Graphical Calculus of the representation theory of quantum Lie algebras Dongseok KIM

History of Knot theory Peter Guthrie Tait (1877) The subject is a very much more difficult and intricate than at first sight one is inclined to think. Gottfried Leibniz (17th century) tried to create a branch of mathematics that does not take magnitudes into consideration. Carl Gauss first made a brief study of knots, he was hampered because of his analytical approach.

Johann Listing (1848) Vorstudien zur Topologie Examples of knots and their properties as examples of qualitative geometry, or as he called it topologie. Knots were geometrical objects that could be studied without respect to particular quantities or measures. A property of knots that later become an important tool , the invariant. Early study of knots Enumerating knots and simple invariants, Tait, little, Kirkman.

William Thomson (1867) Group theory A theory of atoms which hypothesized their structure as tiny vortices in the ether The notion of a vortex to include any closed loop or link, in particular, an atom may be a type of knot. Group theory Max Dehn and Heinrich Tietze provided the requisite thinking by applying group theory to knots at the turn of the 20th century. The fundamental group of knot complement, also known as the knot group became the major examples of the study of group presentations

The study of low dimensional manifolds J. W Alexander found the most powerful polynomial invariant, Alexander polynomial of knots. Presenting 3-manifolds by branched covering of 3 dimensional sphere branched along links, surgery along links and the actions of knot groups. The complement of Figure eight knot brought hyperbolic geometry into knot theory. V. F. R. Jones(1984) his epochal knot invariants brought a Renaissance of the study of knots.

The overwhelming new invariants of knots and 3-manifolds Quantum groups by Drinfeld and Jimbo Topological quantum field theory by E. Witten The theory of von Neumann algebras, The theory of Hopf algebras, The representations theory of semisimple Lie algebras. Braided categories derived from quantum groups.

Jones polynomial Let V(L) = q-w(D)[L] where w(D) is the number of positive crossings – the number of negative crossings.

Questions Why did we expand them a linear combination of these two diagrams for Jones polynomial ? Can we find other polynomials which is also an invariant of links ? Can we define these kind of invariants for graphs ? How do we relate this to representation theory of Lie algebras?

A generalization is Let P(L) = q-(n/2)w(D)[L]n where w(D) is the number of positive crossings – the number of negative crossings.

Another generalization is

Representation theory Representation of Braid groups: Yang-Baxter equation, R-matrix, knot invariants, 3 manifold invariants. Representation of Lie algebras: Skein modules Colored Jones polynomials Invariant space of the tensor of representations of Lie algebras, Temperley-Lieb algebra Spiders and Webs spaces.

Colored Jones Polynomials Sl(2,C) Jones-Wenzl Idempotents 3-manifold invariants Colored Jones polynomial Other Lie algebras Clasps Skein module 3-manifold invariants

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

Representation theory of Lie algebras

Representation theory sl(2,C) Dimension of the invariants space Other Lie algebras (sl(3,C)) Temperley-Lieb algebras Chord diagrams Web spaces or generalized TR algebras Webs

A Lie group G is a smooth manifold with smooth group operations. A Lie algebra L over F is a vector space over F, with a bilinear operator [ , ]: L £ L ! L (called a Lie bracket) such that it is anti-commutative and it holds the Jacobi identity. [x, y] = - [y, x], [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0.

A representation  of Lie algebra L is an algebra homomorphism from L into gl(V) which preserves Lie bracket. It is equivalent to say V is a L-module, then one might call V is a representation of L. A representation of L is irreducible if there is no nontrivial subspace W of V which is invariant, (W)½ W.

(Schur’s Lemma) If f : V !W intertwines between irreducible representations x : G ! GL(V) and y : G ! GL(W) then either f = 0 or f is an isomorphism (i.e. x and y are equivalent representations).

Representation theory of sl(2,C) sl(2,C) is the set of all trace zero 2 £ 2 complex matrices which is a Lie algebra denoted by A1. Let be a set of generators for sl(2,C). One can see that [X,Y] = XY- YX = H, [H,X] = 2X, [H,Y] = -2Y.

Let V be a finite dimensional irreducible representation of sl(2,C) Let V be a finite dimensional irreducible representation of sl(2,C). By preservation of Jordan decomposition, one can decompose V into a direct sum of eigenspaces of H V  © V where  runs over a finite set of complex numbers.

H(X(v)) = X(H(v))+[H,X](v) Let v be an eigenvector of H with associated eigenvalue a. One can easily see H(X(v)) = X(H(v))+[H,X](v) = X(a v)+2X(v) = (a +2)X(v) and H(Y(v)) = (a –2)Y(v). There exists n such that Vn¹ 0 and X(Vn) = 0. Let v 2 Vn, then one can easily show W = span{v, Y(v), Y2(v), …} is an invariant subspace of V. So W=V.

X(Ym(v))=m(n-m+1)Ym-1(v), Since X(Ym(v))=m(n-m+1)Ym-1(v), one find Yn¹ 0 but Yn+1=0. This mean eigenvalues of H are integers n, n-2, …, 2-n,-n. So V has the dimension n+1, denoted by V(n). It is known that it is isomorphic to the symmetric algebra Symn(C2) or C[x,y]n, the set of all homogeneous polynomials of degree n in two variables x, y.

H(v ­ w) = H(v) ­ w + v ­ H(w) Let V  © V and W  © Wb be irreducible representation of sl(2,C). Let v, w be eigenvectors of H with associated eigenvalues a, b. H(v ­ w) = H(v) ­ w + v ­ H(w) = (a+b) v ­ w. Thus v ­ w belongs to the eigenspace of H with the associated eigenvalue a + b.

For sl(2,C), V(2) ­ V(3) can be decomposed by comparing its eigenvalues. V(2) has eigenvalues 2, 0 and –2, V(3) has eigenvalues 3, 1, – 1 and –3. 12 eigenvalues of V(2) ­ V(3) are 5 and –5, 3 and –3 (taken twice) and 1 and –1 (taken three times). We can rewrite it as 5, 3, 1, – 1, – 3, – 5 and 3, 1, – 1, – 3 and 1, – 1, so we find that V(2) ­ V(3) = V(5) © V(3) © V(1).

This generalizes to Clebsch-Gordan formula. For nonnegative integers a, b, we have V(a) ­ V(b) = V(a+b) © V(a+b-2) © … © V(|a-b|).

Tensor Product Given two irreducible representations V, W of a Lie algebra L, we want to find the decomposition of V ­ W into irreducible representations. The following are equivalent. An L module U is in the decomposition of V ­ W into irreducible representations. U*­ V­ W has a nontrivial L invariant vector. Dim(Inv(U*­ V­ W )) > 0. HomL(U, V­ W) is nontrivial. Ddd

f(ei*­ fj)(ek)=dikfj. 1) , 4) By Schur’s lemma, HomL(U, V­ W) is nontrivial if and only if U ½ V­ W. 4) , 3) Because HomL(U, V­ W) @ U*­ (V­ W). We have to careful that one of V, W has to be finite dimensional for Hom(V,W)@ V*­ W. In the case both are finite dimensional, we pick a basis ei for V and fj. Then f(ei*­ fj)(ek)=dikfj. It is easy to see its inverse map.

It follows by the definitions. InvL(V)={v2 V| A¢v = 0 for all A2 L}. 2) , 3) It follows by the definitions. InvL(V)={v2 V| A¢v = 0 for all A2 L}. Recalled the action of L is defined by a (f) (a) = a(f(a)) - f(a a).

For sl(2,C), dim(Inv(V(a)­ V(b)­ V(c))) = 1 if a+b+c is even and if there is a triangle of sides length a, b and c, zero otherwise. Remark that (V(a))*@ V(a). There exist i, j, k such that a=(i+j), b=j+k and c=k+I because we can pick i=(a-b+c)/2, j=(a+b-c)/2 and k=(-a+b+c)/2.

For sl(n,C), Littlewood-Richardson formula. [Steinberg] Let l', l'' 2 L+. The the number of time V(l) occurs in V(l') ­ V(l'') is given by

[Brauer] Let l', l'' 2 L+. Let ml(m) be the number of eigenspace Vm in the irreducible representation V(l).

We find the Weyl group W of sl(2,C) is Z/2Z generated by the reflection along the origin and d =1.

Let V1, V2, …, Vn be finite dimensional irreducible representations of sl(2,C). Next question we want to know is the dimension of the invariant space of V1­ V2 ­ … ­ Vn. It is known by Cartan-Weyl character theory that the dimension of the invariant space is the number of copies of V(0) @ C in the decomposition of V1­ V2 ­ … ­ Vn into irreducible representations.

The dimension of the invariant space of (V(1))­ 2n is the n-th Catalan number, or geometrically the number of Chord diagrams of the disc with 2n points. Proof : A walk of length n from a to b is a sequence of integers {w0, w1, …, wn} where w0 = a, wn = b and |wi - wi+1|=1 for all I = 0, 1, … , n-1. A walk is nonnegative if wi > -1 for all i. Let P+(2n:0,0) be the set of all nonnegative walks of length 2n from 0 to 0. Since Vk ­ V1 = Vk+1 © Vk-1 for k>0 and V0 ­V1=V1, the number of copies of V_0 in (V(1))­ 2n is the number of paths in P+(2n:0,0) which is also well known as n-th Catalan number.

For n=3, 3rd Catalan number is There are five chord diagrams of the disc with 6 points

Then one can calculate that V(1) has two eigenvalues 1, -1 and their eigenvectors are the standard basis (1,0)t, (0,1)t. Then one can calculate that X(x)=0, X(y)=x, Y(x)=y, Y(y)=0, H(x)=x, H(y)=-y. For V(1)­ V(1), we pick w=x­ y - y­ x. Then H(w) = H(x) ­ y + x ­ H(y) – ( H(y) ­ x + y ­ H(x) ) = 0 - 0 = 0, X(w) = X(x) ­ y + x ­ X(y) – ( X(y) ­ x + y ­ X(x) ) = x ­ x – x ­ x = 0, Y(w) = Y(x) ­ y + x ­ Y(y) – ( Y(y) ­ x + y ­ Y(x) ) = y ­ y – y ­ y = 0.

We can pick a basis of inv(V(1)­ V(1)­ V(1)­ V(1)) as follows. U1 = x­ y­ z­ w - y­ x­ z­ w + y­ x­ w­ z -x­ y­ w­ z. U2 = x­ y­ z­ w - x­ z­ y­ w + w­ z­ y­ x -w­ y­ z­ x. U3 = x­ y­ z­ w - z­ y­ x­ w + z­ w­ x­ y -x­ w­ x­ y. From the chord diagram, one can find all actual invariant vectors in (V(1))­ 2n by looking at how x ­ y - y ­ x sits in.

eiej = ejei if |i - j| > 1 First we regard the disc as a square with n points on the left edge and n points on the right edge. We can define a product by juxtaposing squares which make it to well known n-th Temperley-Lieb algebras, TLn. It is generated by 1, e1, e2, …, en-1 with relations eiej = ejei if |i - j| > 1 ei ei+1 ei = ei (ei)2 = [2] ei

Because of non-cocommutativity, the switching map :V ­ W Because of non-cocommutativity, the switching map :V ­ W ! W ­ V given by x ­ y ! y ­ x is in general not an (equivariant) map between quantum group representations. Thus, there is no natural symmetric group action on invariant spaces with n tensor factors with n > 2. (There is often a braid group action.) However, the following operations exist and are natural: Tensor product: Inv(V) ­ Inv(W) ! Inv(V ­ W) Cyclic permutation: Inv(V ­ W) ! Inv(W ­ V) Contraction: Inv(V ­ V* ­ W) ! Inv(W)

A spider is an abstraction of a representation theory with these three operations. It is a collection of vector spaces, or perhaps modules or sets, to be thought of as invariant spaces, together with abstract operations called join, rotation, and stitch, to be thought of as tensor product, rotation, and contraction. It is both convenient and conceptually important to depict these operations with certain planar graphs. These graphs are called webs, hence the term ``spider''.

Rotation Join

Stitch

The combinatorial sl(2,C) spider, parameterized by a, is the list of web spaces, Wn, together with all rotation, join, and stitch operations. [Rumer,Teller,Weyl] If a = -2, then the isomorphisms n:Wn ! Inv((V(1))­ n) can be uniquely chosen to send the operations of join to tensor product, stitch to contraction, and rotation to cyclic permutation of tensor factors composed with negation.