Group theory, Lie algebra and representations, Master/PhD course, Lp I-II, 2009/2010
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Course codes: CTH: FFM 480, GU: FY4 860
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Lecturer:Bengt EW Nilsson, tel 772 3160, Origo 6104C,
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Literature: J. Fuchs and C. Schweigert, (referred to as (FS) in the notes)
"Symmetries, Lie algebras and representations" (Cambridge 1997, paperback 2003,
see list of
misprints),
plus some handed out pages copied from
R. Slansky "Group theory for unified model biulding",
Physics Reports Vol 79, No 1 (1981) 1-128,
and lecture notes.
Additional literature:
A: at the level of the course:
1. B.G. Wybourne, "Classical groups for physicists" (Wiley 1974).
2. M. Tinkham, "Group theory and quantum mechanics" (McGraw-Hill 1964).
3. H. Georgi, "Lie algebras in particle physics",
Frontiers in Physics, lecture note series 54 (Benjamin/Cummings 1982).
4. R.N. Cahn: "Semi-simple Lie algebras and their representations",
which is slightly more advanced than the previous three.
5. D.B. Lichtenberg, "Unitary symmetry and elementary particles" (Academic Press 1970).
Contains an introduction to Young tabeaux.
B: advanced standard references:
1. J.E. Humphreys, "Introduction to Lie algebras and representation theory",
Graduate Texts in Mathematics 9 (Springer 1972)
2. V.G. Kac, "Infinite dimensional Lie algebras" (Cambridge 1990, 3rd ed.).
3. D. Olive, "Kac-Moody and Virasoro algebras in relation to quantum physics", International
4. R. Gilmore, "Lie groups, Physics, and Geometry" (Cambridge 2008).
Journal of Modern Physics A, Vol 1, No 2 (1986) 303-414.
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Introductory lecture: Tuesday September 1, 2009 at 15.30 in Origo 6115
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Schedule:
Mondays 10.00-12.00 in Origo 6115 if available: First lecture September 7!
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1ST WEEK: Group theory and symmetries in physics: Introduction to the subject and to finite groups
(calender week 37)
Lecture 1: Read: Lecture notes and have a look at Fuchs and Scweigert (FS) chapter 1 and pages mentioned in the lecture.
(Skip the details if you have not seen some of the topics before, and if you have not yet got the book,
wait until you receive it. )
You may also recapitulate some group theory that you have seen before, e.g., by looking
in Sakurai, Advanced quantum mechanics,
sections 3.1 and 3.6 on spherical harmonics and the action
of the angular momentum operators on them, plus
Appendix A.5 and A.6 together with section 5.3
where you should in particular recall the diagram in Fig. 5.2.
For the classification of finite groups see
Finite groups
More about the mathematician
Sophus Lie
and the short life of
Nils Henrik Abel, both Norwegians.
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2ND WEEK: Finite groups, cont from last week (calender week 38)
Lecture 2: Finite groups: more on cosets and classes.
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3RD WEEK: Finite groups, cont from last week (calender week 39)
Lecture 3: Finite groups: representations and characters. The Great Orthogonality Theorem and
decomposition of reducible representations.
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4TH WEEK: Finite groups, applications: a brief introduction to Galois theory and quantum mechanics (calender week 40)
Lecture 4: Rules for character tables, Galois´ theorem and the solution of cubic equations.
Some basic quantum mechanics and the role of symmetries.
Home problem 1: Solving fourth order polynomial equations.
Use Galois´ theorem to show that any fourth order polynomial equation can be solved. Note that you
do not have to present the solution just show that it is always possible to find it.
a) Give the relevant subgroup diagram.
b) Describe all the subgroups appearing in the diagram. If there is a subgroup in the diagram not already
mentioned in the course you need to construct it!
c) Find a chain of subgroups that satisfy the conditions in Galois´ theorem.
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5TH WEEK: Lie groups (calender week 41)
Lecture 5: Introduction to continuous groups, Lie groups. Starting with matrix groups.
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6TH WEEK: Lie groups, cont. (calender week 42)
Lecture 6: Introduction to continuous groups, Lie groups, cont. Center, simply connected etc.
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NOTE: No lecture week 43 (i.e. on Monday Oct 19)
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7TH WEEK: Introduction to Lie algebras (calender week 44)
NOTE: New time and place for the Monday lecture: 15.30 in Origo 6115
Lecture 7: Some final comments on matrix Lie groups. Introduction to Lie algebras.
Home problem 2: The Lorenz group in disguise.
Consider the group SL(2,C) and the 2x2 complex matrix H(x,y,z,t) defined as the linear combination of the
four 2x2 matrices (unit matrix, the three Pauli matrices) with coefficients (ct,x,y,z). NB Use only plus signs when
doing this!
a) Show that H is hermitian.
b) Show that the most general 2x2 hermitian matrix has this form.
c) Let g in SL(2,C) act on H by g from the right and its hermitian conjugate from the left. Show that the result
is H(x´,y´,z´,t´).
d) Give the relation between the old and the new coordinates. Is there any invariant quantity that explains
the form of this relation?
e) find the subgroup of SL(2,C) that leaves t invariant.
f) An arbitrary g in SL(2,C) can be written g=kh with h in SU(2), ie h=exp(\frac{i}{2} \sigma^i\times\theta^i),
and k=exp(\frac{1}{2}\sigma^i\times ^\beta^i) where \theta^i are three angles and \beta^i are three
boost parameters
(actually \beta=v/c). Construct k^{\dagger}H(x,y,z,t)k=H(x´,y´,z´,t´). If this
is too difficult let the boost vector be
(0,0,b).
g) Show that the answer in the previous subproblem is just a Lorenz transformation.
h) For the boosts (0,0,b) and (0,0,b´) show that applying k(b´) after k(b) results in a a new boost
and two Lorentz transformations.
i) If the two three-vectors b^i and b´^ i and not parallel show that k(b^i)k(b´^ i)=k(b"^i)h(\theta) and determine
b"^i and the angle \theta which is related to the Thomas precession.
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8TH WEEK: More on Lie algebras (calender week 45)
NOTE: New time and place: Wednesdays 15.15 in Origo 6115
Lecture 8: Lie algebras: some examples and intro to representation theory.
Lie group elements as exponentials of Lie algebra elements,
the so(N) Lie algebras, structure constants, the Jacobi identity, the adjoint representation,
and the Baker-Hausdorff formula.
To read in Fuchs and Schweigert: Chapter 1
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9TH WEEK: Intro to Lie algebra representation theory: two low-rank examples (calender week 46)
NOTE:Time and place: Wednesdays 15.15 in Origo 6115
Lecture 9: Introduction to representation theory; two low-rank examples.
Representation theory of sl(2) (read FS Chap 2), and of sl(3) (read FS Chap 3).
Applications in elementary particle physics
The theory of su(3) and sl(3): Representations and tensor decomposition.
Read the lecture notes and FS Chap 3 (note that the orthogonal basis in FS section 3.5 are normalized
slightly differently in the lecture).
To read in Fuchs and Schweigert: Chapters 2 and 3
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10TH WEEK: Cont.:Intro to Lie algebra representation theory (calender week 47)
NOTE:Time and place: Wednesdays 15.15 in Origo 6115
Lecture 10: Introduction to representation theory.
Representation theory of sl(2) (read FS Chap 2), and of sl(3) (read FS Chap 3).
Applications in elementary particle physics
The theory of su(3) and sl(3): Representations and tensor decomposition.
Read the lecture notes and FS Chap 3 (note that the orthogonal basis in FS section 3.5 are normalized
slightly differently in the lecture).
To read in Fuchs and Schweigert: Chapters 2 and 3
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11TH WEEK: Intro to Lie algebra representation theory: two low-rank examples (calender week 48)
NOTE:Time and place: Wednesdays 15.15 in Origo 6115
Lecture 11: Representation theory of sl(3,R) (read FS Chap 3).
Applications in elementary particle physics
The theory of su(3) and sl(3): Representations and tensor decomposition.
Read the lecture notes and FS Chap 3 (note that the orthogonal basis in FS section 3.5 are normalized
slightly differently in the lecture).
To read in Fuchs and Schweigert: Chapters 4 and 5 (skip the sections marked with a * )
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12TH WEEK: Cont: General theory of Lie algebras and their representation theory: (calender week 49)
NOTE:Time and place: Wednesdays 15.15 in Origo 6115
Lecture 12: Roots, weights and their lattices, the Killing form, the Cartan-Weyl basis and the
formulation of Chevalley based on the Cartan matrix and Serre relations.
To read in Fuchs and Schweigert: Chapter 6
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13TH WEEK: Lie algebra theory: classification of finite and affine algebras (calender week 50)
NOTE:Last lecture before Christmas!! Time and place: Wednesdays 15.15 in Origo 6115
Lecture 13: More on the Cartan matrix and the introduction of Dynkin diagrams.
A first look at infinite dimensional affine Lie algebras and some comments on Kac-Moody algebras in general.
To read in Fuchs and Schweigert: Chapter 7
Home problem 3: Cartan matrices, Lie algebras and lattices.
This problem aims at constructing a self-dual lattice that plays an important role in many applications
in both particle physics (eg string theory) and mathematics. You may scan the net for these and other
more commercial applications!
a) Construct all possible rank two Cartan matrices for finite dimensional Lie algebras and derive from them
the corresponding root diagrams and weight lattices.
b) Start from the definition of the Lie group SO(2n) and construct first the generators and the Cartan matrix,
and then the root and weight lattices. Divide the weight lattices into subsets (shifted sublattices) where the
elements in each subset are equivalent up to elements of the root lattice, so called conjugacy classes.
c) Start from the result of b). When the fundamental weights become of the same length
as the roots one can try to construct a larger algebra
by considering the roots together with some of the weights as the roots of a new algebra. Show that this can be
done in such a way that this new algebra has a root lattice that is identical to its weight lattice,
ie it is self-dual! Which Lie algebra is it that is obtained this way?
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14TH WEEK: Lie algebra theory: (calender week 3, 2010)
Lecture 14: Monday January 18 at 10.00 in Origo 6115
To read in Fuchs and Schweigert: Chapters 8, 12.1 and 12.3
Home problem 4a: Fuchs and Schweigert Exercise 8.2 (page 142).
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15TH WEEK: Lie algebra theory: (calender week 4, 2010)
Lecture 15: Monday January 25 at 10.00 in Origo 6115: Last lecture!!
To read in Fuchs and Schweigert: Chapter 12
Home problem 4b: Fuchs and Schweigert Exercise 12.3 (page 219).
Read this interesting article about an application of the strange Lie algebra E8
E8 application.
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Examination:
Home problems and an oral exam for highest mark (CTH: 5; GU: VG)
Limits: the 4 home exam problems give a maximum of 10 points each.
Of the total of 40 points
40% are needed for a 3, or G,
60% for a 4,
70% for a VG and
80% for a 5.
Note: To get the highest mark, that is a 5 or VG, also a successful oral exam is required.
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Do not forget to hand in the problems and to book a time for the oral exam
(last day for oral is March 12, 2010)!
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