Download Dream Map

Dror Bar−Natan: Talks: Kyoto−0705
http://www.math.toronto.edu/~drorbn/Talks/Kyoto−0705
Dror’s Dream Map of Quantum
Groups
Drinfel’d, Etingof−Kazhdan
Knot
Theory
Prehistoric
Hallucinations
Some harsh reality.
about
Khovanov
EK mix tangles and braids
C A T E G O R I F I C A T I O N
and algebras and Verma modules
Night Time
Dreams
gr K̂ /tv
K
K̂ /tv
translucent
vertices
about
Z /tv
Etingof − Kazhdan
Day Time
Dreams
gr K̂V
KV = K(T)VG
Gens and rels?
about Reshetikhin
− Turaev & quasi
triangular quasi
Hopf algebras
A /tv
specific
quantum
groups
W
K̂V
Knotted (trivalent?)
virtual knots
−
→
A
Wsl(2)
Reality
of associators, FT
invariants and
co−commutative
quasi−Hopf
algebras
U (sl(2))
gr K̂
K̂
K = KTG
Knotted Trivalent Graphs
with operations:
Generators
Z
Wsl(n)
GRT
GT
U (sl(n))
A
The unipotent completion
etc.
d
delete
This is the Drinfel’d
theory of Associators!
u
Relations
A 5 term relation,
a 6 term relation,
and lesser ones.
Why should we care?
unzip
1. Usefulness!
2. Beauty!
3. Guidance!
4. Confidence!
5. Grothendieck−Teichmuller!
#
Definable subsets:
Genus g knots, ribbon knots, boundary links and more...
Proof.
A La Carte Drawings
Fushimi−Inari
:= 2 A("3 )
Knotted Trivalent Graphs (KTG’s):
4
1
2
1
The 5−term relation:
2
= ( 1) (1 1)() (1 )
3
=
3
( )
2 A "4
3
4
=
1
2
Claim. With := (,) the above relation
Z
;
is equivalent to the Drinfel’d’s pentagon equation.
Drinfel’d
Etingof
Kazhdan
J. Murakami
Ohtsuki
1
= ( 1 1)() (1 1 )()
Kauffman
Goussarov
Polyak
2
3
( )
4
2 A "4
Viro
D. Thurston
Haviv
4