Group theory and Lie algebra, Master course, 3p, Lp III, 2005

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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 problems and an oral exam for highest mark (CTH: 5; GU: VG)

Limits: the X home exam problems give a maximum of three points each. Of the total of 3X 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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Introductory lecture: Tuesday January 18, 2005 at 15.15 in Origo 6115
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NEW Schedule:
Tuesdays 8.00-10.00 (instead of 15.15-17.00) and Thursdays 10.00-12.00
from January 18 to March 10, 2005
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1ST WEEK: Group theory and symmetries in physics: introduction (calender week 3)

Lecture 1: Read: Lecture notes and have a look at FS chapter 1
(skip the details if you have not seen some of the topics before).

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.

Lecture 2:Cont. from previous lecture: finite groups

Read handed out copies of Chap 2 in Tinkham, "Group theory and quantum mechanics",
and the lecture notes on matrix representations.

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.

Home exam problem 1: Solve problem 2.1 in Tinkham (see handed out copies of Chap 2 in Tinkham).

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2ND WEEK: Finite groups, cont from last week, and indroduction to Lie groups (calender week 4)

Lecture 3:Finite groups, more on cosets and classes, representations and characters
Read Tinkham chapter 2, and lecture notes and copies from Tinkham Chap 3 handed out during Lecture 4
on representations and characters (note that the proofs of the theorems in Tinkham Chap 3 are not
included in the course).

Lecture 4: Lie groups and some of their properties, lattices and dual vector spaces
Read the lecture notes.
You may also have a look at Chap 2 in FS to be prepared for next week.

Home exam problem 2: The symmetry group, D3, of the equilateral triangle has six elements.
Derive the character table for the possible matrix representations of this group.
Verify that the two orthogonality theorems are satisfied for the characters.
Is D3 a simple group?

Home exam problem 3: Show that SU(2) is not an invariant subgroup of SL(2,C).
In fact, SL(2,C) is a simple Lie group in the sense that it has no invariant Lie subgroups.
(As we will see later SL(2,C) has the same Lie algebra as the Lorentz group SO(1,3).)

NOTE: Dead-line for handing in the solutions to home exam problems 1 to 3 is February 15.

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3RD WEEK: Introduction to Lie algebras and representation theory, index notation and tensors (calender week 5)
The lectures this week will cover Chap 2 in FS, plus the connection between
Lie Groups and Lie algebras.

Lecture 5: 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.

Lecture 6: Representation theory of sl(2) (read FS Chap 2)

Home exam problem 4: Prove that the Witt and Virasoro algebras satisfy the Jacobi identity.
Why is the product between two vector fields in the Witt algebra not an acceptable product,
while the commutator is? The constant c in the Virasoro algebra is a real constant; thus the extra c-term,
known as a central extension in mathematics and a conformal anomaly in physics (e.g. in string theory
and condensed matter physics), commutes with all the generators L_n.

NOTE: Dead-line for handing in the solutions to home exam problems 4 is February 22.

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4th WEEK: Introduction to the theory of Lie algebras and representations: sl(3) (calender week 6)

Lecture 7: Lie algebra theory: construction of sl(3) (read FS Chap 3).
Applications in elementary particle physics

Extra lecture: Tuesday Febr 8 at 15.15 in Origo 6115: The tensor concept.

Lecture 8: The theory of su(3) and sl(3): Representations and tensor decomposition.
Read the notes and FS Chap 3 (note that the orthogonal basis in FS section 3.5 are normalized
slightly differently in the lecture).

Suggested exercises: FS Exercises 3.1, 3.2, and 3.3.

Home exam problem 5: Use tensor notation for sl(3,R) to derive the tensor products
(for 'tensor product' see Sakuarai's book) discussed in FS section 3.6 and 3.7: that is,
denote a tre-dimensional representation by a lower i-index and let an upper i-index correspond
to the representation obtained from two antisymmetric lower indices contracted with the epsilon
tensor as discussed in the lecture.
a)(1p) Decompose into irreducible representations a tensor with two lower indices, and a tensor
with one lower and one upper index. How does this decomposition change if we consider instead so(3)?
b)(2p) Decompose into irreps an sl(3,R) tensor with three lower indices. Chapter 6 in Sakurai
may be helpful.

NOTE: Dead-line for handing in the solutions to home exam problems 5 is March 1.

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5th WEEK: The theory of Lie algebras; the sl(3) story continued (calender week 7)

Lecture 9: Cont: Representation theory of su(3) and sl(3) (read FS Chap 3)

Lecture 10: Algebras and Lie algebras, representations.
Read notes, plus the parts of FS Chapters 4 and 5 that we have discussed in the course so far.
Try to make sense of the more formal language used in FS by comparing to the lecture notes.

Home exam problem 6: Solve Exercise 3.4 in FS.

Home exam problem 7: Construct the weight diagram for the (2,0) representation of the
Lie algebra sl(3,R). Using weight diagram techniques, derive the decomposition into irreps
of the tensor product of three three-dimensional representations of sl(3,R), i.e. 3x3x3.
Repeat this for the tensor product of 3 and 3bar.

NOTE: Dead-line for handing in the solutions to home exam problems 6 and 7 is March 8.

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6th WEEK: Lie algebras: the general theory (calender week 8)

Lecture 11: cont. from last lecture: Chapters 4 and 5 in FS.

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

Lecture 12: The Cartan-Weyl basis and the Killing form
Read notes and chapter 6 in FS.

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 8: FS Exercise 6.7

NOTE: Dead-line for handing in the solutions to home exam problems 8 is March 8.

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7th WEEK: Classification of finite and affine Lie algebras, FS chapter 7 (calender week 9)

Lecture 13: cont from last week: Chevalley basis, FS chapter 6

Lecture 14: FS Chapter 7

Home exam problem 9: a) Construct all 3x3 Cartan matrices with detA>0 and detA=0,
and identify the corresponding Dynkin diagrams and hence the Lie algebras obtained.
b)Explain also how one new vertex (node) 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 contain all possible Cartan matrices with det>0 and det=0 that can be obtained this way.

NOTE: Dead-line for handing in the solutions to home exam problem 9 is March 15.

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8th WEEK: FS parts of chapters 8, 12, 13, 14 and 18, plus comments on chapters 9, 10 and 11. (calender week 10)

Lectures rescheduled this week
Lecture 15: Thursday 10.00-12.00: cont last week

Lecture 16: Friday 09.00-12.00: FS chapter 8, and part of chapter 12 (see lecture notes)

Recommended exercises:FS 8.3 and 8.6

Home exam problem 10: FS Exercise 8.2

NOTE: Dead-line for handing in the solutions to home exam problem 10 is March 17.

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9th WEEK: Examination weeks: Tuesday 15 March - Friday 1 April (calender weeks 11-13)

Please schedule the oral exam with me as soon as possible.
Oral exams:
March 17: 09.00 Erik P.G. Johansson

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The 2005 course is very similar to the one given last year, see below, apart from a number of
applications that will added to the lecture notes. The home problems are also updated in the 2005 version.
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----------------------------------------------------------------------------
Course codes: CTH: FFM 480, GU: FY4 860
-----------------------------------------------------------------------------
Lecturer:Bengt EW Nilsson, tel 772 3160, Origo 6104C,
-------------------------------------------------------------------
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.
----------------------------------------------------------------------------
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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