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 aA. The homotope algebra Aa is the algebra A with a new multiplication given by xy=xay,for x,yA. If p,q are invertible elements of A then the map ApaqAaxqxpis an algebra isomorphism.

The radical of a homotope algebra

Let R be a PID and let A=MnR and let ϵA. The smith normal form says that there exist p,qGLnRsuch thatpϵq=diagϵ1,,ϵk,0,,0,,0,withϵ1ϵ2ϵk. Thus, MnϵMnδ,whereδ=diagϵ1,,ϵk,0,,0,,0, and RadMnδ=xMnif xST0 then S>k or T>k,Rad2Mnδ=xMnif xST0 then S>k and T>k,andRad3Mnδ=0.

RadAa=xAaxaRadA and Rad3AaRadA.

Proof.

AB, both split semisimple

Assume AB 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=Pμ be an index set for a basis of the simple A-module Aμ, for each μAˆ (the composite Pμ 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 μλ if Aμ appears with multiplicity mμλ in ResABBλ. 1.1
If λBˆ then
Bˆλ=PμλμAˆ,PμAˆμ and μλ 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=S3 and B=S4 is picture goes here

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

{aPQμμAˆ,Pμ,QμAˆμ}and{bPQμνλλBˆ,Pμλ,QνλBˆλ} 1.3
respectively, so that
aPQμaSTν=δμνδQSaPTμandbPQμγλbSTτνσ=δλσδQSδγτbPTμνλ, 1.4
and such that
aP QμbS Tμ τλ=δQ Sμ γbP Tμ τλandbS Tσ τλaP Qμ=δTPτμbS Qσ μλ. 1.5
Then
1=bSSμμλ 1.6
and
aPQμ=1aPQμ1=(bR Rρ ρλ)aPQμ(bS Sσ σγ)=bP Qμ μλ 1.7
where the sum is over all edges μλ in the graph Γ.

Now assume that B is a subalgebra of an algebra C and there is an element eC such that for all bB,

  1. ebe=ϵ1b, with ϵ1bA, and
  2. ϵ1a1ba2=a1ϵ1ba2 for all a1,a2A, and
  3. ea=ae, for all aA.
Note that the map ϵ1:BBAb1b2ϵ1b1b2is 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 λμ,λBˆ,μCˆ, for each edge μλ,μAˆ,λBˆ. 1.8
For each νCˆ define
Cˆν=PμλνμAˆ,λBˆ,νCˆ,PμAˆ and μλ and λμ 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 τσρbP Tμ γλϵ1(bT Xγ πσ)ebX Sπ ξη=δQ Rν τσρbP Tμ γλδγπϵγσaT XγebX Sπ ξη=δQ Rν τσρδπγϵγσbP Xμ γλebX Sγ ξη=δQ Rν τσρδπγϵγσcP Sμ ξλ ηγ. Define
eP Qμ νλ σγ=1ϵγσcP Qμ νλ σγ,whenverϵγσ0.TheneP Qμ νλ σγeR Sτ ξρ ηπ=δQ Rν τσρδγπeP Sμ ξλ ηγ 1.12
so that these are matrix units. Furthermore
bP Qμ νλeR Sτ ξρ ηπ=δQ Rν τλρeP Sμ ξλ ηπandeP Qμ νλ σγbR Sτ ξρ=δQ Rν τσρeP 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=1e1=(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=λγeP Qμ νλ λγ and eeP Aμ νλ σγ=δμγϵγλγτγeP Qγ ντ σγ.

RAL, for A semisimple.

Fix isomorphisms
LμAˆAμLμandRμAˆRμAμ 1.15
where Aμ,μAˆμ are the simple left A-modules, Aμ, μAˆ, are the simple right A-modules, and Lμ,Rμ,μAˆ are vector spaces. In other words, if A has matrix units {aP QμμAˆ,Pμ,QμAˆμ}thenLhas a basislPXPAˆμ,XLˆμRhas a basisrYQQAˆμ,YLˆμ such that
aP QμlR Xν=δμνδQRlP XμandrY SνaP Qμ=δμνδSPlY Qμ. 1.16
The map ϵ:L𝔽RA is determined by the constants ϵXYμ𝔽 given by
ϵ(lQ XμrY Pμ)=ϵXYμaQ Pμ 1.17
and ϵXYμ does not depend on Q and P since ϵ(lS XλrY Pμ)=ϵ(aS QλlQ XλrY PμaP Tμ)=aS Qλϵ(lQ XλrY Pμ)aP Tμ=δλμaS QμϵXYμaQ PμaP Tμ=ϵXYμaS Tμ. For each μAˆ construct a matrix
μ=ϵXYμ 1.18
and use row reduction (Smith normal form) to find invertible matrices
Dμ=DSTμandCμ=CZWμsuch thatDμμCμ=diagϵXμ 1.19
is a diagonal matrix with diagonal entries denoted ϵXμ. The ϵPμ are the invariant factors of the matrix μ.

Using notation as in (???) define elements of RAL by

mX Yμ=rX PμlP Yμ,andnX Yμ=Q1,Q2CQ1XμDYQ2μmQ1Q2μ 1.20
where μAˆ,XRˆμ,YLˆμ.
  1. The sets {mX YμμAˆ,XRˆμ,YLˆμ}and{nX YμμAˆ,XRˆμ,YLˆμ} are bases of RAL, 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 RAL is RadRAL=𝔽-span{nY TμϵYμ=0orϵTμ=0} and the images of the elements eY Tμ=1ϵTμnY Tμ,for ϵYμ0 or ϵTμ0, are a set of matrix units in RAL/RadRAL.

Proof.

The structure of Zϵ

Let ϵ:LDRCbe a C,C bimodule homomorphism. The left radical Lϵ and the right radical Rϵ of ϵ are defined by Lϵ=lLϵlrRadC,for all rR,Rϵ=rRϵlrRadC,for all lL, The map ϵ is nondegenerate if RadC=0, Lϵ=0, and Rϵ=0. Let C=C/RadC,L=L/Lϵ,R=R/Rϵ,andφ:RCLRCLrlrl Then kerφ is generated by RCLϵ and RϵCL, and we have that kerφRRϵ and LkerφLϵ. THne I=RadC+LC+kerφis a nilpotent ideal of Aϵ, and AϵIAϵwhere the mapϵ:LDRC 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 homomorphism τ:RL*rλr:LClϵlrso thatϵ=evidτ and AϵA(evL).

If C,D,L,R are finite dimensional vector spaces over 𝔽 and D=𝔽 then ϵ=ϵ0evP:L0P*DR0PC, with P projective and imϵ0RadC.

If ϵ=ϵ0evP with P finitely generated and projective then Aϵ-modAϵ0-modMeMwheree=1-ipiαi.

If imϵRadC then RadAϵ0=I=RadCRadDL0R0CL0 and Aϵ0RadAϵ0CRadCDRadD.

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:LZL*Cλlλl and the centralizer map is the (Z,Z) bimodule homomorphism ξ:L*CLZλlzλ,l:LLmλml Recall that [Bou, Alg. II §4.2 Cor.]
  1. If 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-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 biL and bi* such that if lLthenl=ibi*lbiso thatξibi*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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