On the definition of Kac-Moody 2-category

0.26 MB·English

by  Jonathan Brundan

ON THE DEFINITION OF KAC-MOODY 2-CATEGORY
JONATHANBRUNDAN
5
1
0 Abstract. We show that the Kac-Moody 2-categories defined by Rouquier
2 andbyKhovanovandLaudaarethesame.

n
a
J

2 1. Introduction

] Assume that we are given the following data:
T
• a (not necessarily finite) index set I;
R
• integers a for each i,j ∈ I such that a = 2, a ≤ 0 for all i 6= j, and
ij ii ij
h. aij =0 if and only if aji =0;
t • positive integers d such that d a =d a for all i,j ∈I.
a i i ij j ji
m Thus A=(a ) is a symmetrizable generalizedCartanmatrix. Set d :=−a
ij i,j∈I ij ij
for short. Fix also the additional data:
[

• a complex vector space h;
1
• linearly independent vectors α ∈h∗ for each i∈I called simple roots;
v i
0 • linearly independent vectors hi ∈h for each i∈I such that hhi,αji=aij.
5 LetgbetheassociatedKac-MoodyalgebrawithChevalleygenerators{e ,h ,f }
i i i i∈I
3 and Cartan subalgebra h. Let P :={λ∈h∗|hh ,λi∈Z for all i∈I} be its weight
0 i
lattice. In [R], Rouquier has associated to g a certain 2-category A(g), which we
0
. will denote instead by U(g). It depends also on:
1
0 • a ground ring k;
5 • units tij ∈k× for i,j ∈I with i6=j such that tij =tji if dij =0;
1 • scalarsspq ∈k for i,j ∈I and 0≤p<d ,0≤q <d suchthat spq =sqp.
ij ij ji ij ji
:
v (Often one assumes further that these parameters are homogeneous in the sense
i that spq = 0 unless pd +qd = d d , but we do not insist on that here.) The
X ij i j i ij
following is the definition from [R, §4.1.3] formulated in diagrammatic terms.
r
a
Definition 1.1. The Kac-Moody 2-category U(g) is the strict additive k-linear
2-category with object set P, generating 1-morphisms E 1 : λ → λ + α and
i λ i
F 1 : λ → λ−α for each i ∈ I and λ ∈ P, and generating 2-morphisms x :
i λ i
E 1 → E 1 ,τ : E E 1 → E E 1 , η : 1 → F E 1 and ε : E F 1 → 1 ,
i λ i λ i j λ j i λ λ i i λ i i λ λ
subject to certain relations. To record these, we adopt a diagrammatic formalism
likein[KL],representingtheidentity2-morphismsofEi1λ andFi1λ byλ+αi↑λ and
i
λ−αi↓λ, respectively, and the other generators by
i
i λ
x= • λ, τ = λ , η = , ε= . (1.1)
λ i
i i j
2010Mathematics Subject Classification: 17B10,18D10.
ResearchsupportedinpartbyNSFgrantDMS-1161094.
1
2 J.BRUNDAN
We stress that our diagrams are simply shorthands for algebraic expressions built
by horizontally and vertically composing generators;they do not satisfy any topo-
logical invariance other than the “rectilinear isotopy” implied by the interchange
law. First, we have the quiver Hecke relations:
• • λ if i=j,
i• jλ − i jλ = i jλ − i •jλ = i j (1.2)
0 otherwise,

0 if i=j,
 tij λ if dij =0,
i jλ = tij diij•j λ +tji •dλji + spijq p• •qλ otherwise, (1.3)
 i j i j 00≤≤Xpq<<ddjiji i j
λ − λ = r+srX=,sd≥i0j−1tijr•i j •ksλ +00≤≤rX,pqs<<≥dd0jijsipijq r•i •jq•ksλ if i=k 6=j,
i j k i j k r+s=p−1
0 otherwise.
 (1.4)
(Note in the above relations that we represent powers of x by decorating the dot
with a multiplicity.) Next we have the right adjunction relations
i i
= , = , (1.5)
λ λ λ λ
i i
whichimplythatF 1 istherightdualofE 1 . Finallytherearesomeinversion
i λ+αi i λ
relations. To formulate these, we first introduce a new 2-morphism
i i
σ = λ := λ :EjFi1λ →FiEj1λ. (1.6)
j
j
Then we require that the following 2-morphisms are isomorphisms:
i
∼
λ :EjFi1λ →FiEj1λ if i6=j, (1.7)
j
i hhi,λi−1 λ
λ ⊕ ni• :EiFi1λ →∼ FiEi1λ⊕1⊕λhhi,λi if hhi,λi≥0, (1.8)
i Mn=0
i −hhi,λi−1
λ ⊕ i •n :EiFi1λ⊕1λ⊕−hhi,λi →∼ FiEi1λ if hhi,λi≤0. (1.9)
i Mn=0 λ
(This means formally that there are some additional as yet unnamed generators
which serve as two-sided inverses to the 2-morphisms in (1.7)–(1.9).)
KAC-MOODY 2-CATEGORIES 3
Ourmaintheoremidentifiesthe2-categoryU(g)justdefinedwiththeKhovanov-
Lauda 2-category from [KL]. Actually Khovanov and Lauda worked just with the
choice of parameters in which t = 1 and spq = 0 always. Subsequently, Cautis
ij ij
and Lauda [CL] generalized the definition to incorporate more general choices of
these parameters as above. By the “Khovanov-Lauda 2-category” we really mean
the more general version from [CL].
Main Theorem. Rouquier’s Kac-Moody 2-category U(g) is isomorphic to the
Khovanov-Lauda 2-category.
Theproofisanelementaryrelationchase. Toexplainthestrategy,recallthatthe
Khovanov-Lauda2-categoryhas the same objects and 1-morphisms as U(g). Then
there are generating 2-morphisms represented by the same diagrams as x,τ,η and
ε above, plus additional generating 2-morphisms x′ : F 1 → F 1 , τ′ : F F 1 →
i λ i λ i j λ
F F 1 , η′ :1 →E F 1 and ε′ :F E 1 →1 represented diagrammatically by
j i λ λ i i λ i i λ λ
i j i i λ
x′ = • λ, τ′ = λ , η′ = , ε′ = . (1.10)
λ i
These satisfy further relations which we will recall in more detail later in the in-
troduction. It is evident that all of the defining relations of U(g) recorded above
are satisfied in the Khovanov-Lauda 2-category. Hence there is a strict k-linear
2-functor from U(g) to the Khovanov-Lauda 2-category which is the identity on
objects and 1-morphisms, and maps the generating 2-morphisms x,τ,η and ε to
the corresponding 2-morphisms from [KL, CL].
To see that this functor is an isomorphism, we construct a two-sided inverse.
In order to do this, we need to identify appropriate 2-morphisms x′,τ′,η′ and
ε′ in U(g) that will be the images of the additional generators (1.10) under the
inverse functor. The definitions of η′ and ε′ that follow are essentially the same as
Rouquier’s“candidates”forsecondadjunctionfrom[R,§4.1.4],exceptthatwehave
renormalizedbythesign(−1)hhi,λi+1 inordertobeconsistentwiththeconventions
of [KL, CL]. We will also define a leftward crossing
i
σ′ = λ :FiEj1λ →EjFi1λ, (1.11)
j
which we have chosen to normalize differently from the leftward crossing in [CL].
Definition 1.2. Define the downwarddots and crossingsx′ and τ′ to be the right
mates of x and τ (up to the factor t−1 in the latter case):
ij
i i
x′ = • λ:= • λ , (1.12)
j i
j i j i
τ′ = λ :=t−ij1 λ (1=.6)t−ij1 λ . (1.13)
4 J.BRUNDAN
Then to define η′,ε′ and σ′, we assume initially that hh ,λi > 0. Thinking of
i
(1.8) as a column vector of morphisms, its inverse is a row vector. We define the
2-morphisms σ′ and η′ so that −σ′ is the leftmost entry of this row vector and η′
is its rightmost entry:
i i i hhi,λi−1• λ −1
− λ ⊕···⊕ λ :=(cid:18) λ ⊕···⊕ i (cid:19) . (1.14)
i i
Instead,ifhh ,λi<0,themorphism(1.9)isarowvectoranditsinverseisacolumn
i
vector. We define σ′ and ε′ so that −σ′ is the top entry of this column vector and
ε′ is its bottom one:
i λ i −1
i
− λ ⊕···⊕ i :=(cid:18) λ ⊕···⊕ •−hhi,λi−1 (cid:19) . (1.15)
λ
i i
To complete the definitions of σ′,η′ and ε′, it remains to set
i i −1
λ :=− λ if hhi,λi=0, (1.16)
(cid:18) (cid:19)
i i
λ
:=− hhi,λi• if hhi,λi≥0, (1.17)
i λ
i
i
i
:= λ if hh ,λi≤0, (1.18)
λ •−hhi,λi i
i i −1
λ := λ if i6=j. (1.19)
(cid:18) (cid:19)
j j
Now to prove the Main Theoremwe must show that allof the defining relations
for the Khovanov-Lauda 2-category from [CL] are satisfied by the 2-morphisms
in U(g) just introduced. First we will show that Lauda’s infinite Grassmannian
relation holds inU(g). This assertsthat, aswellas the dottedbubble 2-morphisms
r• λ , λ •s ∈ End(1 ) already defined for r,s ≥ 0, there are unique dotted
λ
i i
bubble 2-morphisms for r,s<0 such that
r• λ =0 if r <hh ,λi−1, r• λ =1 if r =hh ,λi−1, (1.20)
i i i 1λ i
λ •s =0 if s<−hh ,λi−1, λ •s =1 if s=−hh ,λi−1, (1.21)
i i i 1λ i
r• λ •s =0 for all t>0. (1.22)
i i
rX,s∈Z
r+s=t−2
KAC-MOODY 2-CATEGORIES 5
Using the infinite Grassmannian relation, we deduce that the inverses of the 2-
morphisms from (1.8) and (1.9) are
i hhi,λi−1 i
− λ ⊕ r• λ (1.23)
i Mn=0 Xr≥0 •−n−r−2
i
and
i −hhi,λi−1 i
− λ ⊕ −n−r−2• λ , (1.24)
i Mn=0 Xr≥0 •r
i
respectively. Severalother of the Khovanov-Laudarelations follow from this asser-
tion;see(3.16)–(3.18)below. Afterthat,wewillshowthatthe2-morphismsη′ and
ε′ define a unit and a counit making the right dual F 1 of E 1 also into its
i λ+αi i λ
left dual:
i i
= , = . (1.25)
λ λ λ λ
i i
Consequently,U(g)isrigid, i.e. allofits1-morphismsadmitbothaleftandaright
dual. Finally we willshow that the other generating2-morphismsare cyclic (up to
scalars):
j i
i i j i j i
• λ= • λ , λ = λ =t−ji1 λ . (1.26)
Acknowledgements. The diagrams in this paper were created using Till Tantau’s
TikZ package.
2. Chevalley involution
In this section we introduce a useful symmetry. First though we record some of
the most basic additional relations that hold in the 2-categoryU(g).
Lemma 2.1. The following relations hold:
i i λ λ
• = • , • = • , (2.1)
λ λ i i
i i
λ = λ , λ = λ . (2.2)
i j i j j j
j j j j j
i •λ − i• λ = i• λ − i •λ = i λ if i=j, (2.3)
0 otherwise,

6 J.BRUNDAN
i i
i − i = r+srX=,sd≥i0j−1tijj r•k•s λ +00≤≤Xpq<<ddjijsipijq j r•k••sq λ if i=k 6=j,
λ λ r,s≥0
j k j k r+s=p−1
 0 otherwise.(2.4)
Proof. The first two relations follow from the definition (1.12) of the downward
dot using the adjunction relations (1.5). The second two follow similarly from the
definition of the rightward crossing (1.6). For (2.3), attach a rightward cap to the
top right strand and a rightward cup to the bottom left strand of (1.2), then use
(2.1) and the definition (1.6). Finally for (2.4), attach a rightward cap to the top
right strand and a rightwardcup to the bottom left strand in (1.4), then use (2.2)
and the definition of the rightwardcrossing. (cid:3)
Taking notation from [CL], we define new parameters from
′t :=t−1, ′spq :=t−1t−1sqp. (2.5)
ij ji ij ij ji ji
The next lemma explains the significance of these scalars.
Lemma 2.2. The following relations hold:
i •j i j i j i• j i j
λ − • λ = •λ − λ = λ if i=j, (2.6)

0 otherwise,

0 if i=j,
i j
 i j
λ = ′tij iλj i j i j if dij =0, (2.7)
′tij dij• λ +′tji •dλji + ′spijq p• •qλ otherwise,
 00≤≤Xpq<<ddjiji
i j k i j k
i j k i j k  ′tijr• •sλ + ′spijq r• •q•sλ if i=k 6=j,
λ − λ =r+srX=,sd≥i0j−1 00≤≤Xpq<<ddjiji
r,s≥0
r+s=p−1
0 otherwise.(2.8)
Proof. Put rightward caps on the top and rightward cups on the bottom of the
relations (1.2)–(1.4), then use (1.5), the definitions (1.6), (1.12), (1.13), and (2.1)–
(2.2). (cid:3)
For any strict k-linear 2-category C, we write Copp for the 2-category with the
same objects as C but with morphism categories defined from HomCopp(λ,µ) :=
Hom (λ,µ)opp, so the vertical composition in Copp is the opposite of the one in C,
C
while the horizontal composition in Copp is the same as in C.
KAC-MOODY 2-CATEGORIES 7
Theorem 2.3. Let ′U(g) be the Kac-Moody 2-category defined as in Definition 1.1
but using the primed parameters from (2.5) in place of t and spq. Then there is
ij ij
an isomorphism of strict k-linear 2-categories
T:′U(g)→∼ U(g)opp
defined on objects by T(λ):=−λ, on generating 1-morphisms by T(E 1 ):=F 1
i λ i −λ
and T(F 1 ):=E 1 , and on generating 2-morphisms by:
i λ i −λ
i i j i −λ λ i
• λ 7→ • −λ, λ 7→− −λ , 7→ , 7→ .
λ i i −λ
i i j
The effect of T on the other named 2-morphisms in ′U(g) is as follows:
i j i i −λ λ i
• λ 7→ • −λ, λ 7→− −λ , 7→ , 7→ ,
λ i i −λ
i j i
j i j i
λ 7→−t−ij1 −λ , λ 7→−tij −λ .
i j i j
Note in particular that T2 =id.
Proof. ToseethatTiswelldefinedweneedtoverifythattheimagesunderTofthe
relations (1.2)–(1.5) and (1.7)–(1.9) with primed parameters hold in U(g)opp. For
the firstthree, this followsfromLemma 2.2, while (1.5) is clear. Forthe remaining
i i
ones, we first note that λ 7→ − −λ by the definition (1.6). Then for
i i
i hhi,λi−1
i
example for (1.8), we must show for hhi,λi ≥ 0 that −λ ⊕ • n is
i Mn=0 −λ
invertible in U(g). This follows by composing (1.9) with diag(−1,1,...,1), using
also (2.1). The rest of the theorem is a routine check from the definitions (1.12)–
(1.19). (cid:3)
We will often appeal to Theorem 2.3 to establish mirrorimages of relations in a
horizontal axis. For example, applying it to (2.2), we obtain the following relation
(whichcouldalsobededuceddirectlyfromthedefinitionofthedownwardcrossing):
Corollary 2.4. The following relations hold:
i j i j j j
λ =tji λ , λ =tji λ . (2.9)
i i
3. The infinite Grassmannian relation
The goalin this sectionis to show thatthe infinite Grassmannianrelationholds
in U(g). We begin by introducing notation for the other entries of the row vector
8 J.BRUNDAN
thatisthe two-sidedinverseof(1.8)andofthe columnvectorthatisthe two-sided
inverse of (1.9): let
i hhi,λi−1 i i hhi,λi−1 λ −1
− λ ⊕ ♠ λ := λ ⊕ n• if hhi,λi≥0,
i Mn=0 n (cid:18) i Mn=0 i (cid:19)
(3.1)
i −hhi,λi−1 ♣n λ i −hhi,λi−1 i −1
− λ ⊕ i :=(cid:18) λ ⊕ •λn (cid:19) if hhi,λi≤0.
i Mn=0 i Mn=0
(3.2)
Wewillgiveamoreexplicitdescriptionofthese2-morphismsin(3.12)–(3.13)below.
Note right away comparing the present definitions with (1.14)–(1.15) that
i i −hhi♣,λi−1
λ = λ if hh ,λi>0, λ = λ if hh ,λi<0. (3.3)
♠ i i
hhi,λi−1 i i
Also for all admissible values of n we have that
n n
i ♣ −λ ♣ λ i
T λ = , T = −λ , (3.4)
♠ ♠
(cid:18) n (cid:19) i (cid:18) i (cid:19) n
asfollowsonapplyingtheisomorphismTfromTheorem2.3tothedefinitions(3.1)–
(3.2) and using (2.1).
Lemma 3.1. The following relations hold:
i hhi,λi−1 i i i −hhi,λi−1i i
= ♠n λ − λ , = •nλ − λ , (3.5)
λ nX=0 n• i λ nX=0 ♣n i
i i
i i
i
λ =0, n• λ =0, n•i λ =δn,hhi,λi−111λ all assuming 0≤n<hhi,λi,
i
(3.6)
i
=0, =0, λ •n =δ 1 all assuming 0≤n<−hh ,λi.
λ •λn i n,−hhi,λi−1 1λ i
i
(3.7)
Proof. This follows from (3.1)–(3.3). (cid:3)
The dottedbubbles r• λ , λ •s defineendomorphismsof1 forr,s≥0. We
λ
i i
alsogive meaning to negatively dotted bubbles by making the followingdefinitions
KAC-MOODY 2-CATEGORIES 9
for r,s<0:
−r−1
i ♣
r• λ := − λ •−hhi,λi if r >hhi,λi−1, (3.8)
i  11λ if r =hhi,λi−1,
0 if r <hh ,λi−1,
i
λ •s := − hhi,λi−i•s♠−1λ if s>−hhi,λi−1, (3.9)
i  11λ if s=−hhi,λi−1,
0 if s<−hh ,λi−1.
i

Note by Theorem 2.3, (2.1) and (3.4) that
T r• λ = −λ •r , T λ •r = r• −λ , (3.10)
(cid:18) i (cid:19) i (cid:18) i (cid:19) i
for all r,s∈Z. Given (3.6)–(3.9), the following theorem implies the infinite Grass-
mannian relation as formulated in formulae (1.20)–(1.22) in the introduction.
Theorem 3.2. The following holds for all t>0:
r• λ •s =0. (3.11)
i i
rX,s∈Z
r+s=t−2
Proof. We prove this under the assumption that hh ,λi ≥ 0; the result when
i
hh ,λi≤0 then follows using Theorem 2.3 and (3.10). We have that
i
r• λ •s (3=.8) hhi,λi n+t−1• λ •−n−1 + r• λ •s
rX,s∈Z i i (3.9) nX=0 i i r≥−X1,s≥0 i i
r+s=t−2 r+s=t−2
(3=.9) hhi,λi+t−1• −hhi,λi−1n+t−1• hhi,λii• λ + r• λ •s
i nX=0 i ♠n r≥−X1,s≥0 i i
r+s=t−2
i
hhi,λi •
(3=.5)− + r• λ •s
λ i i
r≥−X1,s≥0
t−1• r+s=t−2
i i
(1.=17) λ + r• λ •s (2=.3) •λt−1 + −1• λ •t−1 .
t−1• r≥−X1,s≥0 i i i i
r+s=t−2
Itjustremainstoshowthatthe finalexpressionhereiszero. Whenhh ,λi>0this
i
follows by (3.6) and (3.8). Finally if hh ,λi=0 then
i
i
i •t−1
•λt−1 + −1•i λ i•t−1 ((13.=.188)) λ + λ i•t−1 (3=.5)0.
(cid:3)
10 J.BRUNDAN
Corollary 3.3. The following relations hold:
i i
λ = r• if 0≤n<hh ,λi, (3.12)
♠ λ i
n Xr≥0 •−n−r−2
i
♣n λ i
= −n−r−2• if 0≤n<−hh ,λi. (3.13)
i λ i
Xr≥0 •r
i
Proof. We explain the proof of (3.12); the proof of (3.13) is entirely similar. Re-
membering the definition (3.1), it suffices to show that the vertical composition
consisting of (1.8) on top of (1.23) is equal to the identity. Using (3.5)–(3.6), this
reduces to checking that
i
λ
=0 if 0≤n<hh ,λi, (3.14)
i
r•
Xr≥0
•−n−r−2
i
i i
m+r• •−n−r−2 =δ 1 if 0≤m,n<hh ,λi. (3.15)
λ m,n 1λ i
Xr≥0
For (3.14), each term in the summation is zero: if r ≥hh ,λi the counterclockwise
i
dotted bubble is zero by (3.9); if 0≤r<hh ,λi we have that
i
i i i
(1=.2) •r − s• (3=.6)0.
λ
r• sX,t≥0 t•
λ λ s+t=r−1 i
i i
To prove (3.15), note by (3.6) and (3.9) that in order for m+r• •−n−r−2 to
λ
benon-zerowemusthavem+r≥hh ,λi−1and−n−r−2≥−hh ,λi−1. Adding
i i
these inequalities implies that m ≥ n. Moreover if m = n the only non-zero term
in the summation is the term with r = hh ,λi−1−m, which equals 1 by (3.6)
i 1λ
and (3.9) again. Finally if m>n the left hand side of (3.15) can be rewritten as
r• λ •s ,
i i
rX,s∈Z
r+s=m−n−2
which is zero by (3.11). (cid:3)
Onsubstituting(3.12)–(3.13)intothedefinitions(3.1)–(3.2),thisestablishesthe
assertions about the 2-morphisms (1.23)–(1.24) made in the introduction.
Corollary 3.4. The following relations hold:
i i
hhi,λi−1 r• i
λ
λ = i •−n−r−2 − λ , (3.16)
nX=0 Xr≥0 n• i
i i