greg kuperberg- spiders for rank 2 lie algebras

8/3/2019 Greg Kuperberg- Spiders for Rank 2 Lie Algebras

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C o m m u n M a t h P h y s 180, 109- 151 ( 1996) CommunicationsIMathema t i ca lPhysics Springer- Verlag 1996

Spiders for Rank 2 Lie AlgebrasGreg KuperbergD e p a r t m e n t ofM ath em at i cs, U n iversity ofC aliforn ia, D avis , C A95616, USAE - m a i l : greg@m ath ucdavis.eduReceived: 19 D ecem ber 1995/ Accepted: 26 J a n u a r y 1996

Abstract: A spider is an axiomatization of the representation theory of a group,q u a n t u m group, Lie algebra, or other group or group- like object. It is also known asa spherical category, or a strict, monoidal category with a few extra properties, orby several other names. A recently useful point of view, developed by other authors,of the representation theory of sl(2) has been to present it as a spider by generatorsa n d relations. That is, one has an algebraic spider, defined by invariants of linearrepresentations, and one identifies it as isomorphic to a combinatorial spider, givenby generators and relations. We generalize this approach to the rank 2 simple Liealgebras, namely A2, B2, and G2. Our combinatorial rank 2 spiders yield bases forinvariant spaces which are probably related to Lusztig's canonical bases, and theyare useful for computing quantities such as generalized 67- symbols and quantumlink invariants. Their definition originates in definitions of the rank 2 quantum linkinvariants that were discovered independently by the author and Francois Jaeger.

1. IntroductionO n e of the problems of classical invariant theory is to characterize, for all ^- tuplesV\,...,Vn of finite- dimensional, irreducible representations over C of a compactgroup G or simple Lie algebra g, the vector space of multilinear functions

/ : V x V2 x x Vn - >


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110 G Kuperbergcharacter theory. But it is also useful to consider operations on invariant spaces suchas tensor products and contractions. Even for finite- dimensional, simple Lie algebrasover 2. (There is often a braid group action.) However, the following operationsexist and are natural:

1. Tensor product:I n v ( F ) Inv(W) --> Inv(K W) ,

2. Cyclic permutation:I n v ( F J r ) - > I n v( J r K ) ,

3. Contraction:F * W) -> \ n \ ( W ) .

Since V and W may themselves be tensor products, cyclic permutation of twotensor factors yields cyclic permutation of n tensor factors, but not general permuta-tions. Also, contraction must be interpreted carefully, because in a quantum group,one must reverse order when taking duals: (V W)* = P 0 F *.

A spider is an abstraction of a representation theory with these three operations.I t is a collection of vector spaces, or perhaps modules or sets, to be thought of asinvariant spaces, together with abstract operations called join, rotation, and stitch,t o be thought of as tensor product, cyclic permutation, and contraction. It is bothconvenient and conceptually important to depict these operations with certain planargraphs. These graphs are called webs, hence the term "spider."

Another motivation of the spider operations is that they describe the entire equi-variant tensor category of representations of a group, Lie algebra, or quantum group.In general,

t h e tensor product of two homomorphisms can be defined in terms of tensor prod-uct and cyclic permutation of invariants, and composition of homomorphisms canbe defined in terms of tensor product and contraction. Contrariwise, the spideroperations can be defined in terms of tensor product and composition of morphisms.


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Spiders for Rank 2 Lie Algebras 111F o r this reason, spiders are sometimes defined as a type of (non- symmetric) tensorcategory. To construct such a category, one must divide the tensor factors of aninvariant space

I n v ( F ] ... Vn)i n t o smaller tensor products which serve as the domain and target of a space of mor-phisms. This involves arbitrary choices that are extraneous to most of the argumentsin this paper, so we will not usually treat spiders as categories.

In this paper, we will define certain spiders in terms of generators and relations,a n d we will show that they are isomorphic to the representation theories of ranktwo Lie algebras and the quantum deformations of these representation theories.These results generalize a well- known construction for A\ = sl(2) that first arose ina paper of Rumer, Teller, and Weyl [18], that was developed later by Temperley andLieb [22], and that was greatly developed recently by Jones, Kauffman, Lickorish,Masbaum, and Vogel [5, 6, 10, 12]. Moreover, Frenkel and Khovanov have recentlyestablished that the bases of invariant spaces that arise when one constructs the A\spider by generators and relations are dual to the canonical bases of Lusztig [11].We conjecture that a similar phenomenon holds in the rank 2 cases.

2. The Ax Spider and sl(2, F )Let V = F 2 be the defining representation of sl(2, F ), where F is some field ofcharacteristic 0. (The complex numbers C are a good choice for F .) Choose aparameter a e F . Since the vector space V is a self- dual representation, there existsa non- degenerate, invariant contraction operation : V 0 V > F . F or each , choosen points on the boundary of a disk in the plane, and let Bn, the set of basis webs,be the set of crossingless matchings of the n points. For example, if n is odd, Bnis empty, while B6 has the following 5 elements:

7* < > 7^

By convention, Bo has a single element, the empty disk. Let W n, the web space,be the vector space of formal linear combinations of elements of Bn with coeffi-cients in F .

Theorem 2.1. There exist isomorphisms n : W n > \mr{Vn).T h e o r e m 2.1 has few consequences in isolation; it only says that web spaceshave the same dimension as invariant spaces. By the Weyl character formula,dimlnvF ") is the number of non- negative lattice paths of length n in onedimension, and there is a standard combinatorial bijection between such paths and

crossingless matchings. (Idea: Both combinatorial sets are equivalent to balancedlists of parentheses of length n, such as " ( ( ) ) ( ) " when n = 6.)


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112 G KuperbergA rotation operation is the linear extension to a web space of rotation of basis

webs:

A join operation is a bilinear operation on a pair of web spaces. Given two basiswebs, their join is given by connecting their disks by a band:

A join operation extends bilinearly to arbitrary webs. Finally, a stitch operation isa linear transformation between web spaces. Given a basis web, its stitch at anadjacent pair of vertices is given by connecting the vertices by an arc:

If the result produces a closed loop, then strictly speaking, it is not a matching ofvertices on the boundary and therefore it is not a basis web. In this case, eraset h e closed loop and replace it by a factor of a, the element of F chosen at thebeginning:

The factor of a is a reminder that stitch is a linear operation whose value on a basisweb might be a non- basis web. A stitch also extends linearly from a family of mapBn - > W n- 2 to a map W n - + W n- 2The combinatorial A\ spider, parameterized by a, is the list of web spaces,together with all rotation, join, and stitch operations.Theorem 2.2 (Rumer, Teller, Weyl). If a = 2, then the isomorphisms n : W n I n v( 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 composedwith negation.


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Spiders for Rank 2 Lie Algebras 113( T h e difference in sign between rotation and cyclic permutation of tensor factors

is explained in Sect. 3.)T h e conversion of join to tensor product requires some explanation. Tensor prod-

ucts depend on a linear ordering of tensor factors, but the vertices in a web are onlycyclically ordered. To realize an explicit association between webs and invariant ten-sors, it is necessary to refine the cyclic ordering to a linear ordering. Then join ortensor product becomes a process of concatenation:

1 J o i n

T h e general operation of join corresponds to tensor product modified by cyclicp e rm u t a t io n s .We explicitly construct the n's to demonstrate their uniqueness. Assume somen o n - z e r o value r 6 V (g) V for >2 of a line segment:

r - \i i I = ,

Any other basis web is obtained from a line segment by repeated joins. Thereforeits image under n is, up to sign, a combination of cyclic permutations and tensorproducts of r's. For example, / \ of the basis web

H L./7*-\-J /while

where p4 is cyclic permutation of four tensor factors. It remains only to determiner. The stitch of two arcs is one arc:

Applying ^ and to the two sides yields

r) =


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114 G KuperbergT h i s e q u a t i o n says t h a t r is the i n v e r s e of in an i n d i r e c t s e n s e . M o r e e x p li ci t ly ,g i v e n a b a s i s {ei,e2} for F , one n a t u r a l c h o i c e for r and is:

0 e\) = (e2 0 e2) = 0 , 2 '

B u t in a m o r e d ir e c t se n s e, r is not the i n v e r s e of , b e c a u s e w i t h t h i s c h o i c e of ra n d , as w it h e ve r y o t h e r c o m p a t ib le p a ir of c h o i c e s , a c l o s e d l o o p , or (r\ is 2.

C h e c k i n g t h a t the n's t a k e s p i d e r o p e r a t i o n s to t e n s o r o p e r a t i o n s r e d u c e s toc o m p a r i n g d efin i t io n s and s im p l e c a l c u l a t i o n s . The l e a s t t r i v i a l p a r t of T h e o r e m 2.2is the a s se r t io n t h a t e a c h n is an i s o m o r p h i s m . Sin c e the d o m a i n an d t a r g e t of nh a v e the s a m e d im e n s io n , it suff ices to e s t a b l i s h s u r j e c t i v i t y . To p r o v e s u r j e c t i v i t y ,w e use the i s o m o r p h i s m

B y the F u n d a m e n t a l T h e o r e m of I n v a r i a n t T h e o r y of S c h u r an d Weyl, t he e n d o m o r -p h i sm s of Vn are s p a n n e d by p e r m u t a t i o n s of t e n s o r fa c t o r s. S u c h a p e rm u t a t io nc a n be d e p i c t e d by a d i a g r a m of m a t c h e d d o t s :

I t is a c o m p o s it io n of m a n y c o p ie s of the s w i t c h i n g map :xy\- ^y(&x inE n d ( F 0 F ). A c a lc u l a t io n d e m o n s t ra t e s t h a t l i e s in the i m a ge of $\

= ] ( I' v VIt follows by multilinear expansion that every permutation is in the image of inSince the permutations span, 2n is surjective.What if a is not 2? The specialization of the spider should describe the repre-sentation theory of some object related to sl(2, F ) . The quantum group Uq(sl(2, F ))was invented essentially for this purpose. One parameterization of the A\ spider isa q l2 q~x/ 1 over the field


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Spiders for Rank 2 Lie Algebras 115n closed loops denotes the web which is an times the web obtained by erasing allclosed loops.

Given this meaning for all embedded 1-manifolds, basis webs such as

W =

ca n be embedded in larger disks to form 1-manifolds which are therefore otherwebs:

= a (1)

If the Wi's are arbitrary webs, t h e n a diagram such as the one in Eq. (1) denotes amultilinear expansion and is called a compound web. For example, if

'w) = 2[) (j +3{*t h e n

Another view of a compound web is that it is a sequence of joins and stitches oft h e component webs. It can be realized by many different such sequences, but theyall have the same final value.


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116 G Kuperberg

An important class of examples of compound webs are those generated by aweb called a crossing:

: X : = - y < 8 > n

Since the composition is a non- zero idempotent, it is the equivariant projection fromVn to its highest- weight irreducible summand.

2.3. Isomorphism and equnumeration. In this section, we prove Theorem 2.3. It isfairly easy to construct each N and to prove that is surjective. If n = n\ - \ h ,there is a projectionand let N = KN n *V> where i^ is the inclusion W(N) C Wn. Suppose thatt h e / th clasp of a web w has k tZ- turns, and let m = n 2k. Then the invariantK N n(w) l i e s in the image of some map which has a tensor factor of the formVm > Vnn where m < ni9 and any such map must be zero.Thus, % N n = N J'N, where j ^ is the projection Wn > W{N). From this,it is routine to show that the maps N take spider operations to spider operations.Moreover, each ^ is surjective, because both ^ and n are surjective and ncomplements the kernel of N o n.


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120 G KuperbergWe complete the proof that N is an isomorphism by demonstrating that itsd o m a in and target have the same dimension, thereby generalizing Theorem 2.1 to

t h e clasped case:Theorem 2.4. If N is a multi- index, then

dim W(N) = imlrw(VN).Proof. The proof is by induction on \N\, the number of indices in N. The relationis straightforward for \N\ ^ 2 , so we first assume that \N\ = 3.T h e Clebsch- Gordan theorem states that

d im I n v ( F ; Vj Vk) = 1if each of n, rn, and / is less than or equal to the sum of the other two and if thesum of all three is even, and

d im I n v (F ; Fy Vk) = 0otherwise. On the other hand, B(i,j,k) has at most one element:

This web exists when there are non- negative integers x, y, and z such that i = x + y,j = x + z, and k = y + z. These two conditions on /, j , and A: are equivalent.

Now suppose that \N\ > 3 and express the multi- index N as JK, where / and Ke a c h have length at least 2. Since the Vn's constitute all finite- dimensional irreduciblerepresentations, and since they are self- dual, there is a decompositionlnv(Vj V ) = 0 I n v ( F j 0 V{) 0 I n v ( F / K^) ,/

where / is a single index rather than a multi- index. It suffices to establish a bijectionf : B(J,K) ^ \J (B(J,l) x B(

Iwhere the union is disjoint. The bijection / is very easy to define: A basis web ofw G W(JK) has a minimal cut path, where a cut path is a path whose endpointsseparate J from K. A cut path is minimal if it crosses as few strands as possible:

Let w\ and W 2 be the two resulting webs. The cut path crosses some / strands, andsince it is minimal, there can be no U- turns among the / strands in either wi or H > 2 .


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Spiders for Rank 2 Lie Algebras 121T h u s , the relation f(w) (w\ ,W2) defines the desired map / . It is routine to checkt h a t / is both injective and surjective. D

N o t e that minimal cut paths are also useful in practice for generating the basisof a clasped web space.

Exercise 2.5. List the elements of 5(2,2,2,2,2).

3 . Precise Definition of a SpiderIn this section, we give a precise definition of a spider. The main way in whicht h e general notion of a spider is more complicated than the example is that strandsmay be oriented and there may be more than one type of strand. Indeed, even int h e A\ case, we may consider n parallel strands as equivalent to one strand labelledwith n, as is already suggested by the notation.

A spider has a strand set S which is a unital semigroup with unit 0. In mostof the examples in the paper, S is a free, non- abelian semigroup. The strand seth a s an anti- involution * : S > S called duality or orientation reversal. For eachs ( S, there is a web space W(s), which may be just a set. For each , there isa distinguished web a G W(aa*), called a bare strand, and 1 = $ is called theempty web. Finally, there are three operations that exist for every a and b:

1. Join â,b W( ) x W{b) - > W{ab).2. Rotation pa b : W(ab) - > W(ba).3. Stitch a,b :' W{aa*b) - > W{b).T h e subscripts of the operations may be dropped when they are clear from context.A spider may in addition be defined in a (symmetric tensor) category other thant h e category of sets. For example, an additive spider is one in which web spacesa re abelian groups, rotation and stitch are additive, and join is additive, while ina linear spider, the web spaces are vector spaces and the operations are linear orbilinear.

T h e three spider operations must satisfy the following axioms, which aredivided into groups according to which operations they involve. We define b,a,c=Pc,b a,cbPb,aa*c fr brevity in the last three axioms. Rotation only:

l P ,0 = P ,a = id.2. pa,bcPc,abPb,ca= id.3 . Pa,aâ,b V) = V DX U Rotation and stitch:9. Ga,bd aa*b,c,d = ^b,c,dâ,bcc*d

10. j = *,0P ,*


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122 G Kuperberg Join, rotation, and stitch:

12. Ob^ c(u \X*ba,a*c V) = pb,c(Pa*,c(V) ^ca\ab Pb,a(u))The formal spider axioms are embarrassingly complicated, but they can be

phrased in more natural (if less formal) terms. In particular, the following com-pound web principle is equivalent to axioms 1-7 and 10- 12. To state the principle,we first define a pre- spider to be an algebraic object satisfying these 10 axioms, butn o t necessarily axioms 8 and 9.

Let U be the free semigroup generated by a set X, and let * be any involutionof X extended to an anti- involution of U. Let Lu be an abstract set of labels foreach u G U and let L be the disjoint union. Consider a graph G in a disk such thatt h e vertices of G on the boundary are univalent, such that each edge is labelled bya n element of x G X and oriented unless x = x*9 and such that one of the vertices att h e boundary is distinguished as first, and such that one edge of each internal vertexis distinguished as first. The edges incident to an internal vertex v are then linearlyordered going counterclockwise around v; let xz be the label of the / th edge e of v ift h e edge is unoriented or oriented outward, and let Xf be the dual of the label of eif it is oriented inward. Each v should be labelled by an element of LX Xn. Then wedefine the web space W(x\ xn) as the set of all graphs G with boundary labelledx,...,xn (following the same convention of taking the dual when an edge at theboundary is oriented inward), considered up to isotopy and up to the modificationof reversing an edge and dualizing its label.

We define the free pre- spider Sf(X,L) by defining the spider operations in thesame way as for the combinatorial A\ spider: Join is given by band- connected sum,r o t a t i o n is given by changing which boundary point is first, and stitch is given byconnecting two adjacent boundary points by an arc. A tedious but straightforwardc o m p u t a tio n demonstrates that 6f(X,L) satisfies axioms 1-7 and 10- 12. Conversely,suppose that an algebraic object f consists of a strand set S and a collection ofweb spaces {W(s)} with operations of join, rotation, and stitch. Then the compoundweb principle stipulates that any *- preserving map / :X >S and any set of maps{L u W(f{u))} extend to a morphism f preserving join, rotation, andstitch. The compound web principle is equivalent to the statement that f is a pre-spider. Informally, in a pre- spider y7, if a sequence of joins, rotations, and stitchesare denoted by a planar diagram whose connecting arcs are the stitches, the resultingweb depends only on the diagram and not on the order of the individual operations.The remaining two axioms for f can be understood as follows: Axiom 8 says thatif a disconnected component of a compound web is moved from one face of theremaining part of the web to another, it does not change the value of the web.Axiom 9 says that the value of a boundaryless compound web depends only on itsembedding in the sphere and not on its embedding in the disk.

Although the above definitions are not completely standard, a spider can alsobe defined in category- theoretic terms. A spider is a (small) strict monoidal cate-gory, which is a category with an associative but not necessarily commutative tensorproduct . Moreover, a spider must be pivotal (or rigid), which means that thereis a canonical isomorphism F** = V, and spherical, a condition which is equiva-lent to axiom 9. See Barrett and Westbury [1] for a careful exposition of sphericalcategories. Given any spherical category, the strand set of the corresponding spideris precisely the set of objects of the category, and its web space W(V) is definedas In v(F ) = Hom(77,V), where T is the trivial object, which serves as the identity


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Spiders for Rank 2 Lie Algebras 123of the tensor product operation. The spider operations are t h e n defined in terms ofmonoidal category operations in the same way as for the algebraic A\ spider.

We review the spiders defined so far in terms of these definitions. The strand setof the combinatorial and algebraic A\ spiders is a free semigroup with one genera-t o r ; for G Z ^o, a strand with strand type n is synonymous with n parallel strands.The clasped combinatorial A\ spider is a different object and its strand set is a freesemigroup with countably many generators, namely the different clasps.

It is useful to treat an additive or linear spider or pre- spider as a ring, and toconsider the usual constructions with rings such as morphisms, ideals, and quotients.An ideal / i n a linear pre- spider f is a collection of linear subspaces I(s) c W(s)which are closed under rotation and stitch and closed under join with an arbitraryweb in f. Clearly, if J>is an ideal, the quotient spaces W(s)/ I(s) form a pre- spiderf/


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124 G Kuperberg4. The Combinatorial Rank 2 Spiders

F o r convenience, let C with # e (C or (C(q,q/2,q/ 3,...) be the ground field. The un-clasped combinatorial rank 2 spiders describe the invariants of tensor products of thetwo fundamental representations V( \) and V( 2) of each of the Lie algebras A2, B2,a n d G2. These two representations are duals of each other for A2 and are self- dualin the other two cases. The strand set for the combinatorial A2 spider is defined as thefree semigroup of strings of symbols" + " and "," which correspond to V( \) and V( 2), respectively. The dualof a sign string is given by reversing the string and flipping the signs; forexample,

In the B2 and G2 spiders, the strand set is the free semigroup of strings of self- dualsymbols " 1 " and "2, " so that duality is just string reversal.Given a sign string s s\ sn, define the A2 basis web set B(s) to be theset of non- elliptic, bipartite, trivalent graphs properly embedded in a disk with

boundary points labelled s\,...,sn in counterclockwise order. By a trivalent graphproperly embedded in a disk, we mean a 1-dimensional subset of the disk lo-cally modelled by the following five allowed neighborhoods of a point in thedisk:

T h e allowed neighborhoods might be called empty disk, strand, trivalent vertex,empty boundary, and endpoint. Such a graph is bipartite if its endpoints are signeda n d its edges are oriented in such a way that the in- degree at each vertex is either0 or 3, and such that edges point towards positive vertices and away from negativeo n e s . Finally, such a graph is non- elliptic if all internal faces have at least six sides,where an internal face is a component of the complement of the graph that doesn o t touch the boundary of the disk. These graphs, henceforth called basis webs, areconsidered up to isotopy relative to the boundary. For example, the 6 elements ofB(+ + + ) are

+ + + + + + + +


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Spiders for Rank 2 Lie Algebraswhile

125

is an element of B(- \ 1 ).T h e vector space of formal linear combinations of elements of B(s\ - sn) is the 2 web space W(s\ sn). Partly elliptic, bipartite, trivalent graphs in a disk, willd e n o t e webs also, although not basis webs. Specifically, each type of elliptic face

is defined as a linear combination of basis webs according to the rules

= [3]+ =-[2]--*-+ ,

) ( : >< (2)

T h e value of a larger graph which contains an elliptic face is inductively definedby the same rules. For example,

^ L

+

CO( H e r e and below, we may omit orientations of edges when they are clear fromc o n t e x t . ) Rotation, join, and stitch operations are defined in the usual graphicalway: For basis webs, rotation is rotation of the disk, join is band- connected sum,


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126 G Kuperbergand stitch is the operation of connecting two adjacent boundary points by anarc. Stitch is only defined when the two adjacent boundary points have oppositesign, to preserve the orientation structure of A2 webs, and it may produce ellipticfaces, which must be reduced to obtain a linear combination of basis webs. Com-pound webs, and in particular concatenation, are also defined either by extensionof the three basic operations, or directly by the principle of reduction of ellipticfaces.Another way to phrase the definition of the A2 spider is that it is generated bythe two webs

/y

with Eqs. (2 ) as relations. Similarly, the A\ spider is trivially generated with thesole relation that a closed loop yields a factor of a.The B2 and G 2 spiders are also most conveniently defined by generators andrelations. The B2 spider is generated by the single web

- rwith the relations

=0-0,

= 0 ,

-X-X-M.


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Spiders for Rank 2 Lie Algebras 127where a strand is denoted by a double edge if its strand type is "2" and by a plainsingle edge if its strand type is "1 . " The G2 spider is generated by the webs

xwith the relations

= q9+q6+ q5+q4+q3+ q + 2 +q~+q~3+q~4 + q~5+q~6

= - (g3+ q2+ q + q~2+ q~3)

X = "fl2l+9-2 / ' % + 1 + -1 / \ (4 )

To form a basis, we first note in the G2 case that the last relation allows theelimination of all internal double edges, leaving only those with at least one vertexat the boundary. In the B2 spider, we define a tetravalent vertex to achieve the sameend:

X- X- M


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128 G KuperbergThe tetravalent vertex then satisfies the relations

rIn both spiders, we can say that non- elliptic webs with no internal double edges area basis, provided that we define formal angles of

for the vertices in the B2 spider and angles of

in the G spider, and we declare that a face is elliptic if its total exterior angle isless than 360 degrees. For example, in the G2 spider, the pentagon

is elliptic, but the pentagon

is flat rather than elliptic.If we understand the rank two spiders in terms of generators and relations, then

it is clear that non- elliptic webs linearly span all webs, but it is not obvious that


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Spiders for Rank 2 Lie Algebras 129they are linearly independent. Alternatively, if we understand the rank two spidersin terms of the non- elliptic bases, then it is not clear that the elliptic reductionequations are consistent. Put a third way, do two different reductions of a partlyelliptic web always give the same linear combination of non- elliptic webs? Thea u t h o r [9], and independently Jaeger [4], established that the given coefficientsa r e , up to trivial normalization, the only choices for which the equations areconsistent.

Rank 2 spiders also admit crossings that lead to link invariants, but the linkinvariants will not be discussed in this paper. There are two types of crossings int h e A2 spider:

= q

* /

four types in theB2 spider:

\ . .1/2' \

X -

1/6

= q-1/6

, -1 /2

q 2+q- m

= q

-1/3 )(

,,-1/2

^ X

and four types in the G2 spider

\-1/2

\ qu2+q- m~ - 3 / 2 X

qm+q-m

,1/2 Xm ) I


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130

y = q- 3 / 2qu2+q- m / \

G Kuperberg

\, 3 / 2 , , - 3 / 2 \

7 - 1

N o t e that crossings are the first webs whose coefficients are not symmetric in qand q~x.4.1. Clasps and clasped web spaces. The rank 2 spiders also admit clasps andclasped web spaces. As before, the type of a clasp is the same as an unclaspedstrand type, but in the rank 2 cases there are many more combinations. Our notationwill be that if s is an unclasped strand type, c = [s] is the corresponding clasp. Wewill also consider sequences of clasps C = c\C2 cn = [s\][s2] [sn].Clasps and clasped web spaces in the A2 spider are the easiest to describe: Definet h e weight wt(s) of a sign string s n \ - \ - k 2 if there are n plusses and k minuses. Re-call the usual partial ordering of the weight lattice of lattice of A2: it is generated by

a i + b 2 y (a + l) i + (b - 2) 2 ,a \ + b 2 y (a - 2) x + (b + \ ) 2 .

There is a clasped web space W{C) for each possible clasp sequence C. For example,W([- \ h][I][h]) denotes a clasped web space. The basis 5([^iof JF([si][s2] [sk]) is a subset of the set of unclasped basis websconsisting of those non- elliptic webs with non- convex clasps. Here an (external)clasp is non- convex if it has the property that the weight of any path between end-points of the clasp that is transverse to the web (a cut path) is greater than or equalt o the weight of the clasp; the weight of such a path is defined as the number ofstrands that cross it in the direction of the clasp and the number that cross awayfrom the clasp. For example, the following web has a partly convex clasp, becauseits weight is 2 \ + 2, but there is an arc with weight 2 2 that cuts off the clasp:


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S p i d e r s f o r R a n k 2 L i e Algebras 131I t i s t h e r e f o r e n o t a basis w e b o f W ( [ - \ \ - ][- \ h ][H h ] ) . C l a s p e d w e b s p a c e sc a n a g a in b e in t e r p re t e d a s b o t h q u o t ie n t s a n d su b sp a c e s o f u n c la s p e d w e b s p a c e s.A n y w e b w it h a t le a s t o n e p a r t ly c o n ve x c l a sp i s u n d e r s t o o d a s t h e ze r o ve c t o r . B yt h i s c o n ve n t io n a n d E q s . ( 2 ) , a n y t r iva l e n t gr a p h in a d i sk w it h su i t a b le b o u n d a r yc a n b e i n t e r p r e t e d a s s o m e ve c t o r in t h e c la s p e d w e b s p a c e .

T h e n o n - c o n ve xit y c o n d it io n fo r c la s p s m a y se e m u n n e c e s sa r i ly s t ro n g. O n em i gh t a l t e rn a t i ve ly s t ip u la t e t h a t e ve r y c u t p a t h c r o s s a t le a s t a s m a n y st r a n d s a st h e n u m b e r o f st r a n d s in t h e c la s p , o r t h a t t h e we igh t o f n o c u t p a t h b e s tr ic t ly lesst h a n t h e w e igh t o f th e c la s p . H o w e v e r, b y L e m m a 6 .5 , t h e t w o c o n d it io n s a r e b o t he q u iva l en t t o t h e o n e give n . S a y t h a t a c u t p a t h o f a we b is minimal i f i t s w e i g h t i sm i n i m a l w it h r e sp e c t t o t h e p a r t ia l o r d e r in g. T h e n in p a r t ic u l a r , L e m m a 6 .5 im p lie st h a t a ll m in im a l cu t p a t h s w it h a fixed p a i r o f e n d p o i n t s h a v e t h e sa m e w e igh t , t h em i n i m a l c u t w e i g h t .

A s u s u a l , jo i n a n d ro t a t io n in t h e c la s p e d A2 s p i d e r a r e s t r a i g h t f o r w a r d , b u t s t i t c hi n v o l ve s i n t e r n a l c l a s p s . A n internal clasp of type s is d e fin e d a s a n i d e m p o t e n t i nt h e u n c la s p e d w e b s p a c e W (s*s) t h a t a n n ih ila t e s a n y w e b i n W (s*t) w i t h wt(t) - - ( a + 2) 4- (ft - 2) 2 . ( 6 )


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132 G KuperbergThe clasped G2 spider has a more significant difference. A cut path may containa type " 1 " strand in its interior, and its weight is n ] + (k + k f) 2, where n is thenumber of type " 1 " strands that it cuts, k is the number of type "2" strands that

it cuts, and k' is the number of type " 1 " strands that it contains. For example, thefollowing cut path has weight \ + 2 2:

As before, the webs with non- convex clasps, where non- convexity is defined usingt h e partial orderinga x + b 2 y (a- 2) x + (b + l) 2 ,a +b 2 y (a + 3) + (b - 2) 2

in the G2 weight lattice, form a basis of each clasped web space. The more sig-nificant difference is that a basis element of the unclasped web space with a con-vex clasp is not necessarily zero. Rather, if a web, whether non- elliptic or not,has a cut path which cuts off a clasp, which does not contain any type " 1 "strands and whose weight is less than that of the clasp, then the web is zero. Forexample,

I t may not be immediate that the kernel of this quotienting operation comple-ments the subspace defined as the clasped web space. This will be shown inSect. 6.2.This concludes the definition of the combinatorial rank 2 spiders and the defi-n i t i o n of clasped web spaces. Only the operation of stitch, which depends on theexistence of internal clasps, remains to be fully defined.

5. The Morphism from Combinatorial to AlgebraicLet V+ and F_ be the two fundamental representations of A2 = sl(3), with F_ =( F + ) *. Let V\ and V2 be the two fundamental representations of B2 = sp(4) =so(5), and give the two fundamental representations of G2 the same names with


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Spiders for Rank 2 Lie Algebras 133d im V\ < dim V2 in both cases. Then the vector spaces

\nvBl(Vx V 0 F 2 ) ,

In G2(V{ 0 Ki F 2)

are all 1- dimensional. When q = 1, and for each of ^2? #2> and G2, there existsa morphism from the unclasped combinatorial spider to the unclasped algebraicspider with the following property. If s is a generator of the strand set, then (s) Vs, and if T is a trivalent vertex of some type, then (T) is a non- zero elementin one of the above invariant spaces. We sketch the argument for the existenceof in the A2 case (see Ref. [9] for details): Pick any two non- zero elementsx G InvAl(Vf3) and x* G Inv^2(T 3). By counting dimensions of invariant spaces,e a c h of the left sides of Eqs. (2), (3), and (4) must be some linear combinationsof the right sides in the algebraic A2 spider, if x and x* are denoted by the usualtrivalent vertices. But at the same time, a computation shows that the right sidesare linearly independent in the algebraic A2 spider, and that, up to normalization ofx and x*, the given coefficients are the only ones that respect this normalization.T h u s , after rescaling, the invariant tensors x and x* of Uq(A2) must satisfy therelations of the combinatorial A2 spider, although perhaps with a different choiceof q. Another simple computation shows that the choice of q is the same. Thuswe can set (T) =x and (T*) =x*, if T and * are the trivalent vertices thatgenerate A2.Theorem 5.1. The morphism from the combinatorial to the algebraic A2, B2,or G2 spider is surjective when q = 1, and therefore for generic q.

This theorem is proved in Ref. [9], but we give a more conceptual argumenth e r e :Proof Let q = 1, let g be A2, B2, or G2, and let G be the compact, simply-c o n n e c t e d Lie group whose complexified Lie algebra is 9. If s = s\ sn is a string,define

y _ y


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134a n d signs:

G Kuperberg

T h e morphism intertwines the combinatorial Hermitian adjoint with the usualH e r m it ia n adjoint in the algebraic spider. Since End^CK?) is closed under Hermitianadjoint, it must be semisimple.

T h e category 3C does not contain the kernels and co- kernels of its morphisms,but it may be completed to a bigger category S\ in which 9Cis a full subcategory,by adding these vector spaces as new objects. The category C' then satisfies thehypotheses of the Tannaka- Krein duality theorem [8, p. 177], and must be the cat-egory of finite- dimensional representations of some compact group H. On the oneh a n d , H C G, since everything in 9C1 is invariant under G. On the other hand, Hc a n n o t be any bigger than G, because for each choice of g, G is a maximal compactsubgroup of the symmetry group of (t) for a vertex t. (For example, the symmetrygroup of a non- zero element of I n v # 2( F i V\ V2) is sp(4, ( C ) , with maximal com-p a c t subgroup Spin(5).) Therefore C'is equivalent to the representation categoryof / (g), and 9Ccoincides with the algebraic g spider. D

I t is more difficult to show that is injective. We will prove this in Sect. 6by demonstrating that the clasped web spaces and the web spaces of the claspedalgebraic spider have equinumerous bases. If this is so for clasped web spaces,t h e n it is also true for unclasped web spaces, which demonstrates that is anisomorphism between unclasped spiders. We can then define an internal clasp as ~ 1 ( ) , where is the highest- weight projection from any strand in the algebraicg spider to itself. This completes the definition of the clasped spiders, provided weverify the following lemma to show that maps stitch to contraction.Lemma 5.2. Suppose that s and t are strand types and w G W(st) is zero in theclasped web space W([s]t). Then w is annihilated by an internal clasp of type [s].Proof. We assume the injectivity of for unclasped web spaces, and we assumet h a t w e B(st) is a basis web. Let be the weight of s, and let w' be w with ani n t e r n a l clasp attached. The tensor (w) may be interpreted as a homomorphism, inparticular as a composition Vs > V( ) Vu > Vt9 where u is the transverse strandtype of a minimal cut path that separates s from t in w. By hypothesis, the weightof u is lower than that of s. Therefore any map V( )vanish. D

Vu must vanish, so wr must

T h e following result is then a corollary of the discussion of this section:Theorem 5.3. The morphism from the combinatorial to algebraic rank two spi-ders extends to clasped spiders.


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Spiders for Rank 2 Lie Algebras 1356. Equinumeration6.1. The A2 caseTheorem 6.1. Let C be an A2 clasp sequence and let wt C be the correspondingsequence of weights. Then the vector spaces W(C) and Inv(F(wt C)) have thesame dimension.

O n e of the main steps of the proof of Theorem 6.1 is the same as that of theproof of Theorem 2.3: Given two clasp sequences C and D, there is a decompositionof vector spaces

I n v ( F ( w t C) 0 V(wtD)) ^ 0 Inv( (wtS) 0 V()) 0 Inv( ()* 0 V(wtD)) .

We wish to prove the corresponding decomposition of sets of basis webs:Theorem 6.2. Given two clasp sequences C and D,

B(CD)^{J(B(Cc)xB(ctD)),

where for each weight , c is some clasp with weight .T h e o r e m 6.2 is more complicated than its analogue for the A\ spider, because

t h e r e is no longer always a unique minimal cut path separating the clasp sequencesC and D:

Although the two cut paths in this example have the same weight, they have differenttransverse strand types and they yield webs in different clasped web spaces. Yet notevery ordering of the strands is always possible. Nevertheless, there is a way toreconcile these different decompositions and ameliorate the ordering problem.

I n d e e d , order independence is a large part of the combinatorial content ofT h e o r e m 6.1. For example, it implies the following result, which has an independentproof that is another warm- up to the proof of Theorem 6.1.Theorem 6.3. If two sign strings s\ and s2 of the unclasped A2 spider differ onlyin order, then B(s\) and B(s2) are equinumerous.Proof. The web

is the H- web. It suffices to consider the case in which s\ and s2 are the same exceptfor one pair of transposed signs at adjacent positions p and q. Then there is a


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136 G Kuperbergbijection h : B{s\ ) fe ) t h a t has the following effect on non- elliptic webs. If aweb w connects p to q by a bare strand, /*(w) is the same web with the orientationof the strand reversed:

If there is an H- web attached at p and q, h{w) is w with the H- web removed:

If there is no H- web, t h e n h{w) is w with an H- web attached:+ +

I t is easy to check t h a t h(w) is non- elliptic if w is, and t h a t h has an inverse. Infact, the inverse is also an /z-map. D

In the n o t a t i o n of the above proof, we also say t h a t w and h{w) differ by anH- move.Lemma 6.4. If a is a minimal cut path of w G B(ST) separating S from T, itdivides w into two parts w\ G B(Sc) and W 2 B(c*T) with non- convex clasps cand c*.Proof If c were convex in w\ , w\ would have a cut p a t h ' whose weight is lowert h a n t h a t of c, the same as the weight of . But a' is also a cut p a t h in w and hast h e same endpoints as , contradicting the hypothesis t h a t is minimal:

C -

Lemma 6.5. If and are cut paths from p to q of a basis web w and isminimal, then the weight of a is less than or equal to {and not incomparable to)


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Spiders for Rank 2 Lie Algebras 137the weight of . If is also minimal, the two parts of w cut by are the sameas those of w cut by up to H- moves.

Proof The proof is by induction on the complexity of w; assume that w is aminimal counterexample. Assume also that, having chosen w9 and are transverseand intersect minimally. First, we can discard any structure of w not bounded by U ; the new web has no clasps and is non- elliptic since w is non- elliptic. Second,we claim that and do not intersect except at their endpoints. Otherwise, let 'be a segment of between two consecutive intersections x\ and x, and let ' bet h e arc of from x\ to X2 (which are not necessarily consecutive along ). Then theregion between ' and ' is either empty, in which case ' and ' do not intersectminimally; or it constitutes a smaller counterexample than w for some choice of ';o r j '; or and have comparable weight:

x=pA : 1 '

\

Third, w must be connected, for if one of its connected components meets butn o t /?, is not a minimal cut path; if one of its components meets but not , itmay be discarded to produce a smaller counterexample; and if all components meetb o t h and , one of them is a smaller counterexample:

Finally, if e\ and 2 &re adjacent endpoints of w, define the exterior curvature oft h e arc e~e~i to be 180 60, where n is the number of vertices of w connectinge\ to e2:

The total exterior curvature of w is at least 360 since w is non- elliptic. Moreover,t h e curvature at the arcs containing p and q is at most 120, for if it were 180,either w would be disconnected or it would be a single strand. Therefore there must


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138 G Kuperbergbe a segment y of either or with positive curvature. There are three possibilitiesfor the edges of w that bound a face together with :

1 2 3

1. A "U ." In this case, w is either disconnected or a single strand.2. A "Y." If lies on , then is not minimal. If y lies on /} then an isotopyof across the "Y" produces a smaller counterexample.3. An . " In this case, an isotopy of either or produces a smaller coun-terexample by an H- move.

This eliminates all possibilities for the least counterexample. DLemma 6.6. Let C be a sequence of clasps, let c be an arbitrary clasp, and letw e B(Cc) be a basis web. There is a cut path y that separates c such that anyother such cut path lies between y and c.Proof. If and are two transverse, minimal cut paths that separate c and pa n d q are two consecutive transverse intersection points along either path, then theweight of the arc of from p to q must equal the weight of the arc of from pt o q, for otherwise one path would provide a short- cut for the other and either or would not be minimal. Thus the path y that follows the perimeter of U mustalso be minimal:

If we partially order cut paths by their distance from c, there is a unique maximalelement. D

Given w as in Lemma 6.6, define the core of w relative to c to be the webW G B(Cc') obtained by cutting away c along the cut path y guaranteed by thelemma. Here c' is another clasp with the same weight as c. Let B(C; ) be the setof all cores of webs w e B(Cc) for all clasps c of weight . Note that, since c isnon- convex in w, one of the minimal cut paths that separates c is parallel to c.Therefore, by Lemma 6.5, w differs from its core by H- moves. Note also that thecore wr has the property that no H- webs are attached at c1', and that any web withthis property is its own core.

We describe how an arbitrary basis web extends from one of its cores. Considert h e following stair- step construction of a basis web from a core. Suppose thatw e B(C[s]) is a core relative to a clasp [s] of weight = a \ + b i, and supposet h a t |V] is an arbitrary clasp of the same weight. Then the strings s and s1 each


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Spiders for Rank 2 Lie Algebras 139represent paths p and p' from the the upper left comer to the lower right comer ofa n a x b rectangle, where each " + " in s or s' is a step to the right and each ""is a step down:

s' = - + - + - + + -

+P'

+\

1 J - .

+

~ +

P-

+

If an H- web is placed in each square as indicated, the two paths p and p' delineatea sequence of connected webs, separated by points or segments where p and p'meet and may or may not cross. We attach each web bounded below by p to w,a n d we invert and reverse the arrows of each web bounded above by p and thena t t a c h it to w. The result is a web wf e B(C[s']) with core w:

If s\ and 5*2 are arbitrary strings of the same weight, we can set s' to either int h e stair- step construction, which implies that any core w of a web in B(C[s\]) isalso a core of a web in B(C[s2]). Moreover, it is easy to check that, if w is fixed,t h e collection of webs so produced is closed under H- moves. Since every basis webis related to its core by H- moves, the stair- step construction produces all webs inB(Cc) for all c with weight . In particular, we have demonstrated the followinglemma:Lemma 6.7. Let C be a sequence of clasps and let c be a clasp of weight .If we B(C[c]\ let f{w) be its core. Then the map f : B(C[c]) - > B(C ) is abijection.

Using the above lemmas, we can prove Theorem 6.2:Proof Let w\ e B(C; ) and W2 G B(D; A*), and let c\ and C2 be the clasps of w\a n d W 2 of weight and *. Then we can either extend w\ to a basis web in B{Ccl)o r extend W2 to a basis web in B(Dc*) and then sew the two webs together; by thesymmetry of the stair- step construction, the resulting web w G B(CD) is the same


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140 G. Kuperbergin both cases. Define

/ : U (B(C; ) x B(D; )) -+ B(CD)

by the above operation on pairs of cores. By Lemma 6.5 and the fact that H- movespreserve cores, there is also a map

g : B(CD) - (J (B(C; ) x B(D; *))

given by splitting w along a minimal cut path and taking the cores of the twohalves. By Lemma 6.7, / and g are inverses. Using Lemma 6.7 again, we can seeg as the injection claimed by the theorem. D

T o prove Theorem 6.1, we need one additional lemma. Say that a sign string sis segregated if it is of the form

+ + . . . +a sign string with only + 's or only 's is automatically segregated. Likewise, sayt h a t a clasp is segregated if it is of the form [s] for a segregated sign string s.Lemma 6.8. Let c and d be segregated clasps of weight and . Then the webbasis set B([- \ - ]cd) has one element if and only if * = + \ 9 * = + \, or * = + -2 \ 9 and is empty otherwise.Proof Consider first three segregated clasps c9 d, and e (one of which might beempty) of arbitrary weight, and let w G B(cde). We claim that w must consist of an u m b e r of bare strands plus a flat component, as in the following example:

T h e argument is similar to the proof of Lemma 6.5. Assume first that w is notc o n n e c t e d . If p and q are adjacent endpoints of w, we define the curvature of thesegment of the boundary of w from p to q as in Lemma 6.5. Since w is non- elliptic,it must on the one hand have total exterior curvature at most 360. On the otherh a n d , there can be no "U " or "Y" attached along c, d, or e, and in there is onlyo n e place along each clasp where an H is possible. The curvature at this H, if itis present, is at most 60, and the curvature elsewhere along the clasps is at most0. Moreover, unless w is a bare strand, the curvature at the segments between theclasps is at most 60 also. The largest possible total is 360 exactly, so that w is


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Spiders for Rank 2 Lie Algebras 141flat a n d its boundary is qualitatively a hexagon:

If w is not connected, then the above argument applies to each connected componentof w, with the additional observation that at most one component of w can meet allt h r e e clasps c, d, and e. A component that only meets two clasps is necessarily abare strand.In the case of interest, e = [+ ]. In this case, w might consist entirely of barestrands, or it might have a component in the shape of a parallelogram or a trapezoid:

C cThese possibilities exactly match the restrictions on the weights of c and d. D

Finally, we prove Theorem 6.1:Proof. Let + be the set of all weights = a \ -f b 2 with , Z^ 0 Con-sider the abelian group %[ +] of formal sums of elements v( ) for each e +.T h e n we can define a product in Z[A+] using the dimensions of the clasped webspaces:

where in the sum each C{ is some clasp of weight (. By Theorem 6.2, there existbijections

U (B(c,c2d) x B(c3c4d*)) 9 B(Cc2c3c4) 9 (\jB(c2c3d) x B(c4Cd*)) .d \ d /These bijections imply that multiplication is associative, and therefore Z[A+] isa ring. To establish Theorem 6.1, it suffices to check that the map v( ) \ > V( )induces an isomorphism from Z |yl + ] to the Grothendieck ring of A2. Using induc-t i o n , it suffices to check that for all e A+,

v(0)v( ) = v( ),v( )v( ) = v( + ) 4- v( - x + 2) + v( - 2),v( 2) v( ) = ( + 2) + v( - 2 + ) + ( - ),


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142 G Kuperbergwhere v( ) is defined as 0 for $ +, because similar relations hold in theG r o t h e n d i e c k ring. (Note that we cannot argue from an a priori hypothesis thatZ[ +] is the Grothendieck ring of a category, because such a construction assumest h e existence of internal clasps, which in turn depends on Theorem 6.1.) In termsof webs, the first relation states that if c and d are two segregated clasps, thenB(cd) is empty unless c = d*, in which case it has one element. This follows fromLemma 6.5 or from arguments similar to those of Lemma 6.8. Similarly, the otherrelations are equivalent to Lemma 6.8. D

6.2. The B2 and G2 cases

Theorem 6.9. Let C be a B2 clasp sequence and let wt C be the correspondingsequence of weights. Then the vector spaces W{C) and Inv(F(wt C)) have thesame dimension.Theorem 6.10. Let C be an G2 clasp sequence and let wt C be the correspondingsequence of weights. Then the vector spaces W(C) and Inv(F(wt C)) have thesame dimension.

T o establish these two results we mainly need to alter various technical defini-tions given for the A2 case; most of the lemmas leading up to Theorem 6.2 andt h e proof of the theorem then carry over word-for- word. An H- web in the B2 or G2spider is the web: xIn both the B2 and G2 spiders, one can define an H- move on a basis web w withtwo adjacent endpoints labelled 1 and 2. An H- move at two such endpoints consistsof attaching an H- web to w, provided that this operation does not create an ellipticface, and then replacing an internal double edge by a tetravalent vertex in the B2spider or by a perpendicular single edge in the G2 spider:

y=\ * X 0B 2case),( G , case) . (7)XIf attaching an H- web would result in an elliptic face, then there are two alternatives:E i t h e r an H- web can be removed (possibly after introducing an internal type 2 strandby the above operations), or the two endpoints make a "Y:"

> C


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Spiders for Rank 2 Lie Algebrasor, in the G2 case, they make an "H " with only one type 2 strand:

143

In each of these exceptional cases, the operation indicated is the H- move.The operation of cutting along a minimal path is slightly more complicated than

before, because the path might cut diagonally across a vertex in the B2 spider orc o n t a i n a type 1 edge in the G2 spider. The operation is defined by first introducing atype 2 edge, as is sometimes also necessary for an H- move, by reversing the contrac-t i o n operation of Fig. (7). A core is defined in the same way for all three rank 2 spi-ders. The stair- step construction is essentially the same as before. Each square is anH- web, with the result that sewing together two cores with stair steps in between re-sults in many internal type 2 strands, which are removed by the operation of Fig. (7).

The biggest difference between the three rank 2 cases is in the statement andproof of the analogues of Lemma 6.8. We wish to check that, if s e {1,2} and cand d are arbitrary clasps of weight and , then

\B([s]cd)\ = dimInv(F50 V( ) 0 V()). ( 7 )F o r this purpose, one would like to argue that an element of B([s]cd) has no hyper-bolic faces. As before, it is convenient to consider segregated clasps, where here aclasp is segregated if it is of the form 11. . . 122... 2, as well as reverse segregatedclasps of the form 22. . . 211. . . 1. In the B2 case, it is easy to enumerate the elementsof B([l]cd) if c is segregated and d is reverse segregated, as well as the elementsof B([2]cd) if c is reverse segregated and d is segregated. A web w in such a basisweb set cannot have negative curvature, and after removing bare strands, it is eitherempty or the shape of its boundary is one of the following (possibly with c and dswitched):

VIn the (72 case, the computation is again simpler if one of c and d is segregated andt h e other is reverse segregated. No webs in B([l]cd) have any negative curvature,but a web in B([2]cd) might have one negatively curved face, namely one withthree type 2 strands and one type 1 strand incident to it:


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144 G KuperbergAll other G2 webs in B([2]cd) are flat. An exhaustive enumeration of such webs inb o t h spiders verifies Eq. (7). The details are left to the reader.

Finally, in the G2 spider, unlike in the A2 and B2 spiders, web with a partly con-vex clasp does not necessarily vanish in the clasped web space ^ ( [ s i ] ^ ] [sn])Recall that the clasped web space is defined as the subspace of the unclasped webspace W(s\s2 sn) spanned by non- elliptic webs with non- convex clasps and noi n t e r n a l type 2 edges, and that there is also a kernel / (|>i]|>2] [sn]) spanned bywebs w such that there is a cut path transverse to all edges of w which cuts off aclasp of lower weight. In order to show that the G2 spider is well- defined, we needt h e following lemma:

Lemma 6.11. The spaces I([s\][s2]'in W(ss2- sn).

[sn]) and W([s\][s2]" [sn]) are transverse

Proof. Rank the webs in W{s\s2 sn) first by the number of vertices, and secondby the sum of the weights of minimal cut paths cutting off each string Sf. Weconsider a change of basis which is lower- triangular with respect to this ranking:F o r each web w e W(s\s2 - sn), choose a minimal cut path pt cutting off each 57 asclose as possible to the boundary. The cut paths are unique by the G2 analogue ofLemma 6.6, and they cannot cross, although they could in principle share segments.Recall the equation for a double edge from Eqs. (4), rearranged slightly:

1- 2

1q + l + q - 1 K

Applying this equation, we can convert every single edge contained in a path pi toa transverse double edge; the other terms are all lower with respect to the rankingof webs. This produces a new basis for W(s\s2 sn) in which / ( [ . S ] ^ ] *[sn])a n d ^([silfe] [sn]) are manifestly complements. D

7. Explicit Formulas for A2 Clasps

Although internal clasps must exist by Sect. 5, the argument given there is tooindirect for practical computations. In this section, we give explicit formulas for A2clasps. Note first that clasps are sent to each other by composition with H webs:

+ - + -

KT h u s , it suffices to derive a formula for segregated clasps.


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Spiders for Rank 2 Lie Algebras 145Lemma 7.1. If s is a segregated sign string and t* is another sign string of loweror incomparable weight, then a basis web w e B(st) has either a "U" or a "Y"attached to s.

Proof The proof is also similar to that of Lemma 6.5. We assume the minimalcounterexample. First, the web w must be connected, for if a connected componentmeets t but not s9 it may be discarded to produce a smaller counterexample; if itmeets s but not t, then it is a smaller counterexample; and if all components meetb o t h s and , then one of them must be a smaller counterexample. Second, therec a n be no "Y" or "H " attached at t, for otherwise they may be discarded to pro-duce a smaller counterexample. (N ote that a "U " attached to t has already beeneliminated.)Since w is non- elliptic, the total exterior curvature is at least 360. Since thereis no "U , " "Y," or "H " attached at t*9 the total curvature along t is at most 0.Moreover, unless w is a bare strand (which is not a counterexample), the two arcsconnecting s to t have curvature at most 120. By the hypothesis that s has no "Y,"t h e curvature in each segregated segment of S is at most 0 also. Finally, since shas no "U , " the curvature at the single arc connecting opposite signs of s is at most60, for a total of at most 300, a contradiction:

,< ^ 120/

\ I \ /

V /\ w t .

V ^120

By Lemma 7.1, if a web w e W(ss*) annihilates any "U " or "Y," it satisfiest h e annihilation axiom of a clasp. Moreover, among terms in w9 only s (the webconsisting solely of parallel strands) has the weight of s as its cut weight. Thereforeall other terms are annihilated by such w9 and the annihilation property implies thatw is an idempotent under concatenation if the coefficient of the leading term s is1. Given these facts, an argument by induction (see Wenzl [24] and Ohtsuki andYamada [16]) establishes that

n - \ + +n+ n-l +

ll n- 2+ I n]n- n- \ -

n - \ -


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146 G Kuperbergrecursively defines a clasp of weight n \, and a computation shows that

a + b-

ks=0 [a- k]\ [b- k]\ [aa- b +I I

-defines a segregated clasp of weight a \ + Z?A2

8. Applications and Problems8.1. Higher rank. The main open problem related to the combinatorial rank 2 spi-ders is how to generalize them to higher rank. A proper generalization would con-sist of a complete set of generators and relations for the higher- rank spiders; thestrand sets would correspond to the fundamental representations and their tensorp r o d u c t s . It is easy to make a HOMFLY spider which corresponds to the HOM-FLY polynomial, but this spider describes the invariant theory of An only in thestable limit of large n\ the n web spaces for any fixed n are quotients of theH O M F LY web spaces. (Recently, Murakami, Ohtsuki, and Yamada [15] have de-fined t h e HOMFLY spider in terms of trivalent graphs; this is a step toward anexplicit description of the unstable truncation.) The generalization to a higher- rankLie algebra g, if it exists, would also likely involve formal angles related to theCoxeter geometry of g; note that in both the 2 and B2 spiders, a trivalent ver-tex is dual to a Weyl alcove. N ote also that in all three rank 2 spiders, a large,flat basis web (one with neither elliptic nor hyperbolic faces) coincides with theVoronoi tiling of the plane given by the weight lattice of the corresponding Liealgebra.

T h e bases given by the combinatorial rank 1 spider are dual to Lusztig's canoni-cal bases [2]. Those of the rank 2 spiders are almost certainly also dual to canonicalbases or are closely related, because canonical bases have the same symmetry ofcyclic permutation of tensor factors, and because of the integrality and positivityproperties of the coefficients in Eqs. (2), (3), and (4).

8.2. Generalized j symbols. Given four webs wi, W2,W3, and w4 and six claspsen, C13, c\4, c23, c24, and C34 in some spider, their tetrahedron symbol is defined ast h e value of the compound web


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148 G . Kuperbergnested sequence of three tangles a, b, an d c:

We recall some basic identities in the A\ spider:

O-X = - q/ O - < ?

3/4

The tangle a is a right- handed crossing. Itsexpansion leads to anexpansion ofthetangle b:

XX-'")c


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Spiders for Rank 2 Lie Algebras 149Given a knot K and a clasp c in one of the combinatorial spiders presented here,

consider the clasped cabling of K. I.e., replace the strand of K by several strandstied together with c, for example:

(9)

T h e value of a web such as the one in Fig. (9) is also a regular isotopy invari-a n t of K; it is the Reshetikhin- Turaev ribbon graph invariant of K colored by anirreducible representation whose weight is the highest weight of c [17]. By the iso-morphism between combinatorial and algebraic spiders, clasps are a complete setin the sense that these clasped invariants yield all of the information about K thatc a n be obtained from applying the unclasped link invariants to all cablings of Ka n d all other satellites of K. Moreover, the value of a clasped cabling of K can bec o m p u t ed more easily than an unclasped cabling, because clasped web spaces arem u c h smaller vector spaces than their unclasped counterparts.8.4. Combinatorial consequences. Although Theorems 6.1, 6.9, and 6.10 are prop-erly results in representation theory, they are also interesting as results in enumera-tive combinatorics. John Stembridge and Richard Stanley [20] noted that a B2 basisweb w E 5(11. . . 1) is equivalent to a matching of In cyclically ordered points withn o 6- point star, meaning that among the 2n points, there are no six in cyclic ordern o six points p\, p2, pz,q\,qi,qs with each pt matched to qt. On other hand, then u m b e r of such matchings with no 2&-point star is known to be dim Inv( Vln),where V is the defining representation of sp(2& 2) [21]. Thus, this special caseof Theorem 6.9 was known previously.

I t is interesting that ordinary trivalent graphs are related to the exceptional Liealgebra G2. One corollary of this surprising connection is the following enumerativeresult:Theorem 8.1. For n ^ 3, let an be the number of triangulations of a fixed convexn- gon such that at least six triangles meet at each internal vertex, and let

A(x)= 1 +x2 + anxnn

be a generating function. For n ^ 0, let

for n ^ 0, and let


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150 G Kuperbergbe a generating function. Then

B(x)=A(xB(x) ) . (8)

Proof. (Sketch) By Theorem 6.10, bn may equally well be defined as the numberof G2 basis webs with n endpoints. Each triangulation is dual to a connected basisweb. Equation (8.1) becomes a standard relation between the number of connectedgraphs of a certain type and the number of disjoint unions of such graphs. Moreprecisely, consider n boundary points on a disk with one marked, and consider abasis web w G B( 11 1) with this boundary. Then w is given by an ordered treewhose vertices are its connected components: The root is the component with themarked vertex, and the children of each component are the adjacent componentso t h e r than the parent, if there is one. The children of each component are orderedgoing counterclockwise, with the distinguished boundary point separating the firsta n d last children of the root. Equation (8) is the usual generating function forordered trees [3, p. 11- 12]. D

The sequence {bn} is easy to compute using character theory. Thus, Theorem 8.1produces a fast algorithm for computing {an}.I t is easy to show that the radius of convergence of B(x) is 1/7. Then byT h e o r e m 8.1, the radius of convergence of A(x) is at least B(l/ ) / . Furthermore,numerical evidence supports the conjecture that it is exactly i?(l/ 7)/ 7, i.e., thatConjecture 8.2. If an and bn are defined as in Theorem 8.1, then

lim a~ n= co = 6 . 8 1 1 . . . .O u r equinumeration theorems count all basis webs with a fixed boundary. How-

ever, for each n and k, one can also consider the set of G basis webs, for example,with n endpoints of type 1 and k internal vertices. The role of these sets and theircardinality in representation theory is not known.Given an A^ basisweb set B(s) for some sign string s, there are various H- mapscorresponding to different adjacent pairs of signs. But if s' is another sign string oft h e same weight as s, then any two sequences of H- maps permuting s into s' yieldt h e same bijection B(s) B(s'), provided that the first and the last sign of s aren o t considered adjacent. On the other hand, if a sign is moved all the way around

t h e boundary by H- maps, monodromy arises:


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Spiders for Rank 2 Lie Algebras 151I t would be interesting to compute the cycle structure or establish properties of thismonodromy, which is related to a linear action of the braid group on

Swhere the direct sum on the right is taken over all sign strings of length n.Acknowledgement The author would like to thank Sarah Witherspoon, Bruce Westbury, IgorF r en k el , and Mikhail Khovanov for their attention to this paper and the work it presents Also, theT]HX macro package PSTricks [23] was essential for typesetting the equations and figures

References1 Barrett, J.W , Westbury, B W.: Spherical categories hep- th preprint #93101642 Frenkel, I., Khovanov, M : Canonical bases in tensor products and graphical calculus for

Uq( 2). Preprint, 19953. Goulden, I.P. Jackson, D.M.: Combinatorial Enumeration. New York: Wiley, 19834 Jaeger, F .: Confluent reductions of cubic plane maps In: G r a p h Theory and CombinatoricsI n t e r n a t i o n a l Conference Marseille, 19905 Jones, V.FR.: Index of subfactors. Invent Math. 72, 1-25 (1987)6. Kauffman, L.H : State models and the Jones polynomial Topology 26, 395- 407 (1987)7 Kauffman, L H., Lins, S L : Temperley- Lieb recoupling theory and invariants of 3-manifolds.

Ann Math. Studies, Princeton, N J.: Princeton University Press, 19948. Kirillov, A A: Elements of the Theory of Representations Berlin: Springer-Verlag, 19769 Kuperberg, G.: The quantum G2 link invariant Int. J. Math. 5, 61- 85 (1994)10. Lickorish, W.B.R: Calculations with the Temperley- Lieb algebra Comment Math. Helv. 67,

571-591 (1992)11 . Lusztig, G.: Canonical bases arising from quantized enveloping algebras. J. Am. Math. Soc3, 447- 498 (1990)

12 Masbaum, G., Vogel, P.: 3-valent graphs and the Kauffman bracket. Pacific J. Math. 164,361- 381 (1994)13 . Mollard, M.: Personal communication, 1992

14. Moore, G , Seiberg, N : Classical and quantum conformal field theory. Commun Math Phys.123, 177- 254 (1989)

15 Murakami, H , Ohtsuki, T , Yamada, S.: Homfly polynomial via an invariant of colored planargraphs. Preprint, 1995

16. Ohtsuki, T., Yamada, S : Quantum su(3) invariants via linear skein theory Preprint, 199517. Reshetikhin, N.Yu., Turaev, V.G : Ribbon graphs and their invariants derived from quantumgroups Commun. Math. Phys. 127, 1-26 (1990)18 Rumer, G , Teller, E., Weyl, H.: Eine fur die Valenztheorie geeignete Basis der binaren Vek-

torinvarianten Nachr. Ges. Wiss gottingen Math.-Phys. Kl 1932, pp. 499- 50419. Sinha, D .: Personal communication, 199220. Stanley, R.P., Stembridge, J.R.: Personal communication, 199121 . Sundaram, S.: Tableaux in the representation theory of the classical Lie groups In: Invariant

theory and tableaux (Minneapolis, MN, 1988), Vol. 19 of IMA Vol Math Appl., New York:Springer, 1990, pp 191- 225

22. Temperley, H .N .V, Lieb, E.H : Relations between the "percolation" and "coloring" problemand other graph- theoretical problems associated with regular planar lattices: Some exact reultsfor the "percolation" problem. P r o c . Roy Soc. London, Ser. A 322, 251- 280 (1971)

23 Van Zandt, T.: PSTricks: PostScript macros for generic TgX Available atftp:/ / ftp princeton.edu/pub/ tvz/

24. Wenzl, H : On sequences of projections. In C. R. Math Rep. Acad Sci Canada IX, 1987,p p . 5- 9C o m m u n i c a t e d by H Araki


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greg kuperberg- spiders for rank 2 lie algebras

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