign!)
José Miguel Figueroa-O’Farrill
SMSTC Gauge Theory Lecture 1
39 / 40
José Miguel Figueroa-O’Farrill
SMSTC Gauge Theory Lecture 1
40 / 40