Extremal weight projectors

0.25 MB·

by  Hoel Queffelec

EXTREMAL WEIGHT PROJECTORS
HOELQUEFFELECANDPAULWEDRICH
7
1
0 Abstract. We introduce a quotient of the affine Temperley-Lieb category that encodes all weight-
2 preserving linear maps between finite-dimensional sl2-representations. We study the diagrammatic
idempotents that correspond to projections onto extremal weight spaces and find that they satisfy
n
similarpropertiesasJones-Wenzlprojectors,andthattheycategorifytheChebyshevpolynomialsof
a
the first kind. This gives a categorification of the Kauffman bracket skein algebra of the annulus,
J
which is well adapted to the task of categorifying the multiplication on the Kauffman bracket skein
9
moduleofthetorus.

]
T
R
. 1. Introduction
h
t The Lie algebra sl and its universal enveloping algebra U(sl ) are ubiquitous in representation
a 2 2
m theory, topology and physics, and their study typically starts with the description of the monoidal
category Rep(sl ) of finite-dimensional sl -representations. The representationring of sl , that is, the
[ 2 2 2
Grothendieck group K0(Rep(sl2)) with multiplication inherited from the tensor product of representa-
1 tions, is isomorphic to the polynomial ring [X] generated by the class of the vector representation
v Z
X =[V].
6
Two bases of [X], triangularly equivalent (over ) to the monomial basis, play a prominent role,
1 Z Q
3 especially in quantum topological applications:
2 • the ChebyshevpolynomialsofthefirstkindL ∈ [X],recursivelydefinedasL =2,L =X
m 0 1
0 Z
and L =XL −L for m≥3;
. m m−1 m−2
1 • the Chebyshev polynomials of the second kind J ∈ [X], defined to satisfy the same recur-
m
0 rence, but with initial conditions J =1, J =X. Z
0 1
7
1 The basis elements Jm appear in K0(Rep(sl2)) as the classes of the simple representations Symm(V).
: Since tensor powers of V can be decomposed into simple representations, this implies that the basis
v
change in [X] from the monomial basis to {J } has positive coefficients.
Xi It is easZy to see that the basis change from {mJ } to {L } also has positive coefficients, and in fact
m m
r J = L +J for m ≥ 2. The element L of the representation ring, thus, corresponds to the
m m m−2 m
a formal difference of the classes of Symm(V) and Symm−2(V), which we interpret as the class of the
extremal weight space of Symm(V), that is, the direct sum of the highest and lowest weight spaces.
Anotherindicationthatthebasis{L }ismorefundamentalthanthebasis{J }isgivenbycomparing
m m
the multiplication rules:
(1) J J =J +J +···+J and L L =L +L .
m n m+n m+n−2 |m−n| m n m+n |m−n|
TherepresentationcategoryRep(sl )admitsadiagrammaticgenerators-and-relationsdescriptionas
2
the Karoubienvelope ofthe famous Temperley-LiebcategoryTL, which encodes intertwinersbetween
tensor powers of the vector representation, see Section 2.1. In this description, the simple representa-
tions Symm(V) are obtained through the Jones-Wenzl projectors P ∈ TL, which correspond to the
m
Funding: H. Q. was partially supported by the ANR Quantact and a grant from the Universit´e de Montpellier.
P.W.wassupportedbyaFortbildungsstipendiumattheMaxPlanckInstituteforMathematicsandbytheLeverhulme
ResearchGrantRP2013-K-017toDorothyBuck,ImperialCollegeLondon.
1
2 HOELQUEFFELECANDPAULWEDRICH
idempotent intertwiners:
V⊗m ։Symm(V)֒→V⊗m.
Insummary,theKaroubienvelopeoftheTemperley-Liebcategorygivesadiagrammaticcategorification
ofthepolynomialringandtheJones-WenzlprojectorsP categorifythepolynomialsJ . Thepurpose
m m
of this paper is to give a description of analogs of the Jones-Wenzl projectors for the Chebyshev
polynomials of the first kind L , which we call extremal weight projectors.
m
1.1. Main results. We define a diagrammatic extension ATLess of the Temperley-Lieb category,
whichutilizesdiagramsinanannulus. Moreprecisely,themonoidalcategoryATLess isthequotientof
the affine Temperley-Lieb category ATL by an ideal generated by the relation that an essential circle
in the annulus is zero, see Definition 5 for details. In this category, we give a very simple recursive
diagrammatic definition of morphisms T for m ≥ 0, which is reminiscent of, but simpler than the
m
recursive definition of the Jones-Wenzl projectors. Our main results are:
• ATLess describes the category of weight-preserving linear maps between sl -representations,
2
or in other words, the representation category Rep(h)⊃Rep(sl ) of the Cartan h⊂sl ;
2 2
• the elements T are idempotents, corresponding to projections onto extremal weight spaces;
m
• in the Karoubienvelope of ATLess, there are isomorphisms T ⊗T ∼=T ⊕T , which
m n m+n |m−n|
categorify the multiplication rule in (1).
The simple definition of these idempotents makes them natural objects to study. Additionally, we
expect that they will play a central role in categorifying skein algebras, as we will explain next.
1.2. Skein modules and skein algebras. Let M be an orientable 3-manifold and := [q±1/2].
k Z
Following Przytycki [Prz91] and Turaev [Tur91], the Kauffman bracket skein module Sk (M) of M is
q
constructed by forming the free -module spanned by isotopy classes of framed links in M and taking
k
the quotient by the Kauffman bracket skein relations [Kau90] supported in 3-balls in M:
(2) = q1/2 + q−1/2 , = −(q+q−1)
Formanifoldswithboundary,onemaytaketangleswithendpointsinaspecifieddiscretesetofboundary
pointsasspanningsetfortheskeinmodule. Thus,thecategoriesTLandATLess canbeseenasbeing
built from skein modules of thickened squares or thickened annuli with prescribed boundary points,
at q = 1. The multiplication is induced from gluing these 3-manifolds along the vertical part of their
boundary.
We writeSk (Σ)for the skeinmodules ofthickenedsurfacesΣ×[0,1]. Thesebecome skeinalgebras
q
with the multiplication operations obtained by stacking two copies of the thickened surface i.e. gluing
along the horizontal part of their boundary. The skein algebra Sk ( 3) is free of rank one over
q
R k
and the class of a framed link is given by its Kauffman bracket evaluation, which recovers the Jones
polynomial. The skein algebra Sk (A) of the annulus without boundary points is isomorphic to the
q
polynomial ring [X] generated by the core curve of the annulus, and thus to the representation ring
k
K (Rep(sl )) ⊗ . Elements of the skein of the annulus act on framed knots in skein modules by
0 2
k
cablingoperations. Forexample,cablingknotsinSk ( 3)bytheelementcorrespondingtoJ ∈ [X]
q m
R Z
produces the m-th colored Jones polynomial as evaluation.
1.3. Positive bases for skein algebras and categorification. TheKauffmanbracketskeinmodule
ofthetorusT=S1×S1 hasabasisgivenbycollectionsofparallelnon-separatingsimpleclosedcurves
onT. Thus,asa -module,itisisomorphictoadirectsumofcountablymanycopiesofSk (A),onefor
q
k
eachslopem/nonthetorus. However,FrohmanandGelca[FG00]provedthattheskeinmultiplication
has a particular nice form in terms of the basis assembled from the Chebyshev polynomials L in the
d
EXTREMAL WEIGHT PROJECTORS 3
copies Sk (A). In fact, for a pair (r,s) ∈ 2, they define (r,s) to be the basis element L of
q T gcd(r,s)
Z
slope r/s in Sk (T) and show:
q
(3) (m,n) ∗(r,s) =q(ms−rn)/2(m+r,n+s) +q(rn−ms)/2(m−r,n−s)
T T T T
This formula, taken at roots of unity, plays a central role in Bonahon and Wong’s miraculous
cancellations [BW16] and the related work of Le [Lˆe16b], in connection to cluster algebra. Also in
relation with cluster algebras, for arbitrary closed oriented surfaces Σ, Thurston showed that the
elements L ∈ Sk(A), embedded along simple closed curves in Σ, give a basis for the skein algebra
d
Sk (Σ) with positive structure constants, [Thu14]. He also conjectured that the same holds for
q=1
generic q (see also [Lˆe16a]) and asked whether this positivity phenomenon might be the shadow of a
monoidal categorification of Sk (Σ). We propose a definition.
q
Definition 1. AmonoidalcategorificationofSk (Σ)consistsofagraded,monoidal,additivecategory
q
C(Σ) and an isomorphism of unital -algebras ψ: Sk (Σ)∼=K (C(Σ)).
q 0
k
Thurston’s positivity conjecture suggests that one should look for monoidal categorifications of
Sk (Σ) with simple objects categorifying the cablings of simple closed curves by L . This is the main
q d
motivation for our construction of the highest weight projectors. Indeed, in Section 2.7 we describe
a subcategory of ATLess, whose Karoubi envelope gives a monoidal categorification of Sk (A) (after
q
introducing an extra grading), and the idempotents T categorify the elements L .
d d
A monoidal categorification of a skein algebra becomes even more interesting if one can evaluate
links in it. To make this precise, let Link(Σ) denote the monoidal category whose objects are given
by embedded links L in Σ×[0,1] and with morphisms given by compact, oriented bordisms in Σ×
[0,1]× [0,1] between links, modulo isotopy. The composition is the usual gluing of bordisms and
the tensor product is induced by stacking links (and bordisms) along the thickening direction and
rescaling: [0,2] → [0,1]. We now define a categorification of a skein algebra to consist of a monoidal
categorificationasinDefinition1togetherwithaKhovanov-typelinkhomologyfunctor([Kho00])that
is compatible with the monoidal structure.
Definition 2. A categorification of the skein algebra Sk (Σ) consists of a monoidal categorification
q
as in Definition 1, together with a monoidal functor Kh: Link(Σ)→C(Σ), such that the class of each
link L⊂Σ×[0,1] in Sk (Σ) is sent to K (Kh(L)) under ψ.
q 0
RecallthatBar-Natan’sconstructionofKhovanovhomologyadmitsanextensiontothecaseoflinks
in Σ×[0,1], see [BN05][Section 11.6]. The target category C(Σ) of this construction is the homotopy
categoryofchaincomplexeswithchaingroupsgeneratedby1-manifoldsembeddedinΣanddifferentials
generated by cobordisms in Σ×[0,1], modulo certain relations. This gives a categorification of the
skein module Sk (Σ) and a proposal for a (projective) Khovanov functor Kh: Link(Σ) → C(Σ) (see
−q
also[APS04] fora relatedconstruction). However,itis notknownwhether C(Σ)canbe equipped with
a monoidal structure making Kh monoidal.
In [QW] we will examine this problem in the case of the torus Σ = T. In this case, Bar-Natan’s
cobordism category contains what we call slope subcategories — subcategories of 1-manifolds and
cobordismswhichareS1-equivariantalongafixedslope. TakingthequotientbytheS1-action,itisnot
hardto seethatthe degreezeropartofthis subcategoryis closelyrelatedto the affine Temperley-Lieb
category ATL. The additional relation in ATLess is realized by setting the endo-cobordism of ∅ that
is given by the boundary parallel torus equal to zero. The homotopy category over a corresponding
quotient cobordism category, thus, contains idempotents which categorify the Frohman-Gelca basis
elements (m,n) . Moreover, it follows from this translation and our results in Section 2.6 that these
T
idempotents satisfy a categorified version of the Frohman-Gelca formula (3) in the case when they lie
in the same slope. In [QW] we treat the general case of tensoring such categorified Frohman-Gelca
4 HOELQUEFFELECANDPAULWEDRICH
basis elements, althoughwe have to workin a slightly different setting to avoidfunctoriality problems.
We also speculate that a similar construction should be possible for other closed surfaces Σ.
Acknowledgements: We would like to thank Nils Carqueville, Charlie Frohman, Eugene Gorsky,
Mikhail Khovanov, Scott Morrison, Jake Rasmussen, Philippe Roche, Heather Russell and Catharina
Stroppel for interesting discussions and helpful comments.
2. Affine Temperley-Lieb and extremal weight projectors
2.1. Temperley-Lieb categories. We start by recalling classical results about the Temperley-Lieb
category and its affine version at q =1.
Definition 3. Let TL(m,n) denote the free -module spanned by planarmatchings in a box with m
C
boundary points on the bottom and n boundary points on the top (and no closed components). The
Temperley-Lieb category TL is the -linear strict monoidal category with objects n ∈ , morphism
C N
spacesTL(n,m)andmultiplicationTL(m,n)×TL(l,m)→TL(l,n)is defined onthe bases ofmatch-
ingsbycomposingplanartanglesandreducingeachresultingclosedcomponentsto −2. The monoidal
structure is defined on objects by m⊗n=m+n and on morphisms by placing diagrams side by side.
Example 4. We give examples for the composition and the tensor product of two planar matchings.
· =−2 , ⊗ =
Stacking the two displayed matchings produces a closed component, which is reduced to −2.
We write TL :=TL(n,n) for the Temperley-Liebalgebraonn strands. It has a presentationwith
n
generators U for 1≤i≤n−1, which correspond to cap-cup matchings between the i-th and i+1-st
i
strand. A complete set of relations is:
• U U U =U for 1≤i≤n−2 and U U U =U for 2≤i≤n−1,
i i+1 i i i i−1 i i
• [U ,U ]=0 for |i−j|>1 and
i j
• U2 =−2U .
i i
Gluing such boxes along their left and right boundaries, we get Temperley-Lieb diagrams in an
annulus A = S1×[0,1] together with additional information: we remember the corners of the box as
base points on the boundary circles and the glued edge as an arc connecting them.
→
∗
∗
Definition5. TheaffineTemperley-Liebcategory,denotedATL,isthe -linearcategorywithobjects
C
n∈ andwithmorphismspacesATL(m,n)spannedbycrossinglesstanglesinthe annulusAwithm
N
andnendpointsontheinsideandoutsideboundaryrespectively,withoutinessential(i.e. homologically
trivial) circles. The multiplication is defined on basis elements by stacking an annular diagram inside
another one such that endpoints and basepoints match up, and then reducing inessentialcircles to −2.
We write ATL :=ATL(n,n) for the affine Temperley-Lieb algebra on n strands.
n
EXTREMAL WEIGHT PROJECTORS 5
Note that the affine Temperley-Lieb categorycan be obtained from the usual one by simply adding
a new type of invertible morphism:
... ...
D = , D−1 =
∗ ∗
∗ ∗
We write U =DU D−1 = D−1U D for the cap-cup between the first and last strand. A complete
0 1 n−1
set of relations for ATL in terms of the generators U and the invertible generator D is given by:
n i
• U U U =U =U U U ,
i i+1 i i i i−1 i
• [U ,U ]=0 for i6=j±1 ,
i j
• U2 =−2U ,
i i
• U D =DU
i i+1
where all indices are taken modulo n.
Remark6. ThereexistsavastliteratureabouttheTemperley-Liebcategory,Temperley-Liebalgebras
and their affine versions. Our version of ATL agrees with the one defined by Graham and Lehrer
n
[GL98]atq =1. JonesandReznikoff[JR06]alsostudythisversion,althoughtheyuseshadeddiagrams
which encode a parity. Our ATL should not be confused with the Temperley-Lieb quotients of the
n
Iwahori-Hecke algebras of affine type A, which are studied e.g. in [FG99, EG99, Gre98], and which
appearasthepropersubalgebrasofourATL generatedbytheU ,seealsoSection2.7. Alsonotethat
n i
ATLencodescrossinglesstanglesintheannulusuptoisotopieswhicharetheidentityontheboundary.
Algebras of Temperley-Lieb diagrams where isotopies need not be the identity on the boundary have
e.g. been studied in [Jon94].
2.2. Link with sl -representation theory. It is well-known that the Temperley-Lieb category TL
2
describesacategoryofsl -representationsandtheirintertwiners. Thiscanbetracedbackto [RTW32].
2
Indeed, there exists a fully faithful -linear monoidal functor ϕ from TL to Rep(sl ), which sends n
2
C
to the n-th tensor power of the vector representation V = hv ,v i. In the following we describe a
+ −
C
version of the functor ϕ using the representation theory of the associative algebra U(sl ).
2
Recall that U(sl ) is the -algebra generated by E,F,H subject to the following relations:
2
C
(4) HE−EH =2E, HF −FH =−2F, EF −FE =H.
It is a Hopf algebra with coproduct and antipode determined by ∆(x)=1⊗x+x⊗1 and S(x)=−x
for x∈sl .
2
Denote V = hv ,v i the vector representation, with
− +
C
Ev =0, Fv =v , Hv =v ,
+ + − + +
Ev =v , Fv =0, Hv =−v ,
− + − − −
The dual V∗ = hv∗,v∗i is isomorphic to V via the map that send v∗ 7→v and v∗ 7→−v .
C − + − + + −
The -linear monoidal functor ϕ is determined by its images on cap and cup morphisms in TL. It
C
associatesto thosethe negativesofthe naturalevaluationandco-evaluationmapsfor duals,composed
with the isomorphism V ∼=V∗:
V ⊗V →
ϕ →V ⊗V ϕ C
−→ C , −→ v∓⊗v± 7→±1
(17→v+⊗v−−v−⊗v+ v±⊗v± 7→0

6 HOELQUEFFELECANDPAULWEDRICH
ItisaneasyexercisetocheckthattheseintertwinersofU(sl )-representationssatisfytherelationsof
2
the Temperley-Lieb category and so ϕ is a well-defined monoidal functor. The exchange of two tensor
factors s: V ⊗V,v ⊗v 7→v ⊗v is the image under ϕ of the following morphism in TL:
ε1 ε2 ε2 ε1
(5) := +
This suggestive notation encodes useful relations in TL and is compatible with planar isotopies:
= , =
Lemma 7. The following analogs of Reidemeister moves hold in TL:
= − , = , =
ThisimpliesthatTLisasymmetricmonoidalcategoryandthatϕisasymmetricmonoidalfunctor.
Remark 8. The tensor product V⊗m decomposes into a direct sum of irreducible representations,
amongst which Symm(V) appears with multiplicity 1 and has highest weight. This decomposition
can be translated in terms of idempotents in TL and the idempotent corresponding to the copy of
n
Symm(V) is the famous Jones-Wenzl projector P [Jon85, Wen87] with diagrammatic representation
m
recursively defined by P =id and:
1 1
m Pm
(6) P = P +
m+1 m m+1
Pm
Dual to the decomposition into simple representations, one can decompose V⊗m as the direct sum
of its weight spaces. Of course, the projections onto weight spaces are usually not U(sl )-equivariant,
2
and then cannot be realized as the images of idempotents in the Temperley-Lieb category TL.
We will thus extend the functor ϕ from TL to ATL by sending the morphisms D ∈ ATL to
m
certainvector spaceendomorphisms ofV⊗m which respectthe weightspace decomposition,but which
breakthe U(sl )-action1. Since we wantATL to inherit the monoidalstructure ofTL,the ϕ-imageof
2
D ∈ATL actuallydeterminesϕ(D) forallD ∈ATL . Therequirementthatϕ(D)is invertibleand
1 m
respects the weight space decomposition of V implies that ϕ(D)(v ) = c v for some c ∈ ∗. The
± ± ± ±
proofofLemma10showsthatweneedc =c−1 inordertoobtainawell-definedfunctorthatCrespects
+ −
all isotopies of diagrams. The reason for the following choice will become clear in Proposition 11.
Definition 9. Let V ⊗W be the image under ϕ of the domain of D and W ⊗V its co-domain. Then
we define ϕ(D) to be the linear map determined by v ⊗w 7→i±1w⊗v for v ∈V and any w ∈W.
± ± ±
Furthermore we set ϕ(D−1)=ϕ(D)−1.
Let h denote the Cartan subalgebra of diagonal matrices in sl and consider U(h) = hHi ⊂ U(sl ).
2 2
We denote by Rep(h) the category of finite-dimensional U(h)-representations of integral weights, i.e.
vectorspaceswitha -grading. Notethattheinclusionh֒→sl inducesarestrictionfunctorRep(sl )→
2 2
Z
Rep(h) and that ϕ(D) and ϕ(D−1) are morphisms in Rep(h).
Lemma 10. The functor ϕ: ATL→Rep(h) is well-defined.
1Settingϕ(D)tobeacertainU(sl2)-intertwinercanbeusedtoconstructevaluationrepresentations ofcertainaffine
Liealgebras. Westressthatthisisnotwhatwehaveinmind.
EXTREMAL WEIGHT PROJECTORS 7
Proof. All morphisms in ATL are compositions of caps or cups between adjacent strands, as well as
the morphisms D and D−1. Any relation satisfied by compositions of these generating morphisms is
either supported in TL⊂ATL (and is thus respected by ϕ) or involves some generators D and D−1.
Since ϕmapsD andD−1 to inverseisomorphisms,itsuffices tocheckthatϕrespectsthe relationthat
D intertwines adjacentcaps and cups. The only interesting case correspondsto isotoping caps or cups
across the dashed arc. For cups we need the relation:
... ...
ϕ  = ϕ 
∗ ∗
   
 ∗   ∗ 
   
Note that both sides annihilate vectors of theform v ⊗w⊗v . On the other hand, the action on
± ±
v ⊗w⊗v of the left-hand side and right-hand side, respectively, are given by:
± ∓
v ⊗w⊗v 7→i±1w⊗v ⊗v 7→iw and v ⊗w⊗v 7→i±1v ⊗v ⊗w 7→iw
± ∓ ∓ ± ± ∓ ∓ ±
The isotopy relation for cups is checked analogously. (cid:3)
Let ⊗: ATL×ATL → ATL denote the bi-functor given on objects by (m,n) 7→ m+n and on
morphisms by superposing a pair of affine Temperley-Lieb diagrams (W ,W ):
1 2
W1
W2
∗
∗
and resolving all crossings via (5). This is well-defined thanks to Lemma 7 and it induces a strict
symmetric monoidal structure on ATL such that both the inclusion TL → ATL and ϕ: ATL →
Rep(h) become symmetric monoidal functors. Let ι and ι′ denote the endo-functors of ATL given by
superposing a single strand on the right or left:
ι′ ι
←− −→
∗ ∗ ∗
∗ ∗ ∗
Note that these endo-functors can be written as ι′(−)=(−)⊗1 and ι′(−)=D−1ι(−)D.
2.3. A quotient. It is a key observation that the functor ϕ: ATL→Rep(h) is not faithful.
Proposition 11. ···
···
 
(7) ϕ =0.
∗
 
 
 ∗ 
 
We define ATLess as the quotient of ATL by the ideal generated by the elements shown in (7),
with any number of through-strings. Note that the monoidal structure ⊗ and the functor ϕ descend
to the quotient ATLess.
Theorem 12. The functor ϕ:ATLess 7→Rep(h) is fully faithful.
8 HOELQUEFFELECANDPAULWEDRICH
Proof. Weconsidertheisomorphismsf: ATLess(2n−m,m)→ATLess(0,2n)andtheirinverses,which
are given by the following operations on diagrams.
f = , f−1 =
∗ ∗
∗ ∗
Since these operationsaregivenby tensoringwith anidentity morphismandthen pre-composingwith
cups, or post-composing with caps, there are corresponding isomorphisms ϕ(f±1) between the mor-
phismspacesHom (V⊗2n−m,V⊗m)andHom ( ,V⊗2n). Toprovethetheorem,itsufficesto
Rep(h) Rep(h)
C
checkthatϕinducesanisomorphismbetweenthemorphismspacesATLess(0,2n)andHom ( ,V⊗2n).
Rep(h)
C
Notethattheformerisspannedbyannularcrossinglessmatchingswithnoclosedcomponents,ofwhich
there are 2n . To see this, we define a bijection between annular crossingless matchings and the set
n
of labelings of their boundary points by an equal number of labels ‘in’ and ‘out’. This can be done by
(cid:0) (cid:1)
the following inverse rules:
• given a crossingless matching, find an isotopy representative and an orientation of the arcs so
that they all turn clockwise around the annulus. Assign ‘in’ to the tail and ‘out’ to the head.
• givenalabelingofboundarypoints,connectbyanarcthatonlywrapsintheclockwisedirection
aroundtheannulusanypairofadjacentboundarypointswithlabels‘in’and‘out’inthatorder.
Then,iteratetheprocessforallpairsof‘in’-and‘out’-labeledpointsinpositionsk andl whose
neighbors in position k+1 and l−1 have already been connected by arcs.
On the Rep(h) side, Hom ( ,V⊗2n) ≃ (V⊗2n) , the zero weight space, is of dimension 2n ,
Rep(h) C 0 n
and spanned by vectors v := v ⊗···⊗v where ε = (ε ,...,ε ) ∈ {+,−}2n contains as many
ε ε1 ε2n 1 2n (cid:0) (cid:1)
pluses as minuses. We shall argue that ϕ is surjective when restricted to ATLess(0,2n), which also
implies injectivity for dimension reasons.
By post-compositionwith the S -action on both sides of the map ϕ, it is actually enough to show
2n
that one vector v is hit. For if one is hit, since all the other ones lie in the same S -orbit, they will
ε 2n
be hit as well. We consider the case of v , and we will prove the result first for n=1:
+−+−···+−
ϕ (1)=v −v , ϕ (1)=iv +iv
 ∗  +− −+  ∗  +− −+
∗ ∗
   
Denoting w and w the two generators of ATLess(0,2) used above, we see that v is obtained as
1 2 +−
the image of ϕ((w −iw )/2) applied to 1∈ .
1 2
Nowthen-folddiagrammatictensorpowerCof(w −iw )/2givesamorphisminATLess(0,2n)whose
1 2
ϕ image evaluates to v on 1∈ . (cid:3)
+−+−···+−
C
Remark 13. In particular, this proves that crossingless matchings form a basis for the Hom-spaces
ATLess(0,2n) and ATLess(2n,0) that can be used to deduce explicit bases for all Hom-spaces.
Example 14. In ATLess we have D =−D−1 and thus D2 =−1.
1
Proof. This follows directly from resolving the crossing in the defining relation:
0 = = +
∗ ∗ ∗
(cid:3)
∗ ∗ ∗
EXTREMAL WEIGHT PROJECTORS 9
2.4. Extremal weight projectors.
Definition 15. The elements T ∈ ATLess are recursively defined via T = ι(T )s ι(T ) for
m m m+1 m m m
m≥2 with initial conditions T =id and T =id +U /2+U /2:
1 1 2 2 1 0
1 1 1
(8) T = + + =  + 
2 ∗ 2 ∗ 2 ∗ 2 ∗ ∗
 
∗ ∗ ∗  ∗ ∗ 
 
 
The recursive relation can be depicted as follows:
Tm Tm Tm
T
m+1
Tm Tm Tm
(9) := = +
∗ ∗ ∗ ∗
∗ ∗ ∗ ∗
Theorem 16. The element ϕ(T ) is the endomorphism of V⊗m projecting onto the extremal weight
m
space hv ,v i.
+···+ −···−
C
Proof. For m =1 this is tautological and for m =2 we use the expression T = id +U /2+U /2 to
2 2 1 0
compute:
ϕ(T ): v 7→v +0+0=v
2 ±± ±± ±±
v 7→v +(v −v )/2+(−v −v )/2=0
±∓ ±∓ ∓± ±∓ ∓± ±∓
For the induction step, we see immediately from the recursion that ϕ(T ) annihilates v
m+1 ε1ε2···εm+1
unless ε =···=ε . In the remaining cases we have:
1 m
ϕ(ι(T )s ι(T ))(v )=ϕ(ι(T )s )(v )=ϕ(ι(T ))(v )=0
m m m ±···±∓ m m ±···±∓ m ±···∓±
ϕ(ι(T )s ι(T ))(v )=ϕ(ι(T )s )(v )=ϕ(ι(T ))(v )=v
m m m ±···±± m m ±···±± m ±···±± ±···±±
So ϕ(T ) is the extremal weight projector. (cid:3)
m+1
Theorem 16, combined with Theorem 12, shows that ATLess describes weight-preserving maps for
tensorproductsofsl -representations,andthatthe idempotentsT realizeextremalweightprojectors.
2 m
Lemma 17. The elements T satisfy the following properties:
m
(1) T2 =T
m m
(2) T s =s T =T for m≥2.
m i i m m
(3) T U =U T =0 for m≥2
m i i m
(4) T ιm−n(T )=ιm−n(T )T =T for 1≤n<m.
m n n m m
(5) D−1T D =T .
m m
Proof. The proof simply consists in applying ϕ and checking that the result holds in Rep(h). For
example,T U =0holdssinceU containsacupmorphism,whoseimageunderϕmaps17→v −v ,
m i i +− −+
which is clearly annihilated by the extremal weight projector. (cid:3)
We leave it to the interested reader to prove these properties diagrammatically inside ATLess.
10 HOELQUEFFELECANDPAULWEDRICH
2.5. Chebyshev recursion. WenowexplicitlyshowthattheprojectorsT categorifytheChebyshev
m
polynomials of the first kind in the same sense as the classical Jones-Wenzl P projectors categorify
m
the Chebyshev polynomials of the second kind. Note that a different approach to the categorification
of the polynomial ring and to orthogonalpolynomials appears in [KS15].
Definition 18. The split Grothendieck group of an additive category C is the abelian group K (C)
0
defined as the quotient of the free abelian groupspanned by the isomorphism classes [X] of objects X
of C, modulo the ideal generated by relations of the form [A⊕B]=[A]+[B] for objects A, B of C.
If C is monoidal, then K (C) inherits a unital ring structure with multiplication [A]·[B]:=[A⊗B].
0
WehavealreadymentionedtheclassicalfactthatK (Rep(sl ))isisomorphictothepolynomialring
0 2
[X] generated by the class X =[V] of the vector representation. The isomorphisms
Z
Symm(V)⊗V ∼=Symm+1(V)⊕Symm−1(V) for m≥1
imply that classes of the simple representationsare givenby the Chebyshev polynomials of the second
kind [SymmV] = J ∈ [X]. This can also be seen in the Grothendieck group of TL, at the cost of
m
Z
passing to the Karoubi envelope.
Definition 19. The Karoubi envelope of a category C is the category Kar(C) with objects given by
pairs(X,e),whereX isanobjectofC ande∈Hom (X,X)anidempotent. Morphismsbetween(X,e)
C
and (Y,f) are of the form f ◦g◦e with g ∈ Hom (X,Y). If C is additive or monoidal, then Kar(C)
C
inherits these structures.
Recall that TL is equivalent to the full subcategory of Rep(sl ) with objects given by V⊗m. Since
2
any objects of Rep(sl ) appears as a direct summand of some V⊗m, it can be picked out by an
2
idempotentendomorphismofV⊗m. ThusKar(TL)isequivalenttoRep(sl )andK (Kar(TL))∼= [X].
2 0
Analogously, Kar(ATLess) is equivalent to Rep(h). We now explicitly compute that the classes oZf the
Jones-Wenzlprojectorssatisfy the recursionrelationof the Chebyshev polynomials of the secondkind.
Lemma 20. In K (Kar(TL)), we have [P ][P ] = [P ⊗P ] = [P ]+[P ]. Here and in the
0 m 1 m 1 m+1 m−1
following, we abuse notation and write P for the object of Kar(TL) given by the pair (m,P ).
m m
Proof. Werewritetherecurrence(6)bysubtractingthetermwithtwoprojectorsfrombothsides. The
result is a decomposition of the idempotent P ⊗P into a sum of orthogonal idempotents in TL,
m 1
which induces an isomorphism between objects in Kar(TL). After applying K , we get:
0
m Pm
P = P + −
 m   m+1   m+1 
Pm
     
Ontheleft-handsidewealreadysee[P ⊗P ]andthefirsttermontheright-handsideis[P ]. Itnow
m 1 m+1
remainstoprovethatthe idempotentshowninthesecondbracketontheright-handsideisisomorphic
to P in Kar(TL). In order to avoid confusion, we return to the pair-notation in Kar(TL). The
m−1
desired isomorphism in Kar(TL) is given by:
Pm
m Pm
(10) m+1,−m+1  m−1, Pm−1 
Pm
  − m Pm  
m+1