Exercise 4: Some things about roots.
(1) (a) Calculate the roots for types Br , Cr , and Dr .
(b) Draw the roots for B1 , B2 , C1 , C2 , and D2 (these can all be drawn in one or two
dimensions).
Note: compare your pictures to your answers for Exercise 1,
Math 128: Lecture 4
March 31, 2014
Recall if g is a Lie algebra with modules M and N , then x g acts
on m n M N by
x(m n) = xm n + m xn.
So (x) = x 1 + 1 x U g U g.
Recall if g is a Lie algebra with modules M and N , then x g acts
on m n M N by
x(m n) = x
Math 128: Lecture 5
April 2, 2014
Last time: Let M be a nite-dimensional simple sl2 (C)-module.
(1) h has at least one weight vector v M . Use hx = xh + [h, x] to
show that cfw_x v + | Z0 are also w.v.s with distinct weights.
(2) Since the weights of h o
Math 128: Lecture 2
March 26, 2014
A (complex) Lie algebra is a vector space g over C with a bracket
[, ] : g g g satisfying
(a) (skew symmetry) [x, y] = [y, x], and
(b) (Jacobi identity) [x, [y, z] + [y, [z, x] + [z, [x, y] = 0,
for all x, y, z g.
A (com
Math 128: Lecture 6
April 4, 2014
From last time:
Let U be a Hopf algebra with module M .
A bilinear form is a map , : M M C.
A bilinear form is symmetric if m, n = n, m for all
x, y M .
A bilinear form is invariant if xm, n = m, S(x)n for all
x U.
A bili
Math 128: Combinatorial representation theory
of complex Lie algebras and related topics
Recommended reading
For the rst while:
1. N. Bourbaki, Elements of Mathematics: Lie Groups and
Algebras.
2. W. Fulton, J. Harris, Representation Theory: A rst course.
Math 128: Lecture 8
April 9, 2014
Some facts about roots
For R, g = cfw_x g | adh (x) = (h)x = 0.
Let , be the Killing form. Recall, invariant means [x, y], z = y, [x, z] .
1.
2.
3.
4.
5.
6.
The adjoint action of g sends g to g+ .
If x g ( = 0), then x is
Math 128: Lecture 9
April 14, 2014
1.
2.
3.
4.
5.
6.
The adjoint action of g sends g to g+ .
If x g ( = 0), then x is nilpotent.
If = , then g , g = 0.
(Symmetry) If R, then R.
The set cfw_h | R spans h, and so R spans h .
If x g and y g then [x , y ] = x
Math 128: Lecture 7
April 7, 2014
Quick comment on bases of the classicals
Answers to Exercise 1 are up.
Basis of Ar :
cfw_Eii Ei+1,i+1 | i = 1, . . . , r
cfw_Eij , Eij | 1 i < j r + 1.
Basis of Br :
cfw_Ei+1,j+1 Ej+1+r,i+1+r | 1 i, j n
cfw_Ei+1,r+j Ej,r+
Math 128: Lecture 11
April 17, 2014
Last time:
Let V be a nite-dimensional simple g-module. Taking sl2 as a
model, we will classify V as follows:
Step 1: Show that for any weight vector v, xv is also a weight
vector for x a monomial in U n+ .
Step 2: Show
Math 128: Lecture 10
April 16, 2014
Last time:
For any basis B of h consisting of roots, the spaces
h = QB
Q
and
h = R Q h
R
Q
are Euclidean with inner product given by the Killing form (or any
positive rational/real scaling thereof).
Last time:
For any b
Math 128: Lecture 13
April 21, 2014
Existence of bases
A weight h is regular if
/
R
R h .
h1
h1 +2
h2
Existence of bases
A weight h is regular if
/
R
R h .
h1
h1 +2
h2
h
Existence of bases
A weight h is regular if
/
R
R h .
Let R+ () = cfw_ R | , > 0.
Math 128: Lecture 12
April 18, 2014
Last time:
So far we have
1. Finite-dimensional simple g-modules V are highest weight
modules, i.e. there is some v + V satisfying
hv + = (h)v + for some h , and h h, and
n v + = 0.
2. Highest weight modules (of weight
Math 128: Lecture 13
April 21, 2014
So far:
So far we have
1. Finite-dimensional simple g-modules V are highest weight
modules, i.e. there is some v + V satisfying
hv + = (h)v + for some h , and h h, and
n v + = 0.
2. Highest weight modules (of weight )
(
Math 128: Lecture 18
May 7, 2014
Last time:
Were trying to calculate m , the dimension of L() in L(), with
P +.
1. Even though cfw_y11 y
its not very helpful.
m
+
v is a spanning set of weight vectors,
2. First alternative: Freudenthals multiplicity for
Math 128: Lecture 21
May 14, 2014
Decomposing modules
Let A be a semisimple algebra over C.
Let A be an indexing set for the isomorphism classes of simple A-mods.
For A, let A be a representative for the class corresponding to .
Decomposing modules
Let A
Math 128: Lecture 19
May 9, 2014
Last time:
Were trying to calculate m , the dimension of L() in L(), with
P + = Z0 cfw_1 , . . . , r .
1. Even though cfw_y11 y
its not very helpful.
m
+
v is a spanning set of weight vectors,
Last time:
Were trying to c
Math 128: Lecture 17
May 5, 2014
Last time:
Fix a base B = cfw_i , . . . , r and a fund. chamber C = cfw_ h | , i > 0.
R
1
Let si = si and = 2 R+ .
We saw si = i and = r i P + .
i=1
Theorem
1. W acts transitively on Weyl chambers.
2. Fix a base B. For al
Math 128: Lecture 20
May 12, 2014
Weigh space multiplicities:
Were trying to calculate m , the dimension of L() in L(), with
P + = Z0 cfw_1 , . . . , r .
1. First solution: Freudenthals multiplicity
formula.
2
m =
+ i, m .
+i
, + 2 , + 2
+ i=1
R
2. Sec
Answers to Exercise 2: Some things about sl2 .
Recall, L(d) is the irreducible sl2 -module with dimension d + 1.
(1) Calculate (give the matrices for) the adjoint representation of sl2 and decompose it into
irreducible summands.
On the basis cfw_x, h, y,
ll.1. Cartan matrix of <1) 55
11. Let (I) be irreducible. Prove that d) is also irreducible. If (D has all roots
of equal length, so does (1)" (and then (13" is isomorphic to (D). On the
other hand, if (D has two root lengths, then so does (13"; but ia is
Math 128: Lecture Last
May 23, 2014
Recall: Our favorite diagram algebras so far
The group algebra of the symmetric group CSk is the algebra with
basis given by permutation diagrams
with multiplication given by concatenation, subject to the relations
=
=
Exercise 7: Some things about crystals.
(1) Verify that the two formulas for the Weyl denominator a agree for type A2 .
(2) Use the three methods from class (Freudenthals multiplicity formula, the Weyl character
formula, and the path model) to calculate t
Exercise 9: Some things about combinatorial representation theory.
Type up an extended abstract for the course. What were the main topics? For each topic, what
were the main theorems, techniques, and examples? Additionally, make a connection to the outsid
CRM week: Some things about mathematicians.
(1) Pick one of the speakers at the Combinatorial Representation Theory conference and learn
some things about them. What is their research about? What are they interested in?
If you had a chance to talk to them
Exercise 5: Some things about weights and representations.
(1) Let g be a nite-dimensional complex semisimple Lie algebra.
(a) Show that if L() and L() are highest weight modules (of weights and ), show that
L() L() has L( + ) as a submodule with multipli
Exercise 3: Some things about NIBS forms.
(1) Prove that the Killing form is an invariant symmetric bilinear form on any simple nite
dimensional complex Lie algebra.
(2) Show that the trace form on the standard representation of sln is non-degenerate.
(3)
Exercise 2: Some things about sl2 .
Recall, L(d) is the irreducible sl2 -module with dimension d + 1.
(1) Calculate (give the matrices for) the adjoint representation of sl2 and decompose it into
irreducible summands.
(2) Let V = cfw_u, v be the standard
Math 128: Lecture 25
May 22, 2014
(Very quick) introduction to quantum groups
Let q be an indeterminate. To every Lie algebra g we can associate a
Hopf algebra Uq g, called a quantum group associated to g, that is a
deformation of U g in the sense that li