Generalized matrix algebra structure

Generalized matrix algebra structure

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

Last updates: 5 November 2010

Generalized matrix algebras

Let A be an algebra and fix a∈A. The homotope algebra Aa is the algebra A with a new multiplication given by x⋅y=xay,for x,y∈A. If p,q are invertible elements of A then the map Apaq→Aax↦qxpis an algebra isomorphism.

The radical of a homotope algebra

Let R be a PID and let A=MnR and let ϵ∈A. The smith normal form says that there exist p,q∈GLnRsuch thatpϵq=diagϵ1,…,ϵk,0,,0,…,0,withϵ1∣ϵ2∣⋯∣ϵk. Thus, Mnϵ≅Mnδ,whereδ=diagϵ1,…,ϵk,0,,0,…,0, and RadMnδ=x∈Mn∣if xST≠0 then S>k or T>k,Rad2Mnδ=x∈Mn∣if xST≠0 then S>k and T>k,andRad3Mnδ=0.

RadAa=x∈A∣axa∈RadA and Rad3Aa⊂RadA.

Proof.

A⊆B, both split semisimple

Assume A⊆B is an inclusion of algebras and that A and B are split semisimple. Let Aˆ be an index set for the irreducible A-modulesAμ,Bˆ be an index set for the irreducible B-modulesBλ,and letAˆμ=P→μ be an index set for a basis of the simple A-module Aμ, for each μ∈Aˆ (the composite P→μ is viewed as a single symbol). Let Γ be the two level graph
vertices on level A:Aˆ,vertices on level B:Bˆ,andmμλ edges μ→λ if Aμ appears with multiplicity mμλ in ResABBλ. 1.1
If λ∈Bˆ then
Bˆλ=P→μ→λ∣μ∈Aˆ,P→μ∈Aˆμ and μ→λ an edge in Γ 1.2
is an index set for a basis of the irreucible B-module Bλ. We think of Bˆλ as the set of paths to λ and Aˆμ as the "set of paths to μ" in the graph Γ. For example, the graph Γ for the symmetric group algebras A=ℂS3 and B=ℂS4 is picture goes here

Since A and B are split semisimple there exist sets of matrix units in the algebras A and B,

{aP Qμ∣μ∈Aˆ,P→μ,Q→μ∈Aˆμ}and{bP Qμ νλ∣λ∈Bˆ,P→μ→λ,Q→ν→λ∈Bˆλ} 1.3
respectively, so that
aP QμaS Tν=δμνδQSaP TμandbP Qμ γλbS Tτ νσ=δλσδQSδγτbP Tμ νλ, 1.4
and such that
aP QμbS Tμ τλ=δQ Sμ γbP Tμ τλandbS Tσ τλ aP Qμ=δT Pτ μbS Qσ μλ. 1.5
Then
1=∑bS Sμ μλ 1.6
and
aP Qμ=1⋅aP Qμ⋅1=(∑bR Rρ ρλ)aP Qμ∑(bS Sσ σγ)=∑bP Qμ μλ 1.7
where the sum is over all edges μ→λ in the graph Γ.

Now assume that B is a subalgebra of an algebra C and there is an element e∈C such that for all b∈B,

  1. ebe=ϵ1b, with ϵ1b∈A, and
  2. ϵ1a1ba2=a1ϵ1ba2 for all a1,a2∈A, and
  3. ea=ae, for all a∈A.
Note that the map ϵ1:B⊗B→Ab1⊗b2↦ϵ1b1b2is an A,A bimodule homomorphism.

Though it is not necessary for the following it is conceptually helpful to let C=BeB, let Cˆ=Aˆ and extend the graph Γ to a graph Γˆ with three levels, so that the edges between level B and level C are reflections of the edges between level A and level B. In other words, Γˆ has

vertices on level C:Cˆ,andan edge λ→μ,λ∈Bˆ,μ∈Cˆ, for each edge μ→λ,μ∈Aˆ,λ∈Bˆ. 1.8
For each ν∈Cˆ define
Cˆν=P→μ→λ→ν∣μ∈Aˆ,λ∈Bˆ,ν∈Cˆ,P→μ∈Aˆ and μ→λ and λ→μ are edges in Γˆ, 1.9
so that Cˆν is te set of paths to ν in the graph Γˆ. In the previous example Γˆ is PICTURE GOES HERE

The element of A given by ϵ1(bP Qμ τλ)=ϵ1(aP PμbP Qμ τλaQ Qτ)=aP Pμϵ1(bP Qμ τλ)aQ Qτ is zero unless μ=τ and

ϵ1(bP Qμ μλ)=ϵ1(aP RμbR Rμ μλaR Qμ)=aP Rμϵ1(bR Rμ μλ)aR Qμ=ϵμλaP Qμ 1.10
for some constant ϵμλ which does not depend on P or Q (since it depends only on R which can be chosen freely). The element of C give by bP Rμ ρλebT Qτ νσ=bP Rμ ρλaR RρebT Qτ νσ=bP Rμ ρλeaR RρbT Qτ νσ is zero unless R=T and ρ=τ and bP Rμ ρλebR Qρ νσ=bP Sμ ρλaS RρebR Qρ νσ=bP Sμ ρλeaS RρbR Qρ νσ=bP Sμ ρλebS Qρ νσ does not depend on the choice of R. If
cP Qμ νλ σγ=bP Tμ γλebT Qγ νσ 1.11
then cP Qμ νλ σγcR Sτ ξρ ηπ=(bP Tμ γλebT Qγ νσ)(bR Xτ πρebX Sπ ξη)=δQ Rν τσ ρbP Tμ γλϵ1(bT Xγ πσ)ebX Sπ ξη=δQ Rν τσ ρbP Tμ γλδγπϵγσaT XγebX Sπ ξη=δQ Rν τσ ρδπγϵγσbP Xμ γλebX Sγ ξη=δQ Rν τσ ρδπγϵγσcP Sμ ξλ ηγ. Define
eP Qμ νλ σγ=1ϵγσcP Qμ νλ σγ,whenver ϵγσ≠0.TheneP Qμ νλ σγeR Sτ ξρ ηπ=δQ Rν τσ ρδγπeP Sμ ξλ ηγ 1.12
so that these are matrix units. Furthermore
bP Qμ νλeR Sτ ξρ ηπ=δQ Rν τλ ρeP Sμ ξλ ηπandeP Qμ νλ σγbR Sτ ξρ=δQ Rν τσ ρeP Sμ ξλ ργ 1.13
so that the bs are related to the es in the same way that the as are related to the b s. Then
e=1⋅e⋅1=(∑bR Rρ ρλ)e(∑bS Sσ σγ)=∑bR Rρ ρλebR Rρ ρλ=∑cR Rρ ρλ γρ=ϵργeR Rρ ρλ γρ. 1.14
In summary ebP Qμ τλe=δμτϵμλaP QμeP Qμ νλ σγ=1ϵγσbP Tμ γλebT Qγ νσbP Qμ νλ=∑λ→γeP Qμ νλ λγ and eeP Aμ νλ σγ=δμγϵγλ∑γ→τ→γeP Qγ ντ σγ.

R⊗AL, for A semisimple.

Fix isomorphisms
L‾≅⨁μ∈AˆA→μ⊗LμandR‾≅⨁μ∈AˆRμ⊗A←μ 1.15
where A→μ,μ∈Aˆμ are the simple left A‾-modules, A←μ, μ∈Aˆ, are the simple right A‾-modules, and Lμ,Rμ,μ∈Aˆ are vector spaces. In other words, if A has matrix units {aP Qμ∣μ∈Aˆ,P→μ,Q→μ∈Aˆμ}thenLhas a basislPX∣P∈Aˆμ,X∈LˆμRhas a basisrYQ∣Q∈Aˆμ,Y∈Lˆμ such that
aP QμlR Xν=δμνδQRlP XμandrY SνaP Qμ=δμνδSPlY Qμ. 1.16
The map ϵ:L‾⊗𝔽R‾→A‾ is determined by the constants ϵXYμ∈𝔽 given by
ϵ(lQ Xμ⊗rY Pμ)=ϵXYμaQ Pμ 1.17
and ϵXYμ does not depend on Q and P since ϵ(lS Xλ⊗rY Pμ)=ϵ(aS QλlQ Xλ⊗rY PμaP Tμ)=aS Qλϵ(lQ Xλ⊗rY Pμ)aP Tμ=δλμaS QμϵXYμaQ PμaP Tμ=ϵXYμaS Tμ. For each μ∈Aˆ construct a matrix
ℰμ=ϵXYμ 1.18
and use row reduction (Smith normal form) to find invertible matrices
Dμ=DSTμandCμ=CZWμsuch thatDμℰμCμ=diagϵXμ 1.19
is a diagonal matrix with diagonal entries denoted ϵXμ. The ϵPμ are the invariant factors of the matrix ℰμ.

Using notation as in (???) define elements of R⊗AL by

mX Yμ=rX Pμ⊗lP Yμ,andnX Yμ=∑Q1,Q2CQ1XμDYQ2μmQ1 Q2μ 1.20
where μ∈Aˆ,X∈Rˆμ,Y∈Lˆμ.
  1. The sets {mX Yμ∣μ∈Aˆ,X∈Rˆμ,Y∈Lˆμ}and{nX Yμ∣μ∈Aˆ,X∈Rˆμ,Y∈Lˆμ} are bases of R‾⊗A‾L‾, which satisfy mS TλmQ Pμ=δλμϵTQμmS PμandnS TλnQ Pμ=δλμδTQϵTμnS Pμ, where ϵTQμ and ϵTμ are as defined in (1.17) and (1.19).
  2. The radical of the algebra R⊗AL is RadR⊗AL=𝔽-span{nY Tμ∣ϵYμ=0 or ϵTμ=0} and the images of the elements eY Tμ=1ϵTμnY Tμ,for ϵYμ≠0 or ϵTμ≠0, are a set of matrix units in R⊗AL/RadR⊗AL.

Proof.

The structure of Zϵ

Let ϵ:L⊗DR→Cbe a C,C bimodule homomorphism. The left radical Lϵ and the right radical Rϵ of ϵ are defined by Lϵ=l∈L∣ϵl⊗r∈RadC,for all r∈R,Rϵ=r∈R∣ϵl⊗r∈RadC,for all l∈L, The map ϵ is nondegenerate if RadC=0, Lϵ=0, and Rϵ=0. Let C‾=C/RadC,L‾=L/Lϵ,R‾=R/Rϵ,andφ:R⊗CL→R‾⊗C‾L‾r‾⊗l‾↦r⊗l‾ Then ker φ is generated by R⊗CLϵ and Rϵ⊗CL, and we have that ker φ⋅R⋅Rϵ and L⋅ker φ⊂Lϵ. THne I=RadC+LC+ker φis a nilpotent ideal of Aϵ, and AϵI≅Aϵ‾where the mapϵ‾:L‾⊗DR‾→C‾ is a nondegenerate (C‾,C‾) bimodule homomorphism.

If ϵ:L⊗DR→C is nondegenerate and R is a projective C-module then there is a (D,C) bimodule homomorphism τ:R→∼L*r↦λr:L→Cl↦ϵl⊗rso thatϵ=ev∘id⊗τ and Aϵ≅A(evL).

If C,D,L,R are finite dimensional vector spaces over 𝔽 and D=𝔽 then ϵ=ϵ0⊕evP:L0⊕P*⊗DR0⊕P→C, with P projective and im ϵ0⊆RadC.

If ϵ=ϵ0⊕evP with P finitely generated and projective then Aϵ-mod→∼Aϵ0-modM↦eMwheree=1-∑ipi⊗αi.

If im ϵ⊆RadC then RadAϵ0=I=RadC⊕RadD⊕L0⊕R0⊗CL0 and Aϵ0RadAϵ0≅CRadC⊕DRadD.

Duals and Projectives

Let L be a C-module and let Z=EndCL so that L is a (C,Z) bimodule. The dual module to L is the (Z,C) bimodule L*=HomCL,C. The evaluation map is the (C,C) bimodule homomorphism ev:L⊗ZL*→Cλ⊗l↦λl and the centralizer map is the (Z,Z) bimodule homomorphism ξ:L*⊗CL→Zλ⊗l↦zλ,l:L→Lm↦λml Recall that [Bou, Alg. II §4.2 Cor.]
  1. If L is a projective C-module if and only if 1∈im ξ,
  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-1z=∑ibi*⊗zbi, 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 bi∈L and bi* such that if l∈Lthenl=∑ibi*lbiso thatξ∑ibi*⊗bi=1.

References

[Bou] N. Bourbaki, Groupes et Algèbres de Lie, Masson, Paris, 1990.

[GW1] F. Goodman and H. Wenzl, The Temperly-Lieb algebra at roots of unity, Pacific J. Math. 161 (1993), 307-334. MR1242201 (95c:16020)

[GL1] J. Graham and G. Lehrer, Diagram algebras, Hecke algebras and decomposition numbers at roots of unity, Ann. Sci. École Norm Sup. (4) 36 (2004), 479-524. MR2013924 (2004k:20007)

[GL2] J. Graham and G. Lehrer, The two-step nilpotent representations of the extended affine Hecke algebra of type A, Compositio Math. 133 (2002), 173-197. MR1923581 (2004d:20004)

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