Almost semisimple algebras

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

Last update: 11 May 2012

Convolution algebras

The decomposition theorem

Let M be a smooth G-variety and let N be a G-variety with finitely many G-orbits such that the orbit decomposition is an algebraic stratification of N, N = φGxφ, and μ:MN is a G-equivariant projective morphism. Let 𝒞M be the constant perverse sheaf on M. The decomposition theorem [CG, 8.4.12] says that μ*𝒞M = i λ=(φ,χ)M^ L(λ,i) ICλ[i] λM^ L(λ) ICλ, where L(λ) = i L(λ,i) , μ* is the derived functor of sheaf theoretic direct image, λ runs over the indexes of the intersection cohomology complexes ICλ, L(λ) are finite dimensional vector spaces, and &eqdot; indicates an equality up to shifts in the derived category.

Convolution algebras

Let μ:MN be a proper map. The convolution algebra is A = ExtDb(N)* ( μ*𝒞M, μ*𝒞M ) = k Extk ( μ*𝒞M, μ*𝒞M ). The decomposition theorem for μ*𝒞M induces a decomposition of A. Since the intersection cohomology complexes ICφ are the simple objects in the category of perverse sheaves, ExtDb(N)0 (ICλ,ICμ) = δλμ, and ExtDb(N)k (ICλ,ICμ) = 0, for   k<0, and the decomposition of A simplifies to A = λM^ End(L(λ)) k>0 λ,μM^ Hom ( L(λ), L(μ) ) ExtDb(N)k (ICλ,ICμ) . In the context there is a good theory of projective, standard and simple modules, and their decomposition matrices satisfy a BGG reciprocity. View elements of A as sums λ,μ PL^(λ), QL^(μ) cPQλμ aPQλμ where cPQλμ , and aPQλμ k>0 ExtDb(N)k (ICλ,ICμ). The algebra A is completely controlled by the dimensions of the L(λ) and the multiplication in Abasic = Ext* (IC,IC) where IC = λM^ ICλ, an algebra which has all one dimensional simple modules. The radical filtration of A is Radl(A) = λ,μM^ Hom (L(λ),L(μ)) kl ExtDb(N)k (ICλ,ICμ) and the nonzero L(λ)   are the simple   A-modules.

Projective modules

Let eλ be a minimal idempotent in μ End(L(μ)). Then P(λ) = Aeλ = L(λ) k>0 μ L(μ) ExtDb(N)k (ICμ,ICλ) is the projective cover of the simple A-module L(λ). Define an A-module filtration P(λ) P(λ)(1) P(λ)(2) by P(λ)(m) = km μ L(μ) ExtDb(N)k (ICμ,ICλ). Then L(λ) = P(λ) / P(λ)(1) and gr(P(λ)) is a semisimple  A-module. Thus the multiplicity of the simple A-module L(μ) in a composition series of P(λ) is [P(λ) : L(μ)] = dim( Ext* ( IC𝕆,χ,IC𝕆,χ ) ) = k0 dim( ExtDb(N)k (ICμ,ICλ) ).

Standard and costandard modules

Let λ=(φ,χ), x𝕆φ, and let ix: {x}N be the injection. Then ix!μ*𝒞M is the stalk of μ*𝒞M at x and the Yoneda product makes Δφ = H* (ix!𝒞M) = HomDb({x}) ( , ix!μ*𝒞M[*] ) = HomDb(N) ( (ix)![-*], μ*𝒞M ), and, φ = H* (ix*𝒞M) = H* ( {x}, ix*μ*𝒞M ) = HomDb({x}) ( , ix!μ*𝒞M[*] ) = HomDb(N) ( (ix)![-*], μ*𝒞M ), into right A-modules. The action of an element a Extk(μ*𝒞M,μ*𝒞M) = HomDb(N) ( μ*𝒞M, μ*𝒞M[k] ) sends H* ( {x}, ix!μ*𝒞M ) H*+k ( {x}, ix!μ*𝒞M ).

A G-equivariant local system is a G-equivariant locally constant sheaf. The orbit 𝕆φ can be identified with G/Gx where Gx is the stabilizer of x. π0(𝕆φ,x) = Gx/Gx° where Gx° is the connected component of the identity in Gx. There is a homomorphism π1 (𝕆φ,x) π0 (𝕆φ,x) = Gx/Gx° and the representations of π1(𝕆φ,x) on the fibers x of G-equivariant local systems are exactly the pullbacks of finite dimensional representations of C = Gx/Gx° to π1(𝕆φ,x). In this way the irreducible G-equivariant local systems on 𝕆φ can be indexed by (some of the) irreducible representations of Gx/Gx° [CG, Lemma 8.4.11]. There is an action of C = Gx/Gx° on Δφ which commutes with the action of A. Similar arguments apply to φ. As (A,C) bimodules, Δφ = χC^ Δ(φ,χ)χ and φ = χC^ (φ,χ)χ, and the standard and costandard A-modules are Δ(λ) = Δ(φ,χ) and (λ) = (φ,χ). Using the decomposition theorem Δ(λ) = H*( ix!𝒞M )χ = k μ L(μ) Hk( ix!ICμ )χ, where the subscript χ denotes the χ-isotypic component. Define a filtration Δ(λ) Δ(λ)(1) Δ(λ)(2) by Δ(λ)(m) = jm φ L(μ) Hj(( ix!ICμ )χ. Then Δ(λ)(m) is an A-module and gr(Δ(λ)) is a semisimple A-module. This (and a similar argument for (λ)) show that the multiplicity of the simple A-module L(μ) in composition series of Δ(λ) and (λ) are [Δ(λ):L(μ)] = kdim( Hk( ix!ICμ )χ ) and [(λ):L(μ)] = kdim( Hk( ix*ICμ )χ ). Define the standard KL-polynomial and the costandard KL-polynomial of A to be PλμΔ(t) = ktk dim( Hk( ix!ICμ )χ ) and Pλμ(t) = ktk dim( Hk( ix*ICμ )χ ), respectively. Then ??? says that [Δ(λ):L(μ)] = PλμΔ(1) and [(λ):L(μ)] = Pλμ*(1). These identities are analogues of the original Kazhdan-Lusztig conjecture describing the multiplicities of simple 𝔤-modules in Verma modules.

The contravariant form

Note that there is a canonical homomorphism Δ(λ) cλ (λ) coming from applying the functor H* to the composition (ix)! (ix)! μ*𝒞M μ*𝒞M (ix)* (ix)* μ*𝒞M, where the two maps arise from the canonical adjoint functor maps. Use the map cλ to define a bilinear form on Δ(λ) by ,: Δ(λ) Δ(λ) m1m2 m1cλ(m2) Then L(λ) = Δ(λ) / Rad(,).

Contragradient modules

There is an involutive automorphism At: AA on A (coming from switching the two factors in Z = M×NM ). If M is an A-module the contragradient module is M* = Hom(M,) with (aψ)(m) = ψ(at(m)), for   aA,  ψM*,  and   mM. Then (λ) Δ(λ)*.

Reciprocity

If λ=(φ,ρ) define dλ = dim(𝕆φ), and assume that Ext Db(N) dψ+dφ+k (ICφ,ICψ) = 0, for all odd   k. Then [P(λ):L(μ)] = kdim ExtDb(N)k (ICλ,ICμ) = kdim Ext Db(N) dλ+dμ+k (ICλ,ICμ) = k (-1)k dimExt Db(N) dλ+dφ+k (ICλ,ICμ) = (-1)dφ+dψ 𝕆χ( 𝕆, i𝕆! ICφ ! i𝕆! ICψ ) = (-1)dφ+dψ 𝕆χ 𝕆, (-1)dφ α,k [ k i𝕆! (ICφ) : α ] α ! (-1)dψ β,l [ l i𝕆! (ICψ) : β ]β = 𝕆,α,β χ 𝕆, k [ k i𝕆! (ICφ) : α* ]α ! l [ l i𝕆! (ICψ) : β ]β = α,βk dimk (iα!ICφ) 𝕆 χ( 𝕆, α*!β ) l diml (iβ!ICψ) = α,β [α!:Lφ] 𝕆 χ( 𝕆, α*β ) [β!:Lψ] = α,β Pφα(1) Dαβ Pψβ(1) = (PDPt)φψ, where

  1. the third equality follows from the vanishing of Ext groups in odd degrees,
  2. χ denotes the Euler characteristic,
  3. P is the matrix (Pφα(1), and
  4. D is the matrix 𝕆 χ( 𝕆, α*β ) .
This identity is the "BGG reciprocity" for the algebra A.

The Steinberg variety

Let xN and define Z = M×NM = { (m1,m2)M×M | μ(m1) = μ(m2) } and Mx = μ-1(x). There are commutative diagrams Z=M×NM M×M N=NΔ N×N ι Δ μ12 μ1×μ2 and Mx M {x} N ι ix μ μ which (via base change) provide isomorphisms H*(Z) = HomDb(Z12) ( Z12, ( Z12[*] ) ) = HomDb(Z12) ( μ12*N, ι! 𝒞M1×M2 [m1+m2][-*] ) = HomDb(N) ( N, (μ12)*ι! 𝒞M1×M2 [m1+m2-*] ) = HomDb(N)( N, Δ! (μ1×μ2)* ( 𝒞M1 𝒞M2 ) [m1+m2-*] ) = HomDb(N) ( N, Δ!( (μ1)*𝒞M1 (μ2)*𝒞M2 ) [m1+m2-*] ) = Ext Db(N) m1+m2-* ( (μ1)* 𝒞M1, (μ2)* 𝒞M2 ), H*(Mx) = HomDb(Mx) ( Mx, ( Mx [*] ) ) = HomDb(Mx) ( μ*{x}, ( (ι*M)[*] ) ) = HomDb({x}) ( {x}, μ*( ι!M[2m] ) [-*] ) = HomDb({x}) ( {x}, ix!μ*𝒞M [m-*] ) = Hm-* ( ix!μ*𝒞M ), and H*(Mx) = HomDb(Mx) ( Mx, Mx[*] ) = HomDb(Mx) ( μ*{x}, Mx[*] ) = HomDb({x}) ( {x}, μ*Mx[*] ) = HomDb({x}) ( {x}, μ!ι* M[*] ) = HomDb({x}) ( {x}, ix*μ!M[*] ) = HomDb({x}) ( {x}, ix*μ* 𝒞M[*-m] ) = H*-m ( ix*μ*𝒞M ).

The category Db(N)

The category Compb(Sh(N)) is the category of all finite complexes A = ( 0A-m A-m+1 An-1 An0 ), m,n0, of sheaves on N with morphisms being morphisms of complexes which commute with the differentials. The jth cohomology sheaf of A is j(A) = ker(AjAj+1) im(Aj-1Aj) . A morphism in Compb(Sh(N)) is a quasi-isomorphism if it induces isomorphisms on cohomology. The category Db(Sh(N)) is the category Compb(Sh(N)) with additional morhpisms obtained by formally inverting all quasi-isomorphisms.

Assume that N is a G-variety with a finite number of orbits such that the G-orbit decomposition N = φ𝕆φ is an algebraic stratification of  X. A constructible sheaf is a sheaf that is locally constant on strata of N. A constructible complex is a complex such that all of its cohomology sheaves are constructible.

The derived category of bounded constructible complexes of sheaves on N is the full subcategory Db(N) of Db(Sh(N)) consisting of constructible complexes. Full means that the morphisms in Db(N) are the same as those in Db(Sh(N)).

The shift functor [i]: Db(N) Db(N) is the functor that shifts all complexes by i.

The Verdier duality functor A: Db(N) Db(N) is defined by requiring HomDb(N) (A1,A2[i]) = HomDb(N) ( Δ* (A1A2) [-i], N [2dimN] ), for all   i, where Δ:NN×N is the diagonal map.

The Verdier duality functor satisfies the properties (A) = A, (A[i]) = A [-i], and HomDb(N) (A1,A2) = HomDb(N) (A2,A1). Define ExtDb(X)k (A1,A2) = HomDb(X) (A1,A2[k]), Hk(A) = Hk(X,A) = HomDb(X) (X, A[k]), the hypercohomology of   ADb(N), Hk(A) = HomDb(N) (N, N[k]), the cohomology of   N, Hk(N) = HomDb(N) ( N, ( N[k] ) ), the Borel-Moore homology of  N, 𝔻X = X, the dualizing complex, respectively. The Yoneda product ExtDb(N)p (A1,A2) × ExtDb(N)q (A2,A3) ExtDb(N)p+q (A1,A3) is given by HomDb(N) (A1,A2[p]) × HomDb(N) ( A2[p], A3[p+q] ) HomDb(N) (A1, A3[p+q]), using the canonical identification HomDb(N) (A2,A3[q]) HomDb(N) (A2[p],A3[p+q]).

If f:XY is a morphism define f* = derived functor of sheaf theoretic direct image, f* = derived functor of sheaf theoretic inverse image, f!A = ( f*A ),   for   ADb(Y), and f!A = ( f*A ),   for   ADb(X). Then HomDb(X) ( f*A1, A2 ) = HomDb(Y) ( A1, f*A2 ), and, HomDb(X) ( A2, f!A1 ) = HomDb(Y) ( f!A2, A1 ). If f:XZ and g:YZ define the base change formula as X×ZY Y X Z π2 f π1 g g!f*A = (π2)* π1!A, for   ADb(X), where X×ZY = {(x,y)X×Y | f(x)=g(y)}.

The category of perverse sheaves on X is a full subcategory of Db(X) which is abelian. The simple objects in the category of perverse sheaves are the intersection cohomology complexes ICφ indexed by pairs φ = (𝕆,χ), where 𝕆 is a G-orbit on X and χ is an irreducible local system on X. By ???, the local systems χ on 𝕆 can be identified with (some of the) representations of the component group ZG(x) / ZG(x)° where x is a point in 𝕆. If X is smooth the constant perverse sheaf 𝒞X on X is given by |𝒞X|Xi = Xi [dimXi], on the irreducible components of X. Since the intersection cohomology complexes ICφ are the simple objects in the category of perverse sheaves, ExtDb(N)0 (ICφ,ICψ) = δφψ and ExtDb(N)k (ICφ,ICψ) = 0, if   k>0.

Dlab-Ringel algebras

Let C and D be rings, L, a   (C,D)   bimodule, R, a   (D,C)   bimodule, and ε: LDR C, a (C,C) bimodule homomorphism. Define an algebra A = CDL RRCL and product determined by the multiplication in C and D, the module structure of R and L and the additional relations cr=0, dl=0, rd=0, lc=0, and (r1l1)(r2l2) = r1ε(l1r2)l2. Let

  1. eC be the image of the identity of C in A, and
  2. eD be the image of the identity of D in A.
Then, if e=eC then 1=eC+eD, C = eCAeC, L = eCAeD, R = eDAeC, D = eDAeD, so that A = { | c l r d | cC, lL, rR, dD } with matrix multiplication. Then
  1. eDAeD=D+RCL is a subring of A, and
  2. RCL is an ideal in eDAeD, and
  3. RCL = eDAeCAeD.

Structure of Z(ε)

Let ε: LDR C be a   (C,C)   bimodule homomorphism. The left radical L(ε) and the right radical R(ε) of ε are defined by L(ε) = {lL | ε(lr)Rad(C), for all rR}, R(ε) = {rR | ε(lr)Rad(C), for all lL}. The map ε is nondegenerate if Rad(C) = 0,   L(ε) = 0, and R(ε) = 0. Let C_ = C/Rad(C), L_ = L/L(ε), R_ = R/R(ε), and φ: RCL R_C_L_ r_l_ rl_. Then kerφ is generated by RCL(ε) and R(ε)CL, and we have that kerφR R(ε) and Lkerφ L(ε). Then I = Rad(C) + L(ε) + R(ε) + kerφ is a nilpotent ideal of   A(ε), and A(ε)I A(ε_) where the map ε_: L_DR_ C_ lr l_r_ is a nondegenerate (C_,C_) bimodule homomorphism.

If ε:LDRC is nondegenerate and R is a projective C-module then there is a (D,C) bimodule isomorphism τ: R L* r λr: L C l ε(lr) so that ε = ev(idτ) and A(ε) A(evL).

If C,D,L,R are finite dimensional vector spaces over 𝔽 and D=𝔽 then ε = ε0evP: ( L0P* ) D ( R0P ) C, with P projective and imε0 Rad(C).

If ε = ε0evP with P finitely generated and projective then A(ε)-mod A(ε0)-mod M eM where e = 1-i piαi.

If imε Rad(C) then Rad(A(ε0)) = I = Rad(C) Rad(D) L0 R0 R0CL0 and A(ε0) Rad(A(ε0)) C Rad(C) D Rad(D) .

The module category of Z(ε)

Let 𝒞 and 𝒟 be categories F:𝒞𝒟 and G:𝒞𝒟 be functors, and FεG, a natural transformation. Define a category 𝒜 with

  1. Objects: ( M,V; FM GM V εM m n ),   where   M𝒞,  V𝒟,  and   m,nMor(𝒟),
  2. Morphisms: (f,g) with fMor(𝒞), gMor(𝒟) such that FM GM V εM m n FM GM V εM m n commutes.
A fundamental case is when 𝒟 is the category of vector spaces over 𝔽.

The equivalence between the category 𝒜 and the module category of Z(ε) is given by letting 𝒞=C-mod and 𝒟=D-mod and F: 𝒞 𝒟 M RCM and G: 𝒞 𝒟 M HomC(L,M) where the D-action on HomC(L,M) is given by (dφ)(l) = φ(ld), for   dD,  lL,  and   φHomC(L,M). Then let ε:FG be the natural transformation given by ε: F G RCM εM HomC(L,M) rm τ: L M l ε(lr)m. Then 𝒜 A-mod (X,Y,ρ,λ) M where X=eM, Y=(1-e)M, and the L-action and R-action on M define ρ and λ via ly = (λ(y))(l) and rx = ρ(rx), for   lL,  rR,  xX,  and   yY. Note that lx=0   and   ry=0, for   lL,  rR,  xX,  yY, and RCX = FX GX Y εX ρ λ = HomC(L,X) commutes.

Macpherson-Vilonen

Let X be a Thom-Mather stratified space with a fixed stratification such that all strata have even codimension. Let P(X)   be the category of perverse sheaves on X. Let S be a closed stratum such that S is contractible and let ι:X-S X, be the inclusion. Let j:L-KL, where L = the link of   S ∪| K = perverse link of   S,   a closed subset of  L. Let F: P(X-S) {vector spaces} P -d-1 (K;P) and G: P(X-S) {vector spaces} P -d (L,K;P) = -d (L,j! P|L-K). Let 𝒜 be the corresponding category as in the previous section. Then the map P(X) 𝒜 Q |Q|X-S -d-1 (K,Q) -d (L,K;Q) -d (𝔻,K;Q) εX ρ λ is an equivalence of categories, where |Q|X-S = ι*Q, and εQ is the coboundary homomorphism in the long exact sequence for the pair L,K. What is 𝔻????

Examples.

  1. The flag variety.
  2. The nilpotent cone.

Quasihereditary algebras

Let 𝔽 be a field. A separable algebra over 𝔽 is an algebra A such that ARad(A) λ ϵA^ Mdλ (𝔽). Two algebras A and B are Morita equivalent if Mod-A is equivalent to Mod-B (Check this in Gelfand-Manin).

A ring A is semiprimary if there is a nilpotent ideal Rad(A) such that A/Rad(A) is semisimple artinian. Note: If A is finite dimensional then A is semiprimary.

A hereditary ring is a ring A such that every submodule of a projective module is projective.

A hereditary ideal is an ideal J such that

  1. J is projective as a right A-module,
  2. J2=J, and
  3. JRad(A)J=0.
Note: J2=J if and only if there is an idempotent eA with J=AeA.

A quasihereditary ring is a semiprimary ring A with a chain of ideals 0=J0 J1 Jm=A such that Jl Jl-1   is a hereditary ideal of   A Jl-1 for each 1lm-1.

Let A be a quasihereditary algebra 0=J0 J1 Jm=A. Let e be an idempotent in A such that Jm-1 = AeA and eA(1-e) Rad(A). Let C=eAe and D = AAeA = AJm-1 and L D C = eA(1-e) and R D C = (1-e) Ae and let ε: LDR C lr lr. Assume D is a separable k-algebra. Then

  1. D+(1-e)AeA(1-e) = (1-e)A (1-e),
  2. A=C(ε),
  3. C is quasihereditary with heredity chain 0=I0 Im-1 =C, where Il = eJle.

Highest weight categories

Let A be a finite dimensional algebra and let A^ be an index set for L(λ), the simple  A-modules. Let P(λ) be the projective cover of   L(λ),   and I(λ) the injective hull of   L(λ). Let be a partial order on   A^. Let (λ) be the largest subobject of   I(λ)   with composition factors   L(μ)   with   μλ, Δ(λ) be the largest quotient of   P(λ)   with composition factors   L(μ)   with   μλ. Then 𝒜 = A-mod is a highest weight category if P(λ) has a filtration 0= P(λ)(m) P(λ)(1) P(λ), with P(λ) P(λ)(1) Δ(λ) and P(λ)(k) P(λ)(k+1) Δ(μ), with   μ<λ, for 1km-1.

Highest weight categories satisfy BGG-reciprocity, [I(λ):(μ)] = [Δ(μ):L(λ)].

Proof.
Since Ext1(Δ(λ),(μ)) = 0 and Hom(Δ(λ),(μ)) = { End(L(μ)), if   λ=μ, 0, if   λμ, } if follows that Hom(Δ(λ),M) = ( number of (λ) in a -filtration of M ). Thus [I(μ) : (λ)] = dim(Hom(Δ(λ),I(μ))) dim(End(L(λ))) = [Δ(λ) : L(μ)]. How does this proof compare to the proof for convolution algebras in Chriss and Ginzburg?

Examples of highest weight categories.

  1. G=G(𝔽_),  𝒜 the category of finite dimensional rational G-modules, and (λ) = H0(G/B,λ),
  2. 𝒜 the category 𝒪, and (λ) = M(λ).
Vogan, Irreducible characters of semisimple Lie groups II; The Kazhdan-Lusztig conjectures Pyw = i qi dim( Ext l(w)-l(y)-2i (My,Lw) ), for   yw.

Let A be a finite dimensional algebra and let 𝒜=A-mod. Then 𝒜 is a highest weight category if and only if A is a quasihereditary algebra.

Proof.
⇒) Assume 𝒜 is a highest weight category. Let λ be a maximal weight and let P(λ) = Aeλ and JAeλA. Then J is projective as a left A-module, HomA(J,A/J) = 0, JRad(J) = 0. So J is a hereditary ideal. Finally, (A/J)-mod is a highest weight category with (A/J)^ = A^-{λ}.

⇐) Assume A is a quasihereditary algebra, 0=J0J1 Jm=A. Define λ<μ if L(λ)   appears in   Ji/Ji-1 Rad(Ji/Ji-1) and L(μ)   appears in   Jj/Jj-1 Rad(Jj/Jj-1) , with i<j. Suppose i is (the unique integer) such that L(λ) appears in (Ji/Ji-1) / Rad(Ji/Ji-1) and let Δ(λ)   be the projective cover of   L(λ),   as an   A/Ji-1   module. Then L(λ) is the simple head of A(λ) and, since Ji-1 Rad(A/Ji-1) Ji-1 = 0, all other composition factors of A(λ) are lower.

If L(λ) is a simple A-module then there is an idempotent eλA such that P(λ) = Aeλ (eλ is a minimal idempotent). Then 0=J0eλ J1eλ Jmeλ = Aeλ = P(λ) is a good filtration of P(λ).

Duals and Projectives

Let L be a C-module and let Z=EndC(L) so that L is a (C,Z) bimodule. The dual module to L is the (Z,C) bimodule L* = HomC(L,C). The evaluation map is the (C,C) bimodule homomorphism ev: LZL* C lλ λ(l) and the centralizer map is the (Z,Z) bimodule homomorphism ξ: L*CL Z λl zλ,l : L L m λ(m)l. Recall that [Bou, Alg. II §4.2 Cor.]

  1. L is a projective C-module if and only if 1imξ,
  2. If L is a projective C-module then ξ is injective,
  3. If L is a finitely generated projective C-module then ξ is bijective,
  4. If L is a finitely generated free module then ξ-1(z) = i bi*z(bi), where { b1,...,bd } is a basis of L and { b1*,...,bd* } is the dual basis in M*.
Statement (a) says that L is projective if and only if there exist biL and bi*L* such that if   lL   then   l = i bi*(l)bi, so that   ξ i bi*bi = 1.

Cellular algebras

A cellular algebra is an algebra A with a basis {| aSTλ| λA^, S,TA^λ} an involutive antihomomorphism A*: AA, and a partial order on A^ such that

  1. (aSTλ)* = aTSλ,
  2. If A(<λ) = span-{| aSTμ|μ<λ} then aaSTλ = QA^λ Aλ(a)QT aQTλ modA(<λ), for all aA.
Applying the involution A* to (b) and using (a) gives that aTSλ a* = QA^λ Aλ(a)QS aTQλ modA(<λ), for all aA.

The concept of a cellular algebra is not really the "right" one. The "right" one comes from the structure of a convolution algebra whenever the decomposition theorem holds [CG, 8.6.9].

Peter Webb's generalized reciprocity

Let 𝔬 be a complete discrete valuation ring, k=𝔬/𝔭 its residue field and let A𝔬 be an algebra over 𝔬, k 𝔬 𝕂 Ak A𝔬 A𝕂

The diagram K0 (A𝕂 ) G0 (A𝕂 ) K0 (Ak ) G0 (Ak ) cA cλ e=Dt D commutes, where e is defined by lifting idempotents. Furthermore e=Dt.

Proof.
If P is projective, U any finitely generated module, put P,U = dimHom(P,U). This is well defined on K0(A𝕂 ) × G0(A𝕂 ) and K0(Ak ) × G0(Ak ). Then e(P) = 𝕂𝔬P^, where k𝔬P^ = P.

Let U0 be a 𝔬-form of U and let P be projective. Then HomA𝔬 (P^,U0) is an 𝔬-lattice in HomA𝕂 (K𝔬P^,U) and the morphism HomA𝔬 (P^,U0) HomAk (P,U0/𝔭U0) is reduction mod 𝔭.

dimHomA𝕂 (K𝔬P^,U) = rank𝔬 HomA𝔬 (P^,U0) = dimHomAk (P,U0/𝔭U0).

This shows that e and D are the transpose of each other with respect to the forms. The diagram commutes from the definition of e.

The Cartan matrix CAk = DCA𝕂 Dt where CA𝕂 is the Cartan matrix of A.

If A𝕂 is semisimple then CA𝕂 = id.

The category 𝒪

Let U be a graded algebra with

  1. U0 reductive,
  2. Ui finite dimensional,
  3. U semisimple under the adjoint action.
The category 𝒪 is the category of graded U modules which are
  1. U0 semisimple, and
  2. U0 locally finite.
Define 𝒪n = {M𝒪 | Mi=0 if i>n}.

Standard and costandard modules

Let U^0 be an index set for the finite dimensional -graded U0 modules. The Verma module or standard module and the coVerma module or costandard module are given by Δ(λ) = UU0U0λ and (λ) = HomU0(U,U0λ), for   λU^0. Let M𝒪. A Δ-flag for M is an increasing filtration 0=M(0) M(1) M(2) such that M = iM(i), and, for each i1, M(i)/ M(i-1) Δ(λ(i)) for some λ(i) U^0.

  1. Δ(λ) has simple head L(λ).
  2. (λ) has simple socle L(λ).
  3. {L(λ) | λU^0} are the simple objects in 𝒪.

  1. Δ(λ) is the projective cover of L(λ) in 𝒪|λ|.
  2. (λ) is the injective hull of L(λ) in 𝒪|λ|.
  3. Hom𝒪(Δ(μ),(λ)) = { 0, if λμ, , if λ=μ. }
  4. Ext𝒪1 (Δ(μ),(λ)) = 0.

Projectives

If K=Ki is a graded U0 module define τn = K i>nKi = inKi. If λU^0 define Q = UU0τn( U0U0U0λ ), and let Pn(λ) be an indecomposable summand of Q which has L(λ) as a quotient and define Km,n, for mn by the exact sequence 0Km,n Pm(λ) Pn(λ) 0.

  1. Q is projective and QL(λ)0.
  2. Pn(λ) is a projective cover of L(λ) in 𝒪n.
  3. Pn(λ) has a Δ flag.
  4. Km,n has a Δ flag.
  5. L(λ) has a projective cover in P(λ) in 𝒪 if and only if the projective system Pm(λ) Pn(λ) stabilizes, in which case P(λ) Pn(λ), for   n0.

Injective module

Tilting modules

Let λU^0. A tilting module is a module that has both a Δ flag and a flag.

There is a unique indecomposable tilting module T(λ) of highest weight λ.

Blocks

Define on U^0 by μλ if [Δ(μ):L(λ)] 0 or [(μ):L(λ)] 0. Let [λ] denote the equivalence class of λ with respect to the equivalence relation generated by . Define 𝒪[λ] = {M𝒪 | if [M:L(μ)]0 then μ[λ]}, and for M𝒪 define M[λ] = U( im( Pn(λ) φ M ) ), the submodule of M generated by the images of morphisms φ: Pn(λ)M.

𝒪 = 𝒪[λ] and M = M[λ], for   M𝒪.

Multiplicities

Let 𝒜 be an abelian category and let L be simple. Let m𝒜. The multiplicity of L in M is [M:L] = supF Card {i | FiM/Fi+1M L}, where the supremum is over all (finite) filtrations of M. If   0MM M′′0   is exact then   [M:L] = [M:L] + [M′′:L].

If M𝒪n and N𝒪 with a Δ-flag then [M:L(λ)] = dimHom𝒪 (Pn(λ),M) and [N:Δ(μ)] = dimHom(N,(μ)). Thus [Pn(λ):Δ(μ)] = [(μ):L(λ)], for λ,μU^0 and nmax{ |λ|, |μ| }.

The category 𝒪int

Start with U = U<0 U0 U>0. 𝒪int = { MU-mod | MU0ss, MU>0nilp, MU<0nilp }.

Finite dimensional algebras

Let A be a finite dimensional algebra.

  1. The projective indecomposables are Ae for a minimal idempotent e of A.
  2. The simples L(λ) are the simple heads of the projective indecomposables P(λ).
  3. The blocks are Az for a minimal central idempotent z of A.
  4. The Cartan matrix is [P(λ) : L(μ)].

Temperley-Lieb algebras

Computation of the εσγ

The quantum dimensions of the finite dimensional simple Uq𝔰𝔩2 modules are dimq( L(k-2j) ) = b(k-j) [2+c(b)] [h(b)] = i=0k-j-1 [2+i] [k-j-i] = [k-j+1] = [dim(L(k-2j))]. As a ( Uq𝔰𝔩2, TLk(n) ) bimodule Vk j=0k2 L(k-2j) TLk(k-j,j). Thus trq(b) = j=0k2 dimq( L(k-2j) ) χTLk(k-j,j)(b), for   bTLk(n), and trq( aZXσ ) = δZX dimq(L(σ)) and trq( b ZX σμγ ) = δ ZX σμ dimq(L(γ)). If aA then trq (aek) = trq(a) trq(ek) = ntrq(a), and so trq( ε1(b) ) = 1n trq( ε1(b)ek ) = 1n trq( ekbek ) = 1n trq( bek2 ) = trq(bek) = trq( b(Tk-q) ) = (z-q) trq(b) = ( q2n-q ) trq(b) = 1n trq(b). So 1n dimq(L(γ)) = 1n trq( b ZX σμ γ ) = trq( ε1 ( b ZX σμ γ ) ) = trq( εσγ a ZX σ ) = εσγ dimq(L(σ)). Thus εσγ = [dim(L(γ))] n[dim(Lσ))] .

Generators and relations

The Temperley-Lieb algebra, Tk(n), is the algebra over given by generators E1,E2,...,Ek-1 and relations EiEj = EjEi, if |i-j|>1, Ei Ei±1 Ei = Ei, and Ei2 = nEi.

If [2] = q+q-1 = n then q = 12 ( n+ n2-4 ), q-1 = 12 ( n- n2-4 ), since q2-nq+1 = 0. Then [k] = qk-q-k q-q-1 = 12k-1 m=1 k+12 ( k 2m-1 ) nk-2m+1 (n2-4)m-1. The problem with this expression is that it is not clear that [k] is a polynomial in n with integer coefficients (which alternate in sign?).

The Iwahori-Hecke algebra Hk(q) is the algebra over with generators T1,T2 ,...,Tk-1 and relations TiTj = TjTi, if |i-j| >1, TiTi±1Ti = Ti+1TiTi+1, if 2ik-1, Ti2 = (q-q-1)Ti+1. There is a surjective algebra homomorphism φ: Hk(q) Tk(n) given by φ(Ti) = Ei-q-1 and φ(q+q-1) = n, with kerφ = TiTi+1Ti + TiTi+1 + Ti+1Ti + Ti + Ti+1 + 1 . Composing with the surjective homomorphism H˜k(q) Hk(q) Xεi Ti-1 T2T11T2Ti-1 Ti Ti

Murphy elements

Let us write Ti = Ei-q-1, so that Xε1 = 1, and Xεi = Ti-1 Xεi-1 Ti-1 in the Temperley-Lieb algebra. Then define m1,...,mk by m1=0 and (q-q-1)mj = qi-2 Xεi - qi-4 Xεi-1 for   2ik. Solving for Xεi in terms of the mi gives Xεi = (q-q-1) ( q-(i-2)mi + q-(i-2+1)mi-1 + + q-(2i-4)m2 ) + q-2(i-1), from which one obtains q(k-2) ( Xε1 + Xε2 ++ Xεk ) - q[k] = (q-q-1) ( mk+[2]mk-1 ++ [k-1]m2 ). Using the definition of Xεi and substituting for Xεi-1 in terms of the mi gives (q-q-1)mi = qi-2 Xεi - qi-4 Xεi-1 = qi-2 ( Ei-1-q-1 ) Xεi-1 ( Ei-1-q-1 ) - qi-4 Xεi-1 = qi-2 Ei-1 Xεi-1 Ei-1 - qi-3 ( Ei-1 Xεi-1 + Xεi-1 Ei-1 ) = qi-2 Ei-1 ( ( q-q-1 ) ( q-(i-3)mi + q-(i-3+1)mi-1 + + q-(2i-6)m2 ) + q-2(i-2) ) Ei-1 - qi-3 Ei-1 ( ( q-q-1 ) ( q-(i-3)mi + q-(i-3+1)mi-1 + + q-(2i-6)m2 ) + q-2(i-2) ) - qi-3 ( ( q-q-1 ) ( q-(i-3)mi + q-(i-3+1)mi-1 + + q-(2i-6)m2 ) + q-2(i-2) ) Ei-1 = qi-2 (q-q-1) q-(i-3) Ei-1 mi-1 Ei-1 - qi-3 (q-q-1) q-(i-3) ( Ei-1 mi-1 + mi-1 Ei-1 ) + qi-2 (q+q-1) Ei-1 ( ( q-q-1 ) ( q-(i-3)mi + q-(i-3+1)mi-1 + + q-(2i-6)m2 ) + q-2(i-2) ) -2qi-3 Ei-1 ( ( q-q-1 ) ( q-(i-3)mi + q-(i-3+1)mi-1 + + q-(2i-6)m2 ) + q-2(i-2) ) = qi-2 (q-q-1) q-(i-3) Ei-1 mi-1 Ei-1 - qi-3 (q-q-1) q-(i-3) ( Ei-1 mi-1 + mi-1 Ei-1 ) + qi-2 (q-q-1) Ei-1 ( ( q-q-1 ) ( q-(i-3)mi + q-(i-3+1)mi-1 + + q-(2i-6)m2 ) + q-2(i-2) ) since Ei-1 commutes with m2,m3 ,...,mi-1. Thus mi = q-(i-2) Ei-1 + qEi-1 mi-1 Ei-1 - ( Ei-1 mi-1 + mi-1 Ei-1 ) + (q-q-1) ( mi-2 + q-1 mi-3 + q-2 mi-4 + + q-(i-4)m2 ) Ei-1. It seems to me that this formula provides the easiest way to compute mi in terms of the Es. I would not be too worried about the coefficients of E1E4 and E2E4 in m4 looking strange. One expects diagrams that are equal to their own flip to act a bit differently in mk. Note also that [3]-1 = [4] [2] and [3]+1 = [2]2, so these are pretty nice q-versions of 2. Let's have a look at m6 and see if we can get an induction going. It might help to categorize the terms according to what their flip is to see where the next level is coming from.

For n such that Tk(n) is semisimple, the simple Tk(n) are indexed by partitions in the set T^k = {λk | λ has at most two columns }. The irreducible Tk(n) modules have seminormal basis {vT | T is a standard tableau of shape λ} and XεivT = q2c(T(i)) vT. Since c(T(i)) = c(T(i-1))-1 if the boxes T(i) and T(i-1) are in the same column and c(T(i)) + c(T(i-1)) = 3-i if the boxes T(i) and T(i-1) are in different columns it follows that mivT = qi-2q2c(T(i)) - qi-4 q2c(T(i-1)) q-q-1 = cT(i) vT, where cT(i) = { 0, if T(i) and T(i-1) are in the same column, [ i-2+2 c(T(i)) ], if T(i) and T(i-1) are in different columns. }

Now we want to define pseudomatrix units in Tk(n) according to the left and right eigenspaces of the mi. Let pST LSRT, normalized so that the coefficients are in [n] with greatest common divisor 1. Then pST pUV = γT δUV pSV, pST = S+,T+ cS+T+ pS+T+, pST ek pUV = βT- δT-U- pp+T+, ek+1 pST ek+1 = εS+T+ δS(k)T(k) pST ek+1.

Examples

Let's start with generic n. Here eST = [a] [b] eS-U- Ek-1 eU-T-. Then Ek = [b] [a] eST and mk = μk(S)eSS, where the first sum is over all pairs (S,T) such that S=T or S and T differ at the k-1st level.

In T2(n) let p12,12 p 12 , 12 = [2]e12,12 [2]e 12 , 12 In T3(n) let p 1 2 3 , 1 2 3 p 1 2 3 , 1 3 2 p 1 3 2 , 1 2 3 p 1 3 2 , 1 3 2 p 1 2 3 , 1 2 3 = [2]e 1 2 3 , 1 2 3 [3][2] e 1 2 3 , 1 3 2 [2]e 1 3 2 , 1 2 3 [3][2] e 1 3 2 , 1 3 2 [3] e 1 2 3 , 1 2 3 In T4(n) let p 1 2 3 4 , 1 2 3 4 p 1 2 3 4 , 1 3 2 4 p 1 3 2 4 , 1 2 3 4 p 1 3 2 4 , 1 3 2 4 p 1 2 3 3 , 1 2 3 3 p 1 2 3 3 , 1 3 2 2 p 1 2 3 3 , 1 4 2 2 p 1 3 2 2 , 1 2 3 3 p 1 3 2 2 , 1 3 2 2 p 1 3 2 2 , 1 4 2 2 p 1 4 2 2 , 1 2 3 3 p 1 4 2 2 , 1 3 2 2 p 1 4 2 2 , 1 4 2 2 p 1 2 3 4 , 1 2 3 4 = [2]2 e 1 2 3 4 , 1 2 3 4 [2]2 e 1 2 3 4 , 1 3 2 4 [2]2 e 1 3 2 4 , 1 2 3 4 [2]2 e 1 3 2 4 , 1 3 2 4 [3][2]2 e 1 2 3 3 , 1 2 3 3 [3][2]2 e 1 2 3 3 , 1 3 2 2 [3][2]2 e 1 2 3 3 , 1 4 2 2 [3][2]2 e 1 3 2 2 , 1 2 3 3 [3][2]2 e 1 3 2 2 , 1 3 2 2 [3][2]2 e 1 3 2 2 , 1 4 2 2 [3][2]2 e 1 4 2 2 , 1 2 3 3 [3][2]2 e 1 4 2 2 , 1 3 2 2 [3][2]2 e 1 4 2 2 , 1 4 2 2 [4][3][2] e 1 2 3 4 , 1 2 3 4

The special value n=±2, i.e. when [4]=0.
Then p 1 4 2 2 , 1 4 3 2 = p 1 2 3 3 , 1 2 3 3 and we let p 1 4 2 2 , 1 4 2 2 (2) = 1- e 1 2 3 , 1 2 3 . In this basis Rad(T4) = span 0 0 0 0 0 0 1 0 0 1 1 1 1 0 and Rad2(T4) = span 0 0 0 0 0 0 0 0 0 0 0 0 1 0 T1 = {(a)} = { a a } = { a 0 0 a a } and T2 = { a1 a2 } = { a1 0 0 a2 a2 } and T3 = { a11 a12 a21 a22 a3 } = { a11 a12 a21 a22 a11 a12 0 a21 a22 0 0 0 a3 a3 }.

The special value n=±1, i.e. when [3]=0.
Then p 1 3 2 , 1 3 2 = p 1 2 3 , 1 2 3 and we let p 1 3 2 , 1 3 2 (2) = 1- e 1 2 3 , 1 2 3 . In this basis Rad(T3) = span 0 1 1 1 0 and Rad2(T3) = span 0 0 1 0 0 . Then E1 = 1 0 0 0 0 , E2 = 1 1 1 1 0 , 1 = 1 0 0 0 1 , m2 = 1 0 0 0 0 , m3 = -1 0 0 1 0 . T1 = { a a } = { a 0 0 0 a } and T2 = { a1 a2 } = { a1 0 0 0 a2 }.

The special value n=0, i.e. when [2]=0.
Then p12,12 = p 12 , 12 and we let p 12,12 (2) = 1. In the basis p12,12 p12,12(2) e1 = 1 0 , m2 = 1 0 , and Rad(T2) = span 1 0 . With respect to this basis there is a new matrix = e2p12,122e2 e2p12,12 p12,12(2)e2 e2p12,12(2)p12,12e2 e2 ( p12,12(2) )2e2 = n 1 1 n = 0 1 1 0 , which is not diagonal. In T3 the basis elements %%%%%%%%%%%%%%%% p 1 2 3 , 1 2 3 (2) p 1 2 3 , 1 3 2 p 1 3 2 , 1 2 3 (2) p 1 3 2 , 1 3 2 (2) p 1 2 3 , 1 2 3 = p12,12 e2 p12,12(2) p12,12(2) e2 p12,12 (2) p12,12 e2 p12,12 p12,12 e2 p12,12(2) 1- p 1 2 3 , 1 2 3 (2) - p 1 3 2 , 1 3 2 (2) form a set of matrix units. In this basis E1 = 0 1 0 0 0 , E2 = 0 0 1 0 0 , 1 = 1 0 0 1 1 , m2 = 0 1 0 0 0 , m3 = -1 0 0 -1 0 , T1 = {(a)} = { 0 a } = { a 0 0 a a } and T2 = { a2 a1 } = { a1 a2 0 a1 a1 }.

References

[GW1] F. Goodman and H. Wenzl, The Temperley-Lieb algebra at roots of unity, Pacific J. Math. 161 (1993), no. 2, 307-334.

[GL1] J. Graham and G. Lehrer, Diagram algebras, Hecke algebras and decomposition numbers at roots of unity, Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 4, 479-524.

[GL2] J. Graham and G. Lehrer, The two-step nilpotent representations of the extended affine Hecke algebra of type A, Compositio Math. 133 (2002), no. 2, 173-197.

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