An application of power operations
toquantum groups

Yaping Yang
20-24 Feb, 2023

Kyoto, Japan
Joint with Gufang Zhao.

History:⑪AUg(8) A(eg) Nakaj(in)19:Nakajima

Varagnoloit 19)

Heg/Nakay(52).

*21:kontserich-Soibelman

modelof

Cohomological Hallaly.
A Heg)
130 COHA
HCR, w)framed reps of ,Prw)

quitter potentiaare
26:

G. Zhao-Y.

G:

quiver

Q:Davison.

CADE)

W l. 2x, x*3.
=a②.

T1)
&dim-reduction)- >-00Atetriple quiredin 150)
Then,II

Critical cohomology->

D(FCC, wi)T1) = DS44 (H14,WI).
N. W.
->

End (H*(NaKC)-> D(H19-w) -K.S.

alghom.

ecohomology groups
theories-(**,1)A*==

Drinfeld

double ofA* - COHA:④ A)Rep,(a)I
YE

preprojective

Integrm

kep (Ta)-Example:A* k-theory I S2.
==k [Z,zm*...En

UgIStel+g==generators
G:

E E(nxIG*J.

[I8,8*]

overe.

allEcr:-13=k*(pt) [Tg*]/qP1.

of 1.

root==-:-InJgE?

nnDInitaEj

Di *1anResidue
ThmEi-s-I1n [n]ge=>=#1)

Power⑤Operations.

Cohomology theories A*with power operations

e.g:abelianp-gP.,

skip.

IiiU

(e.g.i4pSn,Letit 1==Sp).

Ho-ring theories

calledS(a) Pi id,
=(B)

(c)

=

res/ Pikxs
Pj(P1) vesEY (Pj(x)==-61CompositivePj xxxy)-res)*3,*(x) xPy (y)=(by ordering elts ofitin

Sie: a gp, that

POK 14jxx) xP(x)Sr->

has order equal to a power ofevery elf

xir:

PC.

Map[ir, X]. =xY=operation:#Pir:E4X)+->z

E*(EX)

satisfies certain conditions:***Ei Xi>-3E*(X)

⑭=
ample:H*) -, p), Steenrod operations.

P.

(Steend (2)is, as!
②K-theory, exectrpowers, Adams operations
C Atiyah66)
Elliptic cohomology

istreefx

top:0 4r:kx) -> kCX)

ring howit.
fixxy, 4if*=③④MCL) [L]?FlineGrale=+P(x) aP PP,=+for some =((X).

③Complexcobordism

368, Quillen7):(tom Dieck

multiplicative!
/

4. The theories Ent ofHopkins Miller (93)structEo-ringlift)(Movara E-theory, integral--Ando 95:

En* has power

And09:E(pt)=)height kf,exoviatation(coordinateor~

deformation ofagoWCp) [[Wi-Hring)

Lubin-Tateuniversal-operations-.

overphwp-3].

Wittring of Ipn.

xonGE2)KIPW)

misformal splar
on

E.

7.

fE

cF(xM)afirstchem class oflineGrdlea=>f(cECL),c,E(l)BExx** X

E(X) E)

E

2UA

*E

(Ei*XY)--⑪
X-

Note:&a#*(BI) * E*(X)cohomology-xHkR 7.

theory.

E*(X)E*cpt)&·e.g:ix)(Bi xIIS

E*(X) is again

Excpt)*↳Cyclotonic=:

E*(Bi)(5]*E*(x)(where. intip,
I 9aE(3/28i non-trivie]**extension

Sin EXBI)

gen. byset

xp,=non-trivlk(pt) .
=k(p)(B4p)[3)2(x)/,1 xP) [*]line Gun alles-=-of2(. ] 25p]==

cod p-th roots of1.

ExCX)

IPH:E*(X)Im(95):

*

deF

subzt

⑦- >

*

F

Let H Hom((/pi)". KY
=D.

IaIPH

unique=form-gplaw F on Ex,athe composition

5,t:AdamsK-theoryof
operations
inexists.

② IPH is*ahomomorphineringPr) eEx) fH(,(4)): n(e()
E③=IPH) F) EE*cp) (itw].

==*=

where~F

egFxpNED),
Full

IPP:

Then:

whereEnCX)-EnCX), identity on

IPp* (eL)=[p"]F(CL).

forregp law

onExcpt) s.t(Ibow ofgpew.

Enpt"(13)Inparticular.

f(x) f(13)=Fline Grelles.
X.G C, of=↓IPH( 43) (PH (x) =1P (2)x=↳Quiver with automorphism.

#)

Q (IcH)

quiver,=admissible

A:a:It Ia:H ->Assumptions:p.

-No② 7 minn- --sit

In0[ach)] aCTh3).
=edge join two vertices in theWarning:e.g:permation

HFnEH,

sit=

automaphiom ofG.

-it:iYEI)

are on 60th

Lusztin's setup,

0750

-9

Chot allowed)

I&a-orbits

sameH., UniformH.

"n" doesn'tneed to be uniform.

>--(not allowed)↑(with more arrows↓

↓

O

⑳

00=0-0Ok⑧
alsook).

IOk.
⑧

↳Define:Q: I I,03

Ho H/aI I/a==a-orbits=aof1.

a=-bits ofH.

gep(a).

Hemma:Rep 10%) Rep(Q)=9under the assumptions.
Pf

EmbeddingofReple)-VEvii

Rep1a) avie vi

into-

vis V eRepre··But ifVio

Vi:=

Xh=

asling

Xho a s longasifa-orbi-ashearabi

=

hoof

VeNa

Then theaction ofcanie:

onRepla, V) is by(x,y.z)

permutation.

(2,x,y)1-> (y,z,x)1
this is the action of
<aJ->

a1

↳use

*

67:

the*(Replaf.gl. F.

UseAI

E*(teeple)) -> ExCRepleos)(a)(a)-asPetrole
CVolalg,hom,=>>>E*(8ep19%)

thefalgplar
nF

given by IPca).

⑪In-⑪formula

E*(Rep(a)).
(a)

F.

②S

isan

-*->

algebra(Rep1x([5a)-> E* ( Rep1ao))
E,-bow.

(arg.str.
A

Ex (RepCITa)

In particular

③F

roE

Hallaly structures)

3 Replity)

isan

UI

UE(a) Am,

neca agency

Ea(12ep(a))

E(Rep(π)=>AEx) (NaGY,-y))ofmod.

E* (Nak(y))- >FEnsplusituse.

* eallyhom.

UIframing,

5.8t-how.

66

⑫19

Nof:I.

defEx*y=Eq*(x-y.eV))**(x*y) 0*=q(x-y.eck)

E9,*((*x.(y). **e(k)E=gi)=t*(x). *(3).

(*x)*(y)=ECU).),

whereO WoxD(a)
=I
asa

⑬

liation:Recover Lustig'sgFrob. map.
E* 12-theory.

g Sl2 (illustration==PIn.

Repla n)plu

otherwise<spepey->

cyclotomic

↓* ⑬
Wax
it+

=quiver

extension

automorphism.

asa

Yar(

**So:

aPermutes

theRepcee, 5)z, het,...,z.

Suq I =[8]/(qP1).

Q:

terSgetzeromap.

***t,...

perpose).

rankiefix2,↑i)2p+1.

a)aE1,--⑭E,.....

Agt,...,zeigt2,9, ..., Eia...

Gor.. 1 This is②

Lustig's

anall homomorphism

Take the9: FU,

Zpyt z,gt. -,5.""

subaly gen. byI

1>

then+U11)

Ue(8) Ert
ECRIn,9el)↓Plu
herwise.

o18

Evt

otivation/Ik,charc . M*(Y),X: HeCT, 1kY char. lafice
==x+

9x X)

-*xied.

ch(V):+x

LN /1k.

xired: 9x x +/0x(X.Ci><p>=2dimV(a) M

XtX-=MEXGFN:GATGA&].

Tnt space

By Steberg Thm:x1Vxc

Fvv. (G)1 ->1(Lustig 2.sofG.

-INS

<N.Ci

=&E1.

5+x 10=pi, +..

+

chC2N3 =ch(Lao)->Ex. (EXY)
PPEtig's anj.

chChai)**

PYnm,+91

p-adic expansion

DrEXiedchChse)923.

(Ex) 2]...

...

. . . .

32(x2x=1iX]

Lusztig15'-1↓lino.
#=zTh] z(xep x)↑2&=Ex=Ex Weylchar form a ⑯=Ex 2Pa,==n 1.
=1PM, ExorEx 2=-113),ch((x) Eq.
Lirep. of/=1.
Hz(y),51=

LettxeX+

Li:=irrep ofUs(0),1tX+#hen:ch((x) Ex E,.

[,0)

=.

=

where:qFr:U9(0)Eie.

utx+-n 0:iU1J)->

In particular.

·0,j(i)[Lp",] [h*=13reStarryit is&em/Hopeute......

②->

Frobenius

for quantum groups,

mapUs,) US18)

Hope the->Moravare->⑰

is aU? 18) Es U-18)*U18)

En-theoretical quantum gpsjob.
1ofin*:⑳ Imotswill do theEn* (BYp].
X**P4).

⑱roughly

En*(pt)[(x56/jp3,.
CHKR]=-Enpt)⑯ Stapleton:I

X,*I*
ep-copies.

cl prfree mod of-am[P],*

whereEncpt)-aly:Gu,height- -> p3* (p)

-⑭*

↓-

Spec (n,1
VS:THKR]↓spectheights↓

Spec Ecp+)

->

94p)

sit.

DaS- >

Spec E (pt)->hag htR