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 $a\in A$. The homotope algebra $A\left(a\right)$ is the algebra $A$ with a new multiplication given by $$x\cdot y=xay,\text{for} x,y\in A.$$ If $p,q$ are invertible elements of $A$ then the map $$\begin{array}{ccc}A\left(paq\right)& \to & A\left(a\right)\\ x& \mapsto & qxp\end{array}\text{is an algebra isomorphism.}$$

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 $$p,q\in {GL}_n\left(R\right)\text{such that}pϵq=diag\left({ϵ}_1,\dots ,{ϵ}_k,0,,0,\dots ,0\right),\text{with}{ϵ}_1\mid {ϵ}_2\mid \cdots \mid {ϵ}_k.$$ Thus, $$M_n\left(ϵ\right)\cong M_n\left(\delta \right),\text{where}\delta =diag\left({ϵ}_1,\dots ,{ϵ}_k,0,,0,\dots ,0\right),$$ and $$\begin{array}{l}Rad\left(M_n\left(\delta \right)\right)=\left\{x\in M_n\mid \text{if} x_{ST}\ne 0 \text{then} S>k \text{or} T>k\right\},\\ {Rad}^2\left(M_n\left(\delta \right)\right)=\left\{x\in M_n\mid \text{if} x_{ST}\ne 0 \text{then} S>k \text{and} T>k\right\},\text{and}\\ {Rad}^3\left(M_n\left(\delta \right)\right)=0.\end{array}$$

$Rad\left(A\left(a\right)\right)=\left\{x\in A\mid axa\in Rad\left(A\right)\right\}$ 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 $$\begin{array}{l}\hat{A} \text{be an index set for the irreducible} A\text{-modules}{A}^{\mu },\\ \hat{B} \text{be an index set for the irreducible} B\text{-modules}{B}^{\lambda },\text{and let}\\ {\hat{A}}^{\mu }=\left\{P\to \mu \right\} \text{be an index set for a basis of the simple} A\text{-module} A^{\mu },\end{array}$$ for each $\mu \in \hat{A}$ (the composite $P\to \mu$ is viewed as a single symbol). Let $\Gamma$ be the two level graph

$$\begin{array}{l}\text{vertices on level A:}\hat{A},\\ \text{vertices on level B:}\hat{B},\text{and}\\ m_{\mu }^{\lambda } \text{edges} \mu \to \lambda \text{if} A^{\mu } \text{appears with multiplicity} m_{\mu }^{\lambda } \text{in} {Res}_A^B\left(B^{\lambda }\right).\end{array}$$

1.1

If $\lambda \in \hat{B}$ then

$${\hat{B}}^{\lambda }=\left\{P\to \mu \to \lambda \mid \mu \in \hat{A},P\to \mu \in {\hat{A}}^{\mu } \text{and} \mu \to \lambda \text{an edge in} \Gamma \right\}$$

1.2

is an index set for a basis of the irreucible $B$-module $B^{\lambda }$. We think of ${\hat{B}}^{\lambda }$ as the set of paths to $\lambda$ and ${\hat{A}}^{\mu }$ as the "set of paths to $\mu$" in the graph $\Gamma$. For example, the graph $\Gamma$ for the symmetric group algebras $A=ℂS_3$ and $B=ℂS_4$ is $$\text{picture goes here}$$

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

$$\{a_{\begin{array}{c}P\hspace{0.17em}Q\\ \mu \end{array}}\mid \mu \in \hat{A},P\to \mu ,Q\to \mu \in {\hat{A}}^{\mu }\}\text{and}\{b_{\begin{array}{c}P\hspace{0.17em}Q\\ \mu \hspace{0.17em}\nu \\ \lambda \end{array}}\mid \lambda \in \hat{B},P\to \mu \to \lambda ,Q\to \nu \to \lambda \in {\hat{B}}^{\lambda }\}$$

1.3

respectively, so that

$a_{\begin{array}{c}P\hspace{0.17em}Q\\ \mu \end{array}}{a}_{\begin{array}{c}S\hspace{0.17em}T\\ \nu \end{array}}={\delta }_{\mu \nu }{\delta }_{QS}{a}_{\begin{array}{c}P\hspace{0.17em}T\\ \mu \end{array}}\text{and}{b}_{\begin{array}{c}P\hspace{0.17em}Q\\ \mu \hspace{0.17em}\gamma \\ \lambda \end{array}}{b}_{\begin{array}{c}S\hspace{0.17em}T\\ \tau \hspace{0.17em}\nu \\ \sigma \end{array}}={\delta }_{\lambda \sigma }{\delta }_{QS}{\delta }_{\gamma \tau }{b}_{\begin{array}{c}P\hspace{0.17em}T\\ \mu \hspace{0.17em}\nu \\ \lambda \end{array}},$

1.4

and such that

$$a_{\begin{array}{c}P Q\\ \mu \end{array}}{b}_{\begin{array}{c}S T\\ \mu \tau \\ \lambda \end{array}}={\delta }_{\begin{array}{c}Q S\\ \mu \gamma \end{array}}{b}_{\begin{array}{c}P T\\ \mu \tau \\ \lambda \end{array}}\text{and}{b}_{\begin{array}{c}S T\\ \sigma \tau \\ \lambda \end{array}}\hspace{0.17em}{a}_{\begin{array}{c}P Q\\ \mu \end{array}}={\delta }_{\begin{array}{c}T\hspace{0.17em}P\\ \tau \hspace{0.17em}\mu \end{array}}{b}_{\begin{array}{c}S Q\\ \sigma \mu \\ \lambda \end{array}}.$$

1.5

Then

$$1=\sum b_{\begin{array}{c}S\hspace{0.17em}S\\ \mu \hspace{0.17em}\mu \\ \lambda \end{array}}$$

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 $$\begin{array}{rrcl}{ϵ}_1:& B\otimes B& \to & A\\ & b_1\otimes b_2& \mapsto & {ϵ}_1\left(b_1{b}_2\right)\end{array}\text{is an} \left(A,A\right) \text{bimodule homomorphism.}$$

Though it is not necessary for the following it is conceptually helpful to let $C=BeB$, let $\hat{C}=\hat{A}$ and extend the graph $\Gamma$ to a graph $\hat{\Gamma }$ 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, $\hat{\Gamma }$ has

$$\begin{array}{l}\text{vertices on level C:}\hat{C},\text{and}\\ \text{an edge} \lambda \to \mu ,\lambda \in \hat{B},\mu \in \hat{C}, \text{for each edge} \mu \to \lambda ,\mu \in \hat{A},\lambda \in \hat{B}.\end{array}$$

1.8

For each $\nu \in \hat{C}$ define

$${\hat{C}}^{\nu }=\left\{P\to \mu \to \lambda \to \nu \mid \begin{array}{c}\mu \in \hat{A},\lambda \in \hat{B},\nu \in \hat{C},P\to \mu \in \hat{A} \text{and} \\ \mu \to \lambda \text{and} \lambda \to \mu \text{are edges in} \hat{\Gamma }\end{array}\right\},$$

1.9

so that ${\hat{C}}^{\nu }$ is te set of paths to $\nu$ in the graph $\hat{\Gamma }$. In the previous example $\hat{\Gamma }$ is $$\text{PICTURE GOES HERE}$$

The element of $A$ given by $${ϵ}_1\left(b_{\begin{array}{c}P Q\\ \mu \tau \\ \lambda \end{array}}\right)={ϵ}_1\left(a_{\begin{array}{c}P P\\ \mu \end{array}}{b}_{\begin{array}{c}P Q\\ \mu \tau \\ \lambda \end{array}}{a}_{\begin{array}{c}Q Q\\ \tau \end{array}}\right)=a_{\begin{array}{c}P P\\ \mu \end{array}}{ϵ}_1\left(b_{\begin{array}{c}P Q\\ \mu \tau \\ \lambda \end{array}}\right)a_{\begin{array}{c}Q Q\\ \tau \end{array}}$$ is zero unless $\mu =\tau$ and

$${ϵ}_1\left(b_{\begin{array}{c}P Q\\ \mu \mu \\ \lambda \end{array}}\right)={ϵ}_1\left(a_{\begin{array}{c}P R\\ \mu \end{array}}{b}_{\begin{array}{c}R R\\ \mu \mu \\ \lambda \end{array}}{a}_{\begin{array}{c}R Q\\ \mu \end{array}}\right)=a_{\begin{array}{c}P R\\ \mu \end{array}}{ϵ}_1\left(b_{\begin{array}{c}R R\\ \mu \mu \\ \lambda \end{array}}\right)a_{\begin{array}{c}R Q\\ \mu \end{array}}={ϵ}_{\mu }^{\lambda }{a}_{\begin{array}{c}P Q\\ \mu \end{array}}$$

1.10

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 $$b_{\begin{array}{c}P R\\ \mu \rho \\ \lambda \end{array}}e{b}_{\begin{array}{c}R Q\\ \rho \nu \\ \sigma \end{array}}=b_{\begin{array}{c}P S\\ \mu \rho \\ \lambda \end{array}}{a}_{\begin{array}{c}S R\\ \rho \end{array}}e{b}_{\begin{array}{c}R Q\\ \rho \nu \\ \sigma \end{array}}=b_{\begin{array}{c}P S\\ \mu \rho \\ \lambda \end{array}}e{a}_{\begin{array}{c}S R\\ \rho \end{array}}{b}_{\begin{array}{c}R Q\\ \rho \nu \\ \sigma \end{array}}=b_{\begin{array}{c}P S\\ \mu \rho \\ \lambda \end{array}}e{b}_{\begin{array}{c}S Q\\ \rho \nu \\ \sigma \end{array}}$$ does not depend on the choice of $R$. If

$$c_{\begin{array}{c}P Q\\ \mu \nu \\ \lambda \sigma \\ \gamma \end{array}}=b_{\begin{array}{c}P T\\ \mu \gamma \\ \lambda \end{array}}e{b}_{\begin{array}{c}T Q\\ \gamma \nu \\ \sigma \end{array}}$$

1.11

then $$\begin{array}{rcl}{c}_{\begin{array}{c}P Q\\ \mu \nu \\ \lambda \sigma \\ \gamma \end{array}}{c}_{\begin{array}{c}R S\\ \tau \xi \\ \rho \eta \\ \pi \end{array}}& =& \left(b_{\begin{array}{c}P T\\ \mu \gamma \\ \lambda \end{array}}e{b}_{\begin{array}{c}T Q\\ \gamma \nu \\ \sigma \end{array}}\right)\left(b_{\begin{array}{c}R X\\ \tau \pi \\ \rho \end{array}}e{b}_{\begin{array}{c}X S\\ \pi \xi \\ \eta \end{array}}\right)={\delta }_{\begin{array}{c}Q R\\ \nu \tau \\ \sigma \hspace{0.17em}\rho \end{array}}{b}_{\begin{array}{c}P T\\ \mu \gamma \\ \lambda \end{array}}{ϵ}_1\left(b_{\begin{array}{c}T X\\ \gamma \pi \\ \sigma \end{array}}\right)e{b}_{\begin{array}{c}X S\\ \pi \xi \\ \eta \end{array}}\\ & =& {\delta }_{\begin{array}{c}Q R\\ \nu \tau \\ \sigma \hspace{0.17em}\rho \end{array}}{b}_{\begin{array}{c}P T\\ \mu \gamma \\ \lambda \end{array}}{\delta }_{\gamma \pi }{ϵ}_{\gamma }^{\sigma }{a}_{\begin{array}{c}T X\\ \gamma \end{array}}e{b}_{\begin{array}{c}X S\\ \pi \xi \\ \eta \end{array}}={\delta }_{\begin{array}{c}Q R\\ \nu \tau \\ \sigma \hspace{0.17em}\rho \end{array}}{\delta }_{\pi \gamma }{ϵ}_{\gamma }^{\sigma }{b}_{\begin{array}{c}P X\\ \mu \gamma \\ \lambda \end{array}}e{b}_{\begin{array}{c}X S\\ \gamma \xi \\ \eta \end{array}}={\delta }_{\begin{array}{c}Q R\\ \nu \tau \\ \sigma \hspace{0.17em}\rho \end{array}}{\delta }_{\pi \gamma }{ϵ}_{\gamma }^{\sigma }{c}_{\begin{array}{c}P S\\ \mu \xi \\ \lambda \eta \\ \gamma \end{array}}.\end{array}$$ Define

$$e_{\begin{array}{c}P Q\\ \mu \nu \\ \lambda \sigma \\ \gamma \end{array}}=\left(\frac{1}{{ϵ}_{\gamma }^{\sigma }}\right)c_{\begin{array}{c}P Q\\ \mu \nu \\ \lambda \sigma \\ \gamma \end{array}},\text{whenver}\hspace{0.17em}{ϵ}_{\gamma }^{\sigma }\ne 0.\text{Then}{e}_{\begin{array}{c}P Q\\ \mu \nu \\ \lambda \sigma \\ \gamma \end{array}}{e}_{\begin{array}{c}R S\\ \tau \xi \\ \rho \eta \\ \pi \end{array}}={\delta }_{\begin{array}{c}Q R\\ \nu \tau \\ \sigma \hspace{0.17em}\rho \end{array}}{\delta }_{\gamma \pi }{e}_{\begin{array}{c}P S\\ \mu \xi \\ \lambda \eta \\ \gamma \end{array}}$$

1.12

so that these are matrix units. Furthermore

$$b_{\begin{array}{c}P Q\\ \mu \nu \\ \lambda \end{array}}{e}_{\begin{array}{c}R S\\ \tau \xi \\ \rho \eta \\ \pi \end{array}}={\delta }_{\begin{array}{c}Q R\\ \nu \tau \\ \lambda \hspace{0.17em}\rho \end{array}}{e}_{\begin{array}{c}P S\\ \mu \xi \\ \lambda \eta \\ \pi \end{array}}\text{and}{e}_{\begin{array}{c}P Q\\ \mu \nu \\ \lambda \sigma \\ \gamma \end{array}}{b}_{\begin{array}{c}R S\\ \tau \xi \\ \rho \end{array}}={\delta }_{\begin{array}{c}Q R\\ \nu \tau \\ \sigma \hspace{0.17em}\rho \end{array}}{e}_{\begin{array}{c}P S\\ \mu \xi \\ \lambda \rho \\ \gamma \end{array}}$$

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

$$e=1\cdot e\cdot 1=(\sum b_{\begin{array}{c}R R\\ \rho \rho \\ \lambda \end{array}})e(\sum b_{\begin{array}{c}S S\\ \sigma \sigma \\ \gamma \end{array}})=\sum b_{\begin{array}{c}R R\\ \rho \rho \\ \lambda \end{array}}e{b}_{\begin{array}{c}R R\\ \rho \rho \\ \lambda \end{array}}=\sum c_{\begin{array}{c}R R\\ \rho \rho \\ \lambda \gamma \\ \rho \end{array}}={ϵ}_{\rho }^{\gamma }{e}_{\begin{array}{c}R R\\ \rho \rho \\ \lambda \gamma \\ \rho \end{array}}.$$

1.14

In summary $$\begin{array}{c}e{b}_{\begin{array}{c}P Q\\ \mu \tau \\ \lambda \end{array}}e={\delta }_{\mu \tau }{ϵ}_{\mu }^{\lambda }{a}_{\begin{array}{c}P Q\\ \mu \end{array}}\\ e_{\begin{array}{c}P Q\\ \mu \nu \\ \lambda \sigma \\ \gamma \end{array}}=\left(\frac{1}{{ϵ}_{\gamma }^{\sigma }}\right)b_{\begin{array}{c}P T\\ \mu \gamma \\ \lambda \end{array}}e{b}_{\begin{array}{c}T Q\\ \gamma \nu \\ \sigma \end{array}}\\ b_{\begin{array}{c}P Q\\ \mu \nu \\ \lambda \end{array}}=\sum _{\lambda \to \gamma }{e}_{\begin{array}{c}P Q\\ \mu \nu \\ \lambda \lambda \\ \gamma \end{array}}\\ \end{array}$$ and $$e{e}_{\begin{array}{c}P A\\ \mu \nu \\ \lambda \sigma \\ \gamma \end{array}}={\delta }_{\mu \gamma }{ϵ}_{\gamma }^{\lambda }\sum _{\gamma \to \tau \to \gamma }{e}_{\begin{array}{c}P Q\\ \gamma \nu \\ \tau \sigma \\ \gamma \end{array}}.$$

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

Fix isomorphisms

$$\bar{L}\cong \underset{\mu \in \hat{A}}{⨁}{\overrightarrow{A}}^{\mu }\otimes L^{\mu }\text{and}\bar{R}\cong \underset{\mu \in \hat{A}}{⨁}{R}^{\mu }\otimes {\overleftarrow{A}}^{\mu }$$

1.15

where ${\overrightarrow{A}}^{\mu },\mu \in {\hat{A}}^{\mu }$ are the simple left $\bar{A}$-modules, ${\overleftarrow{A}}^{\mu }$, $\mu \in \hat{A}$, are the simple right $\bar{A}$-modules, and $L^{\mu },R^{\mu },\mu \in \hat{A}$ are vector spaces. In other words, if $A$ has matrix units $$\{a_{\begin{array}{c}P Q\\ \mu \end{array}}\mid \mu \in \hat{A},P\to \mu ,Q\to \mu \in {\hat{A}}^{\mu }\}\text{then}\begin{array}{ccc}L& \text{has a basis}& \left\{l_{PX}\mid P\in {\hat{A}}^{\mu },X\in {\hat{L}}^{\mu }\right\}\\ R& \text{has a basis}& \left\{r_{YQ}\mid Q\in {\hat{A}}^{\mu },Y\in {\hat{L}}^{\mu }\right\}\end{array}$$ such that

1.16

The map $ϵ:\bar{L}{\otimes }_{𝔽}\bar{R}\to \bar{A}$ is determined by the constants ${ϵ}_{XY}^{\mu }\in 𝔽$ given by

$ϵ(l_{\begin{array}{c}Q X\\ \mu \end{array}}\otimes r_{\begin{array}{c}Y P\\ \mu \end{array}})={ϵ}_{XY}^{\mu }{a}_{\begin{array}{c}Q P\\ \mu \end{array}}$

1.17

and ${ϵ}_{XY}^{\mu }$ does not depend on $Q$ and $P$ since $$\begin{array}{rcl}ϵ(l_{\begin{array}{c}S X\\ \lambda \end{array}}\otimes r_{\begin{array}{c}Y P\\ \mu \end{array}})& =& ϵ(a_{\begin{array}{c}S Q\\ \lambda \end{array}}{l}_{\begin{array}{c}Q X\\ \lambda \end{array}}\otimes r_{\begin{array}{c}Y P\\ \mu \end{array}}{a}_{\begin{array}{c}P T\\ \mu \end{array}})=a_{\begin{array}{c}S Q\\ \lambda \end{array}}ϵ(l_{\begin{array}{c}Q X\\ \lambda \end{array}}\otimes r_{\begin{array}{c}Y P\\ \mu \end{array}})a_{\begin{array}{c}P T\\ \mu \end{array}}\\ & =& {\delta }_{\lambda \mu }{a}_{\begin{array}{c}S Q\\ \mu \end{array}}{ϵ}_{XY}^{\mu }{a}_{\begin{array}{c}Q P\\ \mu \end{array}}{a}_{\begin{array}{c}P T\\ \mu \end{array}}={ϵ}_{XY}^{\mu }{a}_{\begin{array}{c}S T\\ \mu \end{array}}.\end{array}$$ For each $\mu \in \hat{A}$ construct a matrix

${ℰ}^{\mu }=\left({ϵ}_{XY}^{\mu }\right)$

1.18

and use row reduction (Smith normal form) to find invertible matrices

$D^{\mu }=\left(D_{ST}^{\mu }\right)\text{and}{C}^{\mu }=\left(C_{ZW}^{\mu }\right)\text{such that}{D}^{\mu }{ℰ}^{\mu }{C}^{\mu }=diag\left({ϵ}_X^{\mu }\right)$

1.19

is a diagonal matrix with diagonal entries denoted ${ϵ}_X^{\mu }$. The ${ϵ}_P^{\mu }$ are the invariant factors of the matrix ${ℰ}^{\mu }$.

Using notation as in (???) define elements of $R{\otimes }_{A}L$ by

$m_{\begin{array}{c}X Y\\ \mu \end{array}}=r_{\begin{array}{c}X P\\ \mu \end{array}}\otimes l_{\begin{array}{c}P Y\\ \mu \end{array}},\text{and}{n}_{\begin{array}{c}X Y\\ \mu \end{array}}=\sum _{Q_1,Q_2}{C}_{Q_{1}X}^{\mu }{D}_{Y{Q}_2}^{\mu }{m}_{\begin{array}{c}{Q}_1 Q_2\\ \mu \end{array}}$

1.20

where $\mu \in \hat{A},X\in {\hat{R}}^{\mu },Y\in {\hat{L}}^{\mu }$.

1. The sets $$\{m_{\begin{array}{c}X Y\\ \mu \end{array}}\mid \mu \in \hat{A},X\in {\hat{R}}^{\mu },Y\in {\hat{L}}^{\mu }\}\text{and}\{n_{\begin{array}{c}X Y\\ \mu \end{array}}\mid \mu \in \hat{A},X\in {\hat{R}}^{\mu },Y\in {\hat{L}}^{\mu }\}$$ are bases of $\bar{R}{\otimes }_{\bar{A}}\bar{L}$, which satisfy $$m_{\begin{array}{c}S T\\ \lambda \end{array}}{m}_{\begin{array}{c}Q P\\ \mu \end{array}}={\delta }_{\lambda \mu }{ϵ}_{TQ}^{\mu }{m}_{\begin{array}{c}S P\\ \mu \end{array}}\text{and}{n}_{\begin{array}{c}S T\\ \lambda \end{array}}{n}_{\begin{array}{c}Q P\\ \mu \end{array}}={\delta }_{\lambda \mu }{\delta }_{TQ}{ϵ}_T^{\mu }{n}_{\begin{array}{c}S P\\ \mu \end{array}},$$ 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 $$Rad\left(R{\otimes }_{A}L\right)=𝔽\text{-span}\{n_{\begin{array}{c}Y T\\ \mu \end{array}}\mid {ϵ}_Y^{\mu }=0\hspace{0.17em}\text{or}\hspace{0.17em}{ϵ}_T^{\mu }=0\}$$ and the images of the elements $$e_{\begin{array}{c}Y T\\ \mu \end{array}}=\frac{1}{{ϵ}_T^{\mu }}{n}_{\begin{array}{c}Y T\\ \mu \end{array}},\text{for} {ϵ}_Y^{\mu }\ne 0 \text{or} {ϵ}_T^{\mu }\ne 0,$$ are a set of matrix units in $\left(R{\otimes }_{A}L\right)/Rad\left(R{\otimes }_{A}L\right)$.

		Proof.
	Since $$(r_{\begin{array}{c}S W\\ \lambda \end{array}}\otimes l_{\begin{array}{c}Z T\\ \mu \end{array}})=(r_{\begin{array}{c}S P\\ \lambda \end{array}}{a}_{\begin{array}{c}P W\\ \lambda \end{array}}\otimes l_{\begin{array}{c}Z T\\ \mu \end{array}})=(R_{\begin{array}{c}S P\\ \lambda \end{array}}\otimes a_{\begin{array}{c}P W\\ \lambda \end{array}}{l}_{\begin{array}{c}Z T\\ \mu \end{array}})={\delta }_{\lambda \mu }{\delta }_{WZ}(r_{\begin{array}{c}S P\\ \lambda \end{array}}\otimes l_{\begin{array}{c}P T\\ \lambda \end{array}})$$ the element $m_{\begin{array}{c}X Y\\ \mu \end{array}}$ doe not depend on $P$ and $$\{m_{\begin{array}{c}X Y\\ \mu \end{array}}\mid \mu \in \hat{A},X\in {\hat{R}}^{\mu },Y\in {\hat{L}}^{\mu }\}\text{is a basis of}R{\otimes }_{A}L$$ 1.22 and hence $R{\otimes }_{A}L$ is a direct sum of generalized matrix algebras. If ${\left(C^{-1}\right)}^{\mu }$ and ${\left(D^{-1}\right)}^{\mu }$ are the inverses of the matrices $C^{\mu }$ and $D^{\mu }$ then $$\begin{array}{rcl}\sum _{x,Y}{\left(C^{-1}\right)}_{XS}^{\mu }{\left(D^{-1}\right)}_{TY}^{\mu }{n}_{\begin{array}{c}X Y\\ \mu \end{array}}& =& \sum _{X,Y,Q_1,Q_2}{\left(C^{-1}\right)}_{XS}^{\mu }{C}_{Q_{1}X}^{\mu }{m}_{\begin{array}{c}{Q}_1 Q_2\\ \mu \end{array}}{D}_{Y{Q}_2}^{\mu }{\left(D^{-1}\right)}_{TY}^{\mu }\\ & =& \sum _{Q_1,Q_2}{\delta }_{S{Q}_1}{\delta }_{Q_{2}T}{m}_{\begin{array}{c}{Q}_1 Q_2\\ \mu \end{array}}=m_{\begin{array}{c}S T\\ \mu \end{array}},\end{array}$$ and so the elements $m_{\begin{array}{c}S T\\ \mu \end{array}}$ can be written as a linear combination of the $n_{\begin{array}{c}X Y\\ \mu \end{array}}$. Thus $\{n_{\begin{array}{c}X Y\\ \mu \end{array}}\mid \mu \in \hat{A},X\in {\hat{R}}^{\mu },Y\in {\hat{L}}^{\mu }\}\text{is a basis of}R{\otimes }_{A}L.$ 1.22 By direct computation, $$\begin{array}{rcl}{m}_{\begin{array}{c}S T\\ \mu \end{array}}{m}_{\begin{array}{c}Q P\\ \lambda \end{array}}& =& (r_{\begin{array}{c}S W\\ \lambda \end{array}}\otimes l_{\begin{array}{c}W T\\ \lambda \end{array}})(r_{\begin{array}{c}Q Z\\ \mu \end{array}}\otimes l_{\begin{array}{c}Z P\\ \mu \end{array}})=ϵ(l_{\begin{array}{c}W T\\ \lambda \end{array}}\otimes r_{\begin{array}{c}Q Z\\ \mu \end{array}})l_{\begin{array}{c}Z P\\ \mu \end{array}}\\ & =& {\delta }_{\lambda \mu }(r_{\begin{array}{c}S W\\ \lambda \end{array}}\otimes {ϵ}_{TQ}^{\lambda }{a}_{\begin{array}{c}W Z\\ \lambda \end{array}}\hspace{0.17em}{l}_{\begin{array}{c}Z P\\ \lambda \end{array}})={\delta }_{\lambda \mu }{ϵ}_{TQ}^{\lambda }(r_{\begin{array}{c}S W\\ \lambda \end{array}}\otimes l_{\begin{array}{c}W P\\ \lambda \end{array}})={\delta }_{\lambda \mu }{ϵ}_{TQ}^{\lambda }{m}_{\begin{array}{c}S P\\ \lambda \end{array}},\end{array}$$ and $$\begin{array}{rcl}{n}_{\begin{array}{c}S T\\ \lambda \end{array}}{n}_{\begin{array}{c}U V\\ \mu \end{array}}& =& \sum _{Q_1,Q_2,Q_3,Q_4}{C}_{Q_{1}S}^{\lambda }{D}_{T{Q}_2}^{\lambda }{m}_{\begin{array}{c}{Q}_1 Q_2\\ \lambda \end{array}}{C}_{Q_{3}U}^{\mu }{D}_{V{Q}_4}^{\mu }{m}_{\begin{array}{c}{Q}_3 Q_4\\ \mu \end{array}}\\ & =& \sum _{Q_1,Q_2,Q_3,Q_4}{\delta }_{\lambda \mu }{C}_{Q_{1}S}^{\lambda }{D}_{T{Q}_2}^{\lambda }{ϵ}_{Q_1{Q}_2}^{\mu }{C}_{Q_{3}U}^{\mu }{D}_{V{Q}_4}^{\mu }{m}_{\begin{array}{c}{Q}_3 Q_4\\ \mu \end{array}}\\ & =& {\delta }_{\lambda \mu }\sum _{Q_1,Q_4}{\delta }_{TU}{ϵ}_T^{\mu }{C}_{Q_{1}S}^{\mu }{D}_{V{Q}_4}^{\mu }{m}_{\begin{array}{c}{Q}_1 Q_4\\ \mu \end{array}}={\delta }_{\lambda \mu }{\delta }_{TU}{ϵ}_T^{\mu }{n}_{\begin{array}{c}S V\\ \mu \end{array}}.\end{array}$$ 1.23 Let $$I=𝔽\text{-span}\{n_{\begin{array}{c}Y T\\ \mu \end{array}}\mid {ϵ}_Y^{\mu }=0\hspace{0.17em}\text{or}\hspace{0.17em}{ϵ}_T^{\mu }=0\}$$ The multipliction rule for the $n_{\begin{array}{c}Y T\\ \mu \end{array}}$ implies that $I$ is an ideal of $R{\otimes }_{A}L$. If $$n_{\begin{array}{c}{Y}_1 T_1\\ \mu \end{array}},n_{\begin{array}{c}{Y}_2 T_2\\ \mu \end{array}},n_{\begin{array}{c}{Y}_3 T_3\\ \mu \end{array}}\in \{n_{\begin{array}{c}Y T\\ \mu \end{array}}\mid {ϵ}_Y^{\mu }=0\hspace{0.17em}\text{or}\hspace{0.17em}{ϵ}_T^{\mu }=0\}$$ then $$n_{\begin{array}{c}{Y}_1 T_1\\ \mu \end{array}}{n}_{\begin{array}{c}{Y}_2 T_2\\ \mu \end{array}}{n}_{\begin{array}{c}{Y}_3 T_3\\ \mu \end{array}}={\delta }_{T_1{Y}_2}{ϵ}_{Y_2}^{\mu }{n}_{\begin{array}{c}{Y}_1 T_1\\ \mu \end{array}}{n}_{\begin{array}{c}{Y}_3 T_3\\ \mu \end{array}}={\delta }_{T_1{Y}_2}{\delta }_{T_2{Y}_3}{ϵ}_{Y_2}^{\mu }{ϵ}_{T_2}^{\mu }{n}_{\begin{array}{c}{Y}_1 T_3\\ \mu \end{array}},$$ since ${ϵ}_{Y_2}^{\mu }=0$ or ${ϵ}_{T_2}^{\mu }=0$. Thus any product $n_{\begin{array}{c}{Y}_1 T_1\\ \mu \end{array}}{n}_{\begin{array}{c}{Y}_2 T_2\\ \mu \end{array}}{n}_{\begin{array}{c}{Y}_3 T_3\\ \mu \end{array}}$ of three basis elements of $I$ is $0$. So $I$ is an ideal of $R{\otimes }_{A}L$ consisting of nilpotent elements and so $I\subseteq Rad\left(R{\otimes }_{A}L\right)$. Since $$e_{\begin{array}{c}Y T\\ \lambda \end{array}}{e}_{\begin{array}{c}U V\\ \mu \end{array}}=\frac{1}{{ϵ}_T^{\lambda }}\frac{1}{{ϵ}_V^{\mu }}{n}_{\begin{array}{c}Y T\\ \lambda \end{array}}{n}_{\begin{array}{c}U V\\ \mu \end{array}}={\delta }_{\lambda \mu }{\delta }_{TU}\frac{1}{{ϵ}_T^{\lambda }{ϵ}_V^{\lambda }}{ϵ}_V^{\lambda }{n}_{\begin{array}{c}Y V\\ \lambda \end{array}}={\delta }_{\lambda \mu }{\delta }_{TU}{e}_{\begin{array}{c}Y V\\ \lambda \end{array}}mod\hspace{0.17em}I,$$ the images of the elements $e_{\begin{array}{c}Y T\\ \lambda \end{array}}$ in (????) form a set of matrix units in the algebra $\left(R{\otimes }_{A}L\right)/I$. Thus $\left(R{\otimes }_{A}L\right)/I$ is a split semisimple algebra and so $I\supseteq Rad\left(R{\otimes }_{A}L\right)$. $\square$

The structure of $Z\left(ϵ\right)$

Let $$ϵ:L{\otimes }_{D}R\to C\text{be a} \left(C,C\right) \text{bimodule homomorphism.}$$ The left radical $L\left(ϵ\right)$ and the right radical $R\left(ϵ\right)$ of $ϵ$ are defined by $$\begin{array}{rcl}L\left(ϵ\right)& =& \left\{l\in L\mid ϵ\left(l\otimes r\right)\in Rad\left(C\right),\text{for all} r\in R\right\},\\ R\left(ϵ\right)& =& \left\{r\in R\mid ϵ\left(l\otimes r\right)\in Rad\left(C\right),\text{for all} l\in L\right\},\end{array}$$ The map $ϵ$ is nondegenerate if $Rad\left(C\right)=0$, $L\left(ϵ\right)=0$, and $R\left(ϵ\right)=0$. Let $$\begin{array}{l}\bar{C}=C/Rad\left(C\right),\\ \bar{L}=L/L\left(ϵ\right),\\ \bar{R}=R/R\left(ϵ\right),\end{array}\text{and}\begin{array}{rccc}\phi :& R{\otimes }_{C}L& \to & \bar{R}{\otimes }_{\bar{C}}\bar{L}\\ & \bar{r}\otimes \bar{l}& \mapsto & \overline{r\otimes l}\end{array}$$ Then $ker\hspace{0.17em}\phi$ is generated by $R{\otimes }_{C}L\left(ϵ\right)$ and $R\left(ϵ\right){\otimes }_{C}L$, and we have that $ker\hspace{0.17em}\phi \cdot R\cdot R\left(ϵ\right)$ and $L\cdot ker\hspace{0.17em}\phi \subset L\left(ϵ\right)$. THne $$I=Rad\left(C\right)+L\left(C\right)+ker\hspace{0.17em}\phi \text{is a nilpotent ideal of} A\left(ϵ\right),$$ and $$\frac{A\left(ϵ\right)}{I}\cong A\left(\overline{ϵ}\right)\text{where the map}\begin{array}{rccc}\overline{ϵ}:& \bar{L}{\otimes }_D\bar{R}& \to & \bar{C}\end{array}$$ is a nondegenerate $(\bar{C},\bar{C})$ bimodule homomorphism.

If $ϵ:L{\otimes }_{D}R\to C$ is nondegenerate and $R$ is a projective $C$-module then there is a $(D,C)$ bimodule homomorphism $$\begin{array}{ccccccc}\tau :& R& \stackrel{\sim }{\to }& L^{*}& & & \\ & r& \mapsto & {\lambda }_r:& L& \to & C\\ & & & & l& \mapsto & ϵ\left(l\otimes r\right)\end{array}\text{so that}ϵ=ev\circ \left(id\otimes \tau \right)$$ and $$A\left(ϵ\right)\cong A\left({ev}_L\right).$$

If $C,D,L,R$ are finite dimensional vector spaces over $𝔽$ and $D=𝔽$ then $$ϵ={ϵ}_0\oplus {ev}_P:\left(L_0\oplus P^{*}\right){\otimes }_D\left(R_0\oplus P\right)\to C,$$ with $P$ projective and $im\hspace{0.17em}{ϵ}_0\subseteq Rad\left(C\right)$.

If $ϵ={ϵ}_0\oplus {ev}_P$ with $P$ finitely generated and projective then $$\begin{array}{ccc}A\left(ϵ\right)-mod& \stackrel{\sim }{\to }& A\left({ϵ}_0\right)-mod\\ M& \mapsto & eM\end{array}\text{where}e=1-\sum _i{p}_i\otimes {\alpha }_i.$$

If $im\hspace{0.17em}ϵ\subseteq Rad\left(C\right)$ then $$Rad\left(A\left({ϵ}_0\right)\right)=I=Rad\left(C\right)\oplus Rad\left(D\right)\oplus L_0\oplus R_0{\otimes }_C{L}_0$$ and $$\frac{A\left({ϵ}_0\right)}{Rad\left(A\left({ϵ}_0\right)\right)}\cong \frac{C}{Rad\left(C\right)}\oplus \frac{D}{Rad\left(D\right)}.$$

Duals and Projectives

Let $L$ be a $C$-module and let $$Z={End}_C\left(L\right)$$ so that $L$ is a $(C,Z)$ bimodule. The dual module to $L$ is the $(Z,C)$ bimodule $$L^{*}={Hom}_C\left(L,C\right).$$ The evaluation map is the $(C,C)$ bimodule homomorphism $$\begin{array}{cccc}ev:& L{\otimes }_Z{L}^{*}& \to & C\\ & \lambda \otimes l& \mapsto & \lambda \left(l\right)\end{array}$$ and the centralizer map is the $(Z,Z)$ bimodule homomorphism $$\begin{array}{ccccccc}\xi :& L^{*}{\otimes }_{C}L& \to & Z& & & \\ & \lambda \otimes l& \mapsto & z_{\lambda ,l}:& L& \to & L\\ & & & & m& \mapsto & \lambda \left(m\right)l\end{array}$$ Recall that [Bou, Alg. II §4.2 Cor.]

1. If $L$ is a projective $C$-module if and only if $1\in im\hspace{0.17em}\xi$,
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 $${\xi }^{-1}\left(z\right)=\sum _i{b}_i^{*}\otimes z\left(b^i\right),$$ 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 $$\text{if} l\in L\text{then}l=\sum _i{b}_i^{*}\left(l\right)b_i\text{so that}\xi \left(\sum _i{b}_i^{*}\otimes b_i\right)=1.$$

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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