Data for quiver Hecke algebras

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 10 April 2010

Data for quiver Hecke algebras

1. is the free algebra generated by fi,iI. Q+=iI0idegfi1fid=i1++id. The symmetric group Sd=1d-1 acts on Id=words of length d=uI* |lu=d (by rearrangements) with orbit decomposition Id=Q+ ,ht=dII=uI*|degu=.
2. Fix a symmetric bilinear form ,:Q+Q+ given by values ij so that A=iji=2iii is a Cartan matrix for a symmetrisable Kac-Moody Lie algebra.
3. is the graph with vertices I and edges i j if ij0. Fix an orientation ij where ij=+1 if i j and ij=-1 if i j and set Qijuv=0,i=j1,ijij=0ijv-ji-u-ijijij0.

Quiver Hecke algebras ,Q+

is the associative -graded algebra given by generators eu,x1eu,,xdeu,1eu,,d-1eu,uI with degrees degeu=0,degxieu=uiui,degieu=-uiui+1 where ui is the ith letter in u, and relations euev=uv,uIeu=1,xixj=xjxi,xieu=euxi,ieu=eiui,ij=ji,ifii,i1i2eu=Qui,ui+1xixi+1eu,i+1ii+1-i+1ii+1eu=1xi+2-xiQui+2,ui+1xi+2xi+1-Qui+1,uixi+1xi,if ui=ui+20,if uiui+2, ixjeu=xijieu-ijeu,if ui=ui+1,xijieu,if uiui+1.

Note: xi=uIxieu,andi=uIieu.

Structure of

For each S fix a reduced word =i1il and set =i1il.

(Khovanov-Lauda Rouquier)
has basis x1n1xdndeu|uI,Sd,n1,,nd0. If Q+ and Q+ then I+ =Sk+lSkSlI.I and there is a homomorphism +euevxieuveuevxieuveuxjevxj+keuvieuevieuveujevj+keuv where k=ht.

(Khovanov-Lauda)
+ is a free (right) -module with basis 1|Sk+lSkSl with 1=I,vIeuv. For M-mod and N-mod, M-N=Ind+MN.

Graded characters

-mod is the category of finite dimensional -graded -modules: M=iMiwithjMiMi+j, where j=elements of degree j. M=iuIMuiwithMui=euMi. The graded character of M is gchM=iuIdimMuiqifu. Then gchMN=gchMgchN where the right hand side is ,-shuffle product.

Let K-mod be the Grothendieck group of -mod. Q+K-modU-MgchMLgbg* is an algebra isomorphism, where bg*|gG is the dual canonical basis of U-, Lg|gG are the simple -modules in -mod. Q+K-modU-MgchMLgbg*gEg* where Lg,gG are the simple -modules, if lL then l=Ll and g=l1lk if g=l1lk with l1,,lkGL and l1lk.

g has uniue simple quotient Lg.

Projective -modules

Proj is the category of finitely generated -generated projective -modules. Let Pi be the unique indecomposable projective i-module: Pi=spanx1ne1|n0x1. For PProj and QProj, define PQ=Ind+PQ. Let KProj be the Grothendieck group of Proj.

(Khovanov-Lauda, Rouquier)

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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