Group theory and Lie algebra, Master course, 3p, Lp III, 2004

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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
Journal of Modern Physics A, Vol 1, No 2 (1986) 303-414.
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Examination:
Home exam and an oral exam for highest grade (CTH: 5; GU: VG)
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Introductory lecture: January 20, 2004 at 10.00 in Origo 6115
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Schedule:
Tuesdays 10.00-12.00 and Thursdays 10.00-12.00
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Lectures
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1ST WEEK: Group theory and symmetries in physics (week 4)

Lecture 1: Read: Lecture notes and FS chapter 1.
For the classification of finite groups see Finite groups
Recommended exercises: See lecture notes, and FS Exercise 1.2, 1.4 and 1.5.

More about the mathematician Sophus Lie
and the short life of Nils Henrik Abel, both Norwegians.

Lecture 2:Cont. from previous lecture

Home exam problem 1: The symmetry group of a equilateral triangle has six elements. Derive the character table for the possible matrix representations of this group.
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2ND WEEK: Lie algebras and representations (week 5)

Lecture 3: Read FS chapters 2 and 3.
Recommended exercises: FS 2.1-2.6, 3.1-3.3.

Lecture 4: cont. from prev. lecture

Home exam problem 2: Prove (without assuming a scalar product) that any number
of vectors in a vector space are linearly independent iff (= if and only if) there
exists a linear operator (i.e. matrix) for which these vectors all have different
eigenvalues. Use the Vandermonde's determinant which is the product of all pairwise
differences between the eigenvalues.
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3RD WEEK: Root and weight spaces (week 6)

Read chapters 3 and 13, sections 1,2 and 3, and handed our tables from Slansky's Physics Reports.
Lecture 5: cont from previous lecture
Recommended exercises: see week 2.

Lecture 6: cont.
Recommended exercises: see week 2

Home exam problem 3: Construct the weight diagram for the (2,0) representation of the
Lie algebra sl(3,R). Derive also the decompositions of the different tensor products of three
three-dimensional representations of sl(3,R), i.e. 3x3x3, 3x3x3bar, etc. Demonstrate that
weight diagram methods and index methods give identical results.
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4TH WEEK: The classification of Lie algebras (week 7)
Chapters 4-6 present the foundations of the subject in a mathematical language.
The for this course most important aspects will be emphasized in the lectures,
and the book should be consulted where this material is discussed. The rest of the material
in these chapters can be skipped or read for the purpose of getting a broader familiarity
with the whole subject.
Lecture 7: Read FS chapter 4, sections 1-10, plus lecture notes on Levi's theorem, and chapter 5.

Recommended exercises: FS 4.2, 4.6, 4.7, 4.8, 4.11 and 4.12 (the following are outside the course:
4.3 contains a rather common definition and 4.14 and 4.15 are interesting
but requires a bit more work, note that the three-dimensional case is the Bianchi classification of
spatially homogeneous but anisotropic cosmologies);
5.1, 5.3, 5.4, 5.7a, 5.8 and 5.9(the first part only).

Lecture 8: Read FS chapter 6, sections 1-12

Recommended exercises: FS 6.1, 6.2, 6.3, 6.6, and 6.12 (6.13 is outside the course but
contains useful information).

Home exam problem 4: FS Exercise 6.7
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5TH WEEK: Classification of simple (and affine) Lie algebras (week 8)

NOTE: this week only: lecture 9 Thursday at 10.00, lecture 10 Friday at 08.00

Lecture 9: Read chapter 7, sections 1-7 (sections 8-10 are not included but will be
briefly commented upon).

Recommended exercises: FS 7.1, 7.2, 7.3 and 7.5

Lecture 10: cont. from previous lecture

Home exam problem 5: a) Construct all 3x3 Cartan matrices with detA>0 and detA=0,
and identify the Dynkin diagram and hence the Lie algebras obtained.
b)Explain also how one new vertex can be added with a single line to the
Dynkin diagram of so(8). Start from the result and repeat this procedure once more.
The answer should give all possible Cartan matrices with det>0 and det=0 obtained this way.

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6TH WEEK: Real Lie algebras (and cont. from last week) (week 9)

NOTE: the lecture that was cancelled last week is rescheduled for Friday this week at 08.00

Lecture 11: cont. chapter 7.

Lecture 12: Isomorphisms between Lie algebras; read chapter 8
(the important points in this chapter will be emphasized in the lecture).

Recommended exercises:FS 8.3 and 8.6

Home exam problem 6: FS Exercise 8.2
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7TH WEEK: Loop algebras, representation theory and Casimir operators (week 10)

Lecture 13: Loop algebras: have a look at chapter 12 sections 1, 3, 4, 5, 6, 7, 10 and 12
and lecture notes on the theory for unitary h.w. representations of the Virasoro algebra.
(The important points will be emphasized in the lecture.)

Recommended exercises: FS 12.1, 12.2 and 12.4

Lecture 14: This and that about representations.
Read FS sections 13.4, 13.6, 14.7, 14.8, 14.9, 17.1-17.6, 17.8.
For spinors and supersymmetry read sections 20.1, 20.2, 20.5, 20.6, 20.7 and 20.9.
Note that a lot of the material in these sections have been discussed at various points
during the course.
Also 18.1, 18.3 and 18.4 are included but were unfortunately not covered
in the lectures except for the branching of the adjoint of sl(3) into sl(2) representations.

Home exam problem 7: Prove FS eqs 14.28 and 14.30.
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8TH WEEK: The CTH official examination week (week 11)

Deadline for handing in the home exam problems is April 1, 2004.
Deadline for the oral exam is May 1, 2004.
Limits: the 7 home exam problems give a maximum of three points each. Of the total of 21 points
8 are needed for a 3, or G,
12 for a 4,
14 for a VG and
16 for a 5.
To get the highest mark, that is a 5 or VG, also an oral exam is required.

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