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Mathematical Tripos Part III Paper 2: Quantum Groups and Cobraided Bialgebras, Exams of Mathematics

Gujarat Ayurved UniversityMathematics

The questions and instructions for paper 2 of the mathematical tripos part iii exam, focusing on quantum groups and cobraided bialgebras. Students are required to define and compute various structures, explain concepts, and verify identities.

Typology: Exams

2012/2013

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MATHEMATICAL TRIPOS Part III
Tuesday 12 June 2007 9.00 to 12.00
PAPER 2
QUANTUM GROUPS
Attempt FOUR questions.
There are SIX questions in total.
The questions carry equal weight.
STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS
Cover sheet None
Treasury Tag
Script paper
You may not start to read the questions
printed on the subsequent pages until
instructed to do so by the Invigilator.
pf3
pf4
pf5

Partial preview of the text

Download Mathematical Tripos Part III Paper 2: Quantum Groups and Cobraided Bialgebras and more Exams Mathematics in PDF only on Docsity!

MATHEMATICAL TRIPOS Part III

Tuesday 12 June 2007 9.00 to 12.

PAPER 2

QUANTUM GROUPS

Attempt FOUR questions.

There are SIX questions in total.

The questions carry equal weight.

STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS

Cover sheet None

Treasury Tag

Script paper

You may not start to read the questions

printed on the subsequent pages until

instructed to do so by the Invigilator.

1 (a) Define SL

q

(2) and give its coalgebra and algebra structure explicitly.

(b) Define the coaction of SL

q

(2) on k

q

[x, y] and compute explicitly 4 x

y.

(c) Explain what is meant by an R-point of k

q

[x, y]. What are the C points of

k

q

[x, y]? If R is the algebra M

n

(C) of n by n matrices, and α is an R-point of k

q

[x, y],

show that α determines a decomposition

C

n

= V

x

⊕ V

y

⊕ U

where V

x

is the subspace on which α(x) has non-zero eigenvalues, V

y

is the subspace on

which α(y) has non-zero eigenvalues, and U is the subspace on which both α(x) and α(y)

act nilpotently. [ Hint: In particular, you must show α(y) acts nilpotently on V

x

.] You

may assume q is not a root of unity.

2 (a) Let V be a 3 dimensional simple U

q

module where q is not a root of unity. Show

that there is a basis of V with respect to which K, E, F are represented by the following

matrices:

E = 

0 [2] 0

F =

[2] 0 0

K = 

q

0 0 q

where  = ±1.

(b) Decompose V

⊗ V

into its simple U

q

modules, indicating which are highest

weight vectors, and giving bases explicitly. You may use a different basis from that in part

(a) if you prefer.

Paper 2

5 (a) If A, B are algebras, define what is meant by a measuring coalgebra for the pair

A, B. Define (by stating its universal property) what is meant by the universal measuring

coalgebra P (A, B).

(b) Let C

q

be the comodule given by

C

q

=< K, K

, I, E, F >

with K, K

, and I all group-like, and comultiplication of E, F given by

4 F = F ⊗ I + K

⊗ F ,

4 E = E ⊗ K + I ⊗ E.

If

p : C

q

−→ End(k

q

[x, y])

where

p(K)(x) = qx, p(K)(y) = q

y

p(K

)(x) = q

x, p(K

)(y) = qy

p(E)(y) = x, p(E)(x) = 0 ,

p(F )(x) = y, p(F )(y) = 0 ,

and p is a measuring map, show that

p(E)x

r

x

s

= [s]x

r+

y

s− 1

(c) Outline the proof that there is a bialgebra homomorphism

ρ : U

q

−→ P (k

q

[x, y], k

q

[x, y]).

Paper 2

6 (a) Define the tangle category.

(b) Draw representatives of the classes:

  • (↑ ∪ ↓

∩ X

  • (∪) ◦ (↓ ∩ ↑) ◦ (X

(c) Write the tangle represented by

in terms of elementary tangles

∪ , X

, X

Paper 2 [TURN OVER