Generalized matrix algebra structure

## Generalized matrix algebras

Let $A$ be an algebra and fix $a\in A$. The homotope algebra $A\left(a\right)$ is the algebra $A$ with a new multiplication given by If $p,q$ are invertible elements of $A$ then the map $Apaq → Aa x ↦ qxp is an algebra isomorphism.$

## The radical of a homotope algebra

Let $R$ be a PID and let $A={M}_{n}\left(R\right)$ and let $ϵ\in A$. The smith normal form says that there exist $p,q∈GLnR such that pϵq= diag ϵ1,…,ϵk ,0,,0,…,0 , with ϵ1∣ϵ2∣ ⋯∣ϵk.$ Thus, $Mnϵ≅ Mnδ, where δ= diag ϵ1,…,ϵk ,0,,0,…,0 ,$ and

$Rad\left(A\left(a\right)\right)=\left\{x\in A\mid axa\in Rad\left(A\right)\right\}$ and ${Rad}^{3}\left(A\left(a\right)\right)\subset Rad\left(A\right).$

 Proof. The set $I=\left\{x\in A\mid axa\in Rad\left(A\right)\right\}$ is an ideal in $A$ since, if $y\in A$ then $a\left(x\cdot y\right)a=axay\in Rad\left(A\left(a\right)\right)$. Why and when is $I=Rad\left(A\right)$??? Or do I care? $\square$

## $A\subseteq B$, both split semisimple

Assume $A\subseteq B$ is an inclusion of algebras and that $A$ and $B$ are split semisimple. Let for each $\mu \in \stackrel{ˆ}{A}$ (the composite $P\to \mu$ is viewed as a single symbol). Let $\Gamma$ be the two level graph
 1.1
If $\lambda \in \stackrel{ˆ}{B}$ then
 1.2
is an index set for a basis of the irreucible $B$-module ${B}^{\lambda }$. We think of ${\stackrel{ˆ}{B}}^{\lambda }$ as the set of paths to $\lambda$ and ${\stackrel{ˆ}{A}}^{\mu }$ as the "set of paths to $\mu$" in the graph $\Gamma$. For example, the graph $\Gamma$ for the symmetric group algebras $A=ℂ{S}_{3}$ and $B=ℂ{S}_{4}$ is $picture goes here$
 ${ a P Q μ ∣ μ∈ A ˆ , P→μ,Q→μ∈ A ˆ μ } and { b P Q μ ν λ ∣ λ∈ B ˆ , P→μ→λ, Q→ν→λ∈ B ˆ λ }$ 1.3
respectively, so that
 ${a}_{\begin{array}{c}P Q\\ \mu \end{array}}{a}_{\begin{array}{c}S T\\ \nu \end{array}}={\delta }_{\mu \nu }{\delta }_{QS}{a}_{\begin{array}{c}P T\\ \mu \end{array}}\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{b}_{\begin{array}{c}P Q\\ \mu \gamma \\ \lambda \end{array}}{b}_{\begin{array}{c}S T\\ \tau \nu \\ \sigma \end{array}}={\delta }_{\lambda \sigma }{\delta }_{QS}{\delta }_{\gamma \tau }{b}_{\begin{array}{c}P T\\ \mu \nu \\ \lambda \end{array}},$ 1.4
and such that
 1.5
Then
 $1= ∑b S S μ μ λ$ 1.6
and
 1.7
where the sum is over all edges $\mu \to \lambda$ in the graph $\Gamma$.

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

1. $ebe={ϵ}_{1}\left(b\right)$, with ${ϵ}_{1}\left(b\right)\in A$, and
2. ${ϵ}_{1}\left({a}_{1}b{a}_{2}\right)={a}_{1}{ϵ}_{1}\left(b\right){a}_{2}$ for all ${a}_{1},{a}_{2}\in A$, and
3. $ea=ae$, for all $a\in A$.
Note that the map

 1.8
For each $\nu \in \stackrel{ˆ}{C}$ define
 1.9
so that ${\stackrel{ˆ}{C}}^{\nu }$ is te set of paths to $\nu$ in the graph $\stackrel{ˆ}{\Gamma }$. In the previous example $\stackrel{ˆ}{\Gamma }$ is $PICTURE GOES HERE$

 1.1
for some constant ${ϵ}_{\mu }^{\lambda }$ 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 is zero unless $R=T$ and $\rho =\tau$ and does not depend on the choice of $R$. If
 1.11
then Define
 1.12
so that these are matrix units. Furthermore
 1.13
so that the $b$s are related to the $e$s in the same way that the $a$s are related to the $b$s. Then
 1.14
In summary and

## $R{\otimes }_{A}L$, for $A$ semisimple.

Fix isomorphisms
 $L ‾ ≅ ⨁ μ∈ A ˆ A → μ ⊗Lμ and R ‾ ≅ ⨁ μ∈ A ˆ Rμ ⊗ A ← μ$ 1.15
where ${\stackrel{\to }{A}}^{\mu },\mu \in {\stackrel{ˆ}{A}}^{\mu }$ are the simple left $\stackrel{‾}{A}$-modules, ${\stackrel{←}{A}}^{\mu }$, $\mu \in \stackrel{ˆ}{A}$, are the simple right $\stackrel{‾}{A}$-modules, and ${L}^{\mu },{R}^{\mu },\mu \in \stackrel{ˆ}{A}$ are vector spaces. In other words, if $A$ has matrix units such that
 1.16
The map $ϵ:\stackrel{‾}{L}{\otimes }_{𝔽}\stackrel{‾}{R}\to \stackrel{‾}{A}$ is determined by the constants ${ϵ}_{XY}^{\mu }\in 𝔽$ given by
 1.17
and ${ϵ}_{XY}^{\mu }$ does not depend on $Q$ and $P$ since For each $\mu \in \stackrel{ˆ}{A}$ construct a matrix
 ${ℰ}^{\mu }=\left({ϵ}_{XY}^{\mu }\right)$ 1.18
and use row reduction (Smith normal form) to find invertible matrices
 ${D}^{\mu }=\left({D}_{ST}^{\mu }\right)\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{C}^{\mu }=\left({C}_{ZW}^{\mu }\right)\phantom{\rule{2em}{0ex}}\text{such that}\phantom{\rule{2em}{0ex}}{D}^{\mu }{ℰ}^{\mu }{C}^{\mu }=diag\left({ϵ}_{X}^{\mu }\right)$ 1.19
is a diagonal matrix with diagonal entries denoted ${ϵ}_{X}^{\mu }$. The ${ϵ}_{P}^{\mu }$ are the invariant factors of the matrix ${ℰ}^{\mu }$.
 1.2
where $\mu \in \stackrel{ˆ}{A},X\in {\stackrel{ˆ}{R}}^{\mu },Y\in {\stackrel{ˆ}{L}}^{\mu }$.
1. The sets are bases of $\stackrel{‾}{R}{\otimes }_{\stackrel{‾}{A}}\stackrel{‾}{L}$, which satisfy where ${ϵ}_{TQ}^{\mu }$ and ${ϵ}_{T}^{\mu }$ are as defined in (1.17) and (1.19).
2. The radical of the algebra $R{\otimes }_{A}L$ is and the images of the elements are a set of matrix units in $\left(R{\otimes }_{A}L\right)/Rad\left(R{\otimes }_{A}L\right)$.

## The structure of $Z\left(ϵ\right)$

Let The left radical $L\left(ϵ\right)$ and the right radical $R\left(ϵ\right)$ of $ϵ$ are defined by The map $ϵ$ is nondegenerate if $Rad\left(C\right)=0$, $L\left(ϵ\right)=0$, and $R\left(ϵ\right)=0$. Let $C ‾ =C/RadC, L ‾ =L/Lϵ, R ‾ =R/Rϵ, and φ: R⊗CL → R ‾ ⊗ C ‾ L ‾ r ‾ ⊗ l ‾ ↦ r⊗l ‾$ Then $ker \phi$ is generated by $R{\otimes }_{C}L\left(ϵ\right)$ and $R\left(ϵ\right){\otimes }_{C}L$, and we have that $ker \phi \cdot R\cdot R\left(ϵ\right)$ and $L\cdot ker \phi \subset L\left(ϵ\right)$. THne and $Aϵ I ≅A ϵ ‾ where the map ϵ ‾ : L ‾ ⊗D R ‾ → C ‾$ is a nondegenerate $\left(\stackrel{‾}{C},\stackrel{‾}{C}\right)$ bimodule homomorphism.

If $ϵ:L{\otimes }_{D}R\to C$ is nondegenerate and $R$ is a projective $C$-module then there is a $\left(D,C\right)$ bimodule homomorphism $τ: R → ∼ L* r ↦ λr: L → C l ↦ ϵl⊗r so that ϵ=ev∘id⊗τ$ and $Aϵ≅ A( evL ).$

If $C,D,L,R$ are finite dimensional vector spaces over $𝔽$ and $D=𝔽$ then $ϵ= ϵ0⊕ evP: L0⊕P* ⊗D R0⊕P →C,$ with $P$ projective and $im {ϵ}_{0}\subseteq Rad\left(C\right)$.

If $ϵ={ϵ}_{0}\oplus {ev}_{P}$ with $P$ finitely generated and projective then $Aϵ-mod → ∼ A ϵ0 -mod M ↦ eM where e=1- ∑ i pi⊗αi.$

If $im ϵ\subseteq Rad\left(C\right)$ then $RadAϵ0 =I=RadC ⊕RadD ⊕L0 ⊕R0 ⊗CL0$ and $Aϵ0 RadAϵ0 ≅ C RadC ⊕ D RadD .$

## Duals and Projectives

Let $L$ be a $C$-module and let $Z=EndCL$ so that $L$ is a $\left(C,Z\right)$ bimodule. The dual module to $L$ is the $\left(Z,C\right)$ bimodule $L*= HomCL,C.$ The evaluation map is the $\left(C,C\right)$ bimodule homomorphism $ev: L⊗ZL* → C λ⊗l ↦ λl$ and the centralizer map is the $\left(Z,Z\right)$ bimodule homomorphism $ξ: L*⊗CL → Z λ⊗l ↦ zλ,l: L → L m ↦ λml$ Recall that [Bou, Alg. II §4.2 Cor.]
1. If $L$ is a projective $C$-module if and only if $1\in im \xi$,
2. If $L$ is a projective $C$-module then $\xi$ is injective,
3. If $L$ is a finitely generated projective $C$-module then $\xi$ is bijective,
4. If $L$ is a finitely generated free module then $ξ-1z = ∑ i bi* ⊗ zbi,$ where $\left\{{b}_{1},\dots ,{b}_{d}\right\}$ is a basis of $L$ and $\left\{{b}_{1}^{*},\dots ,{b}_{d}^{*}\right\}$ is the dual basis in ${M}^{*}$.
Statement (a) says that $L$ is projective if and only if there exist ${b}_{i}\in L$ and ${b}_{i}^{*}$ such that

## 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)