Graphical Calculus of the representation theory of quantum Lie algebras III

1. Graphical Calculus of the representation theory of quantum Lie algebrasIII Dongseok KIM

2. Applications and Discussions • Canonical and dual canonical base • 3j, 6j symbols • Representation theory: tensor, invariant spaces • Multi variable Alexander polynomial and its reduced polynomial • Other simply-laced Lie algebras (clasps, their expansion for other Lie algebras) • manifold invariants by clasps • Categorification and its understanding

3. Canonical and dual canonical base • [Frenkel and Khovanov] sl(2,C) webs by generators and relations are dual to the canonical bases of Lusztig. • [Kuperberg and Khovanov] sl(3,C) webs are not dual to the canonical bases. • [Unknown but very possible problems] sl(4,C) and other rank 2 Lie algebras.

4. 3j, 6j symbols for sl(2,C) • A triple integers (a, b, c) is admissible if a + b + c is even and |a-b|· c · a+b. For sl(2,C), the dimension of invariant space of tensors of V(a),V(b) and V(c) is 1 if (a,b,c) is an admissible triple or 0 otherwise, where Va is an irreducible representation of highest weight a. • Given an admissible triple (a,b,c) we define a trivalent vertex

5. We find the 3j symbol [i, j, k] as follows.

6. Tetrahedron Coefficient

7. Let • a1=(c13+c14+c12 )/2 b_1=(c14+c23+c12+c34 )/2 • a2=(c14+c24+c34 )/2 b_2=(c13+c14+c23+c24 )/2 • a3=(c23+c24+c12 )/2 b_3=(c13+c24+c12+c34 )/2 • a4=(c13+c23+c34 )/2 and then

8. The 6j symbol is

9. 3j, 6j symbols for sl(3,C) • For trivalent vertex, each edge is decorated by an irreducible representation of sl(3,C), let us call them V(a1,b1), V(a2,b2) and V(a3,b3) where ai, bj are nonnegative integers. • Let d= min {a1, a2, a3, b1, b2, b3}. Without loss of generality we assume a1=d.

10. Theorem dim(Inv(V(a1,b1)­ V(a2,b2)­ V(a3,b3))) is d+1 if there exist nonnegative integers k, l, m, n, o, p, q such that a2=d+l+p, a3=d+n+q, b1=d+k+p, b2=d+m+q, b3=d+o and k-n=o-l=m. Otherwise, it is zero. • It is a condition that there is a hexagon with all inner angles are 120 degree and length of sides a1, b1 -p, a2 -p, b2 - q, a3 -q, b3 in cyclic order. • So we define trihedral coefficients as (d+1) by (d+1) matrix. The general shape can be found this way where i, j 2 {0,1,2, …,d}.

11. Here is an example, [(1,2), (3,2), (2,2)].

12. The middle of the filling is usually a hexagon but we could change it to triangle as follows. Then clasps are no longer segregated (separated by direction).

13. Here is an example of a nonsegregated clasp and H’s show the relation with segregated clasp.

15. [Kim] • a1=d=0 and p=0. • a1=d=0 and m=0. • a1=d 0 and m=0, find (0,0) entry. • We know some recursive relations but involve different 3j symbols not in the same matrix. • Rest of cases are open and the same for Tetrahedron Coefficient and 6j symbols. • Positivity, integrality and roundness. • [J. Murakami] Volume conjecture.

19. Representation theory • Tensor and invariant spaces • [Knutson and Tao] sl(n,C), saturation, honeycomb and hives. • dim(inv( tensors of fundamental representations)) is polynomial or rational polynomial ? • A nice basis can develop more expansion of clasps.

20. Webs spaces forUq(sl(4,C)) • For Uq(sl(4,C)), webs are generated by For relations, we only conjecture a set of relations

22. Webs spaces forUq(sl(n,C)) • The relations found for webs spaces forUq(sl(4,C)) has no sign of showing it is indeed complete. • For knot invariants, we would not need to find a complete set of relations. • First, we find generators

25. First we can ask if one can find all relations. • One should be able to prove that the web space of empty boundary is one dimensional just by using these relations (upto retangualr) where knot invariant is defined by

26. Other simply-laced Lie algebras • B2: the general case is missing, an expansion of the clasp of weight (a,b): have not found a nicely ordered basis of the expansion. • A strong possiblities : D4.

27. 3 manifold invariants by clasps • Lickorish first found a quantum Uq(sl(2,C)) invariants of 3-manifolds. • Ohtsuki and Yamada did for Uq(sl(3,C)). • Yokota found for Uq(sl(n,C)). • Uq(sp(4,C)).

28. These invariant found for Uq(sl(2,C)), Uq(sl(3,C)) and Uq(sl(n,C)) are the same invariants • There is not complete understanding of clasp of all weights. • No more is known and it might have found but may not be a new invariant.

29. Multivariable Alexander polynomial and its reduced polynomial • Alexander polynomial also can be defined by a representation of braid groups[Ohtsuki] and a skein module approach (used representation of sl(4,C)). • How do we interpolate topological properties of Alexander polynomial to these approaches ? • Seifert surfaces, genus, signature and crossing numbers. • How about the multivariable Alexander polynomial ? • Links with fixed orientations.

30. Categorification and its understanding • [Khovanov] Categorications of the colored Jones polynomial.A categorification of the Jones polynomial. • [Lee, EunSoo] On Khovanov invariant for alternating links.The support of the Khovanov's invariants for alternating knots. • [Bar Natan] On Khovanov’s Categorication of the Jones polynomial.