L PREPRINT - accepted In Proceedings of the 2023 International Conference on
Hardware/Software Codesign and System Synthesis 4.2cm,1cm1.001.8

# Work-in-Progress: A Universal Instrumentation Platform for Non-Volatile
Memories ††thanks: This work was funded by the Federal Ministry of Education
and Research (BMBF, Germany) in the project NEUROTEC II (16ME0398K, 16ME0399).

Felix Staudigl1, Mohammed Hossein1, Tobias Ziegler3, Hazem Al Indari1, Rebecca
Pelke1,
Sebastian Siegel4, Dirk J. Wouters4, Dominik Sisejkovic2, Jan Moritz Joseph1,
and Rainer Leupers1 {staudigl, hossein, pelke, joseph<EMAIL_ADDRESS>aachen.de
{ziegler, wouters<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
1 Institute for Communication Technologies and Embedded Systems, RWTH Aachen
University, Germany 2 Security, Privacy, and Safety Research Group, Corporate
Research Robert Bosch GmbH, Germany 3 Institut für Werkstoffe der
Elektrotechnik 2 (IWE-2), RWTH Aachen University, Germany 4 Peter Grünberg
Institute (PGI-10), Forschungszentrum Jülich GmbH, German

###### Abstract

Emerging non-volatile memories (NVMs) represent a disruptive technology that
allows a paradigm shift from the conventional von Neumann architecture towards
more efficient computing-in-memory (CIM) architectures. Several
instrumentation platforms have been proposed to interface NVMs allowing the
characterization of single cells and crossbar structures. However, these
platforms suffer from low flexibility and are not capable of performing CIM
operations on NVMs. Therefore, we recently designed and built the
NeuroBreakoutBoard, a highly versatile instrumentation platform capable of
executing CIM on NVMs. We present our preliminary results demonstrating a
relative error $<5\%$ in the range of $1\text{\,}\mathrm{k\SIUnitSymbolOhm}$
to $1\text{\,}\mathrm{M\SIUnitSymbolOhm}$ and showcase the switching behavior
of a $\text{HfO}_{2}$/Ti-based memristive cell.

###### Index Terms:

ReRAM, CIM, LIM, memristor, instrumentation

Figure 1: Overview of the implemented structure and modules.

##  I Introduction

Emerging non-volatile memories (NVMs) represent an ideal substrate for
enabling computing-in-memory (CIM) by offering high density and non-volatility
properties [1]. CIM can be implemented in analog [2] or digital [3] fashion.
Although the former offers the best computational efficiency, analog CIM
requires expensive ADCs/DACs to convert inputs and outputs from the digital to
the analog domain and vice versa. The latter uses so-called logic families to
implement digital gates within the NVM, thereby circumventing the conversion
but suffering from lower computational efficiency. Several instrumentation
platforms have been proposed to characterize NVMs at the device and crossbar
levels. However, most of these platforms only support passive crossbar
structures [4, 5, 6, 7, 8, 9, 10]. To the best of our knowledge, there are
currently no platforms capable of executing neither analog nor digital CIM on
NVMs. While the CIM operation is performed within the memory, the
instrumentation platform must facilitate the surrounding circuitry to generate
synchronized voltage pulses (inputs) and process the resulting currents
(outputs). Hence, we designed and built the NeuroBreakoutBoard (NBB), a
universal instrumentation platform to perform CIM on NVMs. Our platform
supports all memristor-based memories in both passive (1R) and active (1T1R)
crossbar configurations. Furthermore, the unique combination of the
interconnection matrix, custom designed trans-impedance amplifiers (TIAs), and
the implemented firmware enable the NBB to perform both analog and digital CIM
operations by using different NVMs.

Figure 2: Image of the NeuroBreakoutBoard and its main components.

$99.4$$99.6$$99.8$$100$$100.2$$\cdot 10^{3}$$0$$20$$40$$60$$80$Resistance
($\text{\,}\mathrm{\SIUnitSymbolOhm}$)# measurments (a)

1501002003004005006007008001000$0$$1$$2$$3$$4$Resistance
($\text{\,}\mathrm{k\SIUnitSymbolOhm}$)Relative error/$\sigma$ (%)Relative
errorRelative $\sigma$ (b)

$0$$50$$100$$150$$200$$250$$300$$10^{4}$$10^{5}$Measurement no.Resistance
($\text{\,}\mathrm{\SIUnitSymbolOhm}$) (c)

Figure 3: Result overview: (a) single device histogram (1,000 measurements),
(b) relative error and standard deviation of various reference resistances,
and (c) write/read operations performed on the MAD200 NVM chip.

##  II NeuroBreakoutBoard (NBB)

The NBB generates the required input pulses with a multi-channel 12bit DAC
($0\text{\,}\mathrm{V}$-$5\text{\,}\mathrm{V}$,$0\text{\,}\mathrm{V}$-$10\text{\,}\mathrm{V}$,
$\pm$$2.5\text{\,}\mathrm{V}$, $\pm$$5\text{\,}\mathrm{V}$,
$\pm$$10\text{\,}\mathrm{V}$). These inputs can be arbitrarily mapped by the
interconnection matrix to the NVM interface shown in Fig. 2. The
west/east/south multiplexers provide five independent potentials, one external
measurement/supply line, and the ground potential. Additionally, the north
multiplexers connect to the sensing module capable of measuring currents and
voltages. The sensing module consists of an array of programmable TIAs,
together with high-performance 14bit ADCs enabling precise measurements of
resistances in the range of $1\text{\,}\mathrm{k\SIUnitSymbolOhm}$ to
$1\text{\,}\mathrm{M\SIUnitSymbolOhm}$. Each TIA offers four distinct
sensitivity levels subdividing the broad range of input currents/voltages by
four to enhance the overall measurement accuracy. The NBB can control up to 68
lines simultaneously and connects via the NVM interface to various extension
boards implementing additional digital circuitry, chip packages, and measuring
capabilities. The NBB is orchestrated by the controller interface, which
bundles all critical control and data signals. The controller unit connects to
this interface and provides a microcontroller which runs the implemented NBB
firmware. The software offers a Python/C++ interface to issue
write/read/compute operations to be executed on the crossbar array. The
implemented firmware executes operations by assigning the respective voltage
pulses on the corresponding pins. Likewise, the firmware controls the sensing
module by iteratively adjusting the programmable TIAs and triggering the
respective ADCs.

##  III Result

In this section, we discuss our preliminary results of two experiments that
showcase the measurement accuracy and flexibility of the NeuroBreakoutBoard.
Each of the four sensitivity stages has been calibrated with high-precision
resistances to obtain maximum accuracy.

Accuracy: To determine the measurement accuracy of the NBB, we applied a
similar methodology described by Berdan et al. [4]. Fig. 3 (a) illustrates the
result of 1,000 measurements of a reference resistor. Each measurement
incorporates 50 consecutive ADC samplings to omit the impact of noise. We
determined the actual resistance of the resistor using the Keithley DMM7510
(marked red). Furthermore, Fig. 3 (b) depicts the relative error and standard
deviation of various reference resistances calculated based on 10,000
measurements per resistance. Both experiments report a high measurement
precision ($<5\%$ relative error and $<1\%$ of relative $\sigma$) considering
the additional line resistances and parasitic effects of the interconnection
matrix.

Switiching characteristics: Finally, we performed several write/read
operations on a $\text{HfO}_{2}$/Ti 1T1R crossbar array. The crossbar is
fabricated with the MAD200 process offered by CMP/CEA-LETI [11] in
$130\text{\,}\mathrm{nm}$ consisting of a demultiplexer and two $512\times 32$
crossbar structures. To facilitate the chip on the NBB, we manufactured an
extension board connecting to the NVM interface. The extension board offers
the required digital circuitry (I/O expanders) to drive and control the chip.
Fig. 3 (c) illustrates the resistance change of a single cell within the
crossbar array over the course of alternating RESET/SET operations.

##  IV Conclusion

The NeuroBreakoutBoard is a versatile instrumentation platform to characterize
NVMs and execute CIM operations. Our preliminary results indicate a relative
error in the range of $1\text{\,}\mathrm{k\SIUnitSymbolOhm}$ to
$1\text{\,}\mathrm{M\SIUnitSymbolOhm}$ lower than $5\%$ and a high precision
($\sigma<1\%$). Moreover, we performed several read/write operations on a 1T1R
crossbar structure. In the future, we intend to provide the NeuroBreakoutBoard
as a commercial solution, thereby facilitating the accessibility of memristor-
based CIM to both academia and industry. Additionally, we aim to integrate the
NeuroBreakoutBoard into a hardware-in-the-loop simulation environment to
conduct thorough investigations on the reliability of non-volatile memories
(NVMs) under realistic workloads.

## References

  * [1] F. Staudigl _et al._ , “A Survey of Neuromorphic Computing-in-Memory: Architectures, Simulators, and Security,” _IEEE Design & Test_, 2022\.
  * [2] A. Ankit _et al._ , “PUMA: A Programmable Ultra-Efficient Memristor-Based Accelerator For Machine Learning Inference,” in _ACM ASPLOS_ , 2019, pp. 715–731.
  * [3] S. Kvatinsky _et al._ , “MAGIC - Memristor-Aided Logic,” _IEEE TCAS II: Express Briefs_ , vol. 61, no. 11, pp. 895–899, nov 2014.
  * [4] R. Berdan _et al._ , “A uController-Based System for Interfacing Selectorless RRAM Crossbar Arrays,” _IEEE Trans Electron Devices_ , 2015\.
  * [5] J. Cayo _et al._ , “Design Steps towards a MCU-based Instrumentation System for Memristor-based Crossbar Arrays,” in _IEEE MOCAST_ , 2021.
  * [6] R. D. L. Fuente _et al._ , “On the Development of MCU-based ad hoc HW Interface Circuitry for Memristor Characterization,” in _IEEE EECCTD_ , 2020.
  * [7] P. Foster _et al._ , “An FPGA Based System for Interfacing with Crossbar Arrays,” in _IEEE ISCAS_ , 2020.
  * [8] A. Grossi _et al._ , “An Automated Test Equipment for Characterization of Emerging MRAM and RRAM Arrays,” _IEEE TETC_ , 2018.
  * [9] J. Xing _et al._ , “An FPGA-based Manipulation System For ReRAM Characterization,” in _IEEE 3M-NANO_ , 2016.
  * [10] J. Xing _et al._ , “An FPGA-Based Instrument for En-Masse RRAM Characterization With ns Pulsing Resolution,” _IEEE TCAS I: Regular Papers_ , 2016.
  * [11] A. Grossi _et al._ , “Experimental Investigation of 4-kb RRAM Arrays Programming Conditions Suitable for TCAM,” _IEEE VLSI Systems_ , 2018\.


# Partial classification of irreducible modules for loop-Witt algebras

Priyanshu Chakraborty AND S. Eswara Rao  Priyanshu Chakraborty, Harish-Chandra
Research Institute (HBNI), Chhatnag Road, Jhunsi, Prayagraj(Allahabad) 211019,
Uttar Pradesh, India<EMAIL_ADDRESS>S. Eswara Rao, School
of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road,
Colaba, Mumbai 400005, India<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Consider the Lie algebra of the group of diffeomorphisms of a $n$-dimensional
torus which is also known as the derivation algebra of the Laurent polynomial
algebra $A$ over $n$ commuting variables, denoted by $DerA$. In this paper we
consider the Lie algebra $(A\rtimes DerA)\otimes B$ for some commutative
associative finitely generated unital algebra $B$ over $\mathbb{C}$ and
classify all irreducible modules for $(A\rtimes DerA)\otimes B$ with finite
dimensional weight spaces. In particularly, we show that Larsson’s constructed
modules of tensor fields exhausts all irreducible modules for $(A\rtimes
DerA)\otimes B$.

##  1\. Introduction

Importance of the representation theory of infinite dimensional Lie algebras
are well known in both mathematics and physics. Many mathematicians studied
representations for the Lie algebra of the derivation algebra of
$A=\mathbb{C}[t_{1}^{\pm 1},t_{2}^{\pm 1},...,t_{n}^{\pm 1}]$, so called Witt
algebra denoted by $W_{n}$ or $DerA$, for reference [1, 10, 11]. In
particular, for $n=1$ the universal central extension for the derivation
algebra of $A$ is known as the Virasoro algebra, generally denoted by $Vir$.
Representations of Virasoro algebras were well studied in [15, 16, 17] and
references therein. In [15], O.Mathieu proved that a simple module for $Vir$
with finite dimensional weight spaces is either a highest weight module, a
lowest weight module or a uniformly bounded module of intermediate series.
Moreover from an interesting paper ([12]) this result comes out that $DerA$
has no non-trivial central extensions for $n\geq 2$. In [13], Larsson
constructed a large class of modules for $DerA$, so called modules of tensor
fields. In [1], the second author proved that for $n\geq 2$, modules of tensor
fields for $DerA$ exhausts all irreducible weight modules for $A\rtimes DerA$
with finite dimensional weight spaces. In [10], complete classification of
irreducible modules for $DerA$ has been done by Yuly Billig and V. Futorny,
which was conjectured by the second author.

On the other hand, in present days mathematicians are showing interest to
study representations of loop algebras of some well known Lie algebras. For
instance, representation of current algebras (also called loop algebras) for
finite dimensional simple Lie algebras has been studied in [8, 19]. In [7],
representation of current algebras were generalized for finite dimensional
perfect Lie algebras. In [18], loop Affine Lie algebras were studied, in fact
any irreducible module for the well known Toroidal Lie algebras has been
reduced to an irreducible module for the loop Affine Lie algebras. Further, in
[6, 9] modules for loop Affine-Virasoro algebras and in [14] modules for loop
toroidal Lie algebras has been studied. For $B=\mathbb{C}[t^{\pm 1}]$,
complete classification of irreducible $Vir\otimes B$-modules has been done in
[2]. Later this work was generalized by A.Savage for $Vir\otimes B$, where $B$
is a commutative associative finitely generated unital algebra over
$\mathbb{C}$ in [5]. In [5], A.Savage prove that a simple module for
$Vir\otimes B$ with finite dimensional weight spaces is either highest weight
module, lowest weight module or uniformly bounded module. Broadly speaking,
irreducible modules for $Vir\otimes B$ has same kind of behaviour as $Vir$
modules. Classification of irreducible modules for loop-Witt algebras is a
challenging open problem. Our results is one step towards that. In this paper
we study irreducible weight modules for $(A\rtimes DerA)\otimes B$ with finite
dimensional weight spaces, where $B$ is a commutative associative finitely
generated unital algebra over $\mathbb{C}$. It is natural to ask whether
modules of tensor fields exhausts all irreducible modules for $(A\rtimes
DerA)\otimes B$ or not. Finally in Theorem 4.1, we prove that under some
natural conditions modules of tensor fields exhausts all irreducible modules
for $(A\rtimes DerA)\otimes B$.

This paper has been organised as follows. In section 2, we start with basic
definitions and preliminaries. At the end of this section we recall the
Theorem 2.1, proved in [1] regarding classification of irreducible $A\rtimes
DerA$ modules for all $n\geq 2$. In section 3, we define weight modules for
$\tau=(A\rtimes DerA)\otimes B$ and construct an irreducible weight module for
$\tau$. Then we consider an irreducible weight module $V$ for $\tau$ with
finite dimensional weight spaces. In Theorem 3.1, we prove that under some
natural conditions $V$ becomes an irreducible module for $A\rtimes DerA.$ In
section 4, we prove that Theorem 2.1 holds for $n=1$ too. At the end of this
section, in Theorem 4.1, we prove our final classification theorem.

We would like to point out one difference between loop of finite dimensional
simple Lie algebras and loop of any extension of Witt algebras. For the loop
of simple finite dimensional Lie algebras we get irreducible modules as
evaluation modules at multiple points, see [8, 7, 19]. But for the loop of
Witt algebras which has been studied in [2, 5] and the current paper,
irreducible modules comes from a single points evaluation modules. So we
conjecture that for the classification of loop-Witt algebras with uniformly
bounded weight spaces we get a single point evaluation module.

##  2\. Notations and Preliminaries

### (1)

Throughout the paper all the vector spaces, algebras, tensor products are
taken over the field of complex numbers $\mathbb{C}$. Let $\mathbb{Z}$,
$\mathbb{N}$, $\mathbb{Z}_{+}$ denote the sets of integers, natural numbers
and non-negative integers respectively. Let $\mathbb{C}^{n}$ denote the direct
sum of $n$ copies of the vector space $\mathbb{C}$. Let $e_{i}$, $1\leq i\leq
n$ be the standard basis of $\mathbb{C}^{n}$ and $(,)$ be the form on
$\mathbb{C}^{n}$ such that $(e_{i},e_{j})=\delta_{ij}$, for $1\leq i,j\leq n$.
For any Lie algebra $G$, let $U(G)$ denote the universal enveloping algebra of
$G$.

### (2)

Let $n\geq 1$ be a fixed natural number and $A=\mathbb{C}[t_{1}^{\pm
1},...,t_{n}^{\pm 1}]$ denote the Laurent polynomial algebra. Let us denote
$\mathbb{Z}^{n}=\displaystyle{\bigoplus_{i=1}^{n}}\mathbb{Z}e_{i}$. We denote
elements of $\mathbb{Z}^{n}$ by $m,r,s,t$ etc. For
$r=\displaystyle{\sum_{i=1}^{n}}r_{i}e_{i}\in\mathbb{Z}^{n}$, let
$t^{r}=t_{1}^{r_{1}}t_{2}^{r_{2}}...t_{n}^{r_{n}}\in A$ and
$D^{i}(r)=t^{r}t_{i}\frac{d}{dt_{i}}$ be a derivation on $A$. Let $DerA$ be
the Lie algebra of derivations on $A$. It is well known that $D^{i}(r)$ for
$1\leq i\leq n$ and $r\in\mathbb{Z}^{n}$ forms a basis for $DerA.$ For
$u=\displaystyle{\sum_{i=1}^{n}}u_{i}e_{i}\in\mathbb{C}^{n}$, let
$D(u,r)=\displaystyle{\sum_{i=1}^{n}}u_{i}D^{i}(r)$. Then $DerA$ is a Lie
algebra with respect to the bracket operation

(2.1) $\displaystyle\left[D(u,r),D(v,s)\right]=D(w,r+s),$

where $w=(u,s)v-(v,r)u$ for all $r,s\in\mathbb{Z}^{n}$ and
$u,v\in\mathbb{C}^{n}$.

Let $\mathfrak{h}$ be the subspace of $DerA$ spanned by $D^{i}(0)$ for $1\leq
i\leq n.$ Then $\mathfrak{h}$ is a maximal abelian sub-algebra of $DerA$.

In particular when $n=1$, we consider $A=\mathbb{C}[t^{\pm 1}]$ and denote
$d_{n}=t^{n}d$ for all $n\in\mathbb{Z},$ where $d=t\frac{d}{dt}.$ Then we have
$[d_{m},d_{n}]=(n-m)d_{m+n}$ for all $m,n\in\mathbb{Z}.$

### (3)

Let $\mathfrak{g}_{1}$ and $\mathfrak{g}_{2}$ be two Lie algebras and
$\phi:\mathfrak{g}_{2}\to Der(\mathfrak{g}_{1})$ be a Lie algebra homomorphism
from $\mathfrak{g}_{2}$ to the Lie algebra of derivations of
$\mathfrak{g}_{1}$. Then the semi-direct product of the Lie algebras
$\mathfrak{g}_{1}$ and $\mathfrak{g}_{2}$ is denoted by
$\mathfrak{g}_{1}\rtimes\mathfrak{g}_{2}=\\{g_{1}+g_{2}:g_{1}\in\mathfrak{g}_{1},g_{2}\in\mathfrak{g}_{2}\\}$
with the bracket operation

(2.2)
$\displaystyle\left[g_{1}+g_{2},h_{1}+h_{2}\right]=\left[g_{1},h_{1}\right]+\left[g_{2},h_{2}\right]+\phi(g_{2})h_{1}-\phi(h_{2})g_{1}.$

### (4)

Consider the Lie algebra $A\rtimes DerA$ with the Lie bracket given by

(2.3) $\displaystyle\left[t^{r},t^{s}\right]=0,$ (2.4)
$\displaystyle\left[D(u,r),t^{s}\right]=(u,s)t^{r+s}$

for all $r,s\in\mathbb{Z}^{n}$ and $u\in\mathbb{C}^{n}$. Note that $A$ is an
ideal in $A\rtimes DerA$. Let $H=\mathbb{C}\oplus\mathfrak{h}$ be a Cartan
sub-algebra of $A\rtimes DerA.$

### (5)

Let $B$ be a commutative associative finitely generated unital algebra over
$\mathbb{C}$. Consider $\tau=(A\rtimes DerA)\otimes B$ and define a Lie
algebra structure on $\tau$ by

$\left[X\otimes b,Y\otimes b^{\prime}\right]=\left[X,Y\right]\otimes
bb^{\prime},$

for all $X,Y\in A\rtimes DerA$, $b,b^{\prime}\in B$. We will identify
$H\otimes 1$ with $H$. For convenience, we denote $D(u,r)\otimes b=D(u,r)b$
and $t^{r}\otimes b=t^{r}b$ for all $u\in\mathbb{C}^{n}$,
$r\in\mathbb{Z}^{n}$, $b\in B.$

### (6)

Let $\mathfrak{gl}_{n}$ be the Lie algebra of $n\times n$ matrices over
$\mathbb{C}$. Then $\mathfrak{gl}_{n}=\mathfrak{sl}_{n}\oplus\mathbb{C}I$,
where $\mathfrak{sl}_{n}$ be the subalgebra of $\mathfrak{gl}_{n}$ consisting
of trace zero matrices and $I$ be the $n\times n$ identity matrix. Let
$V(\mu)$ be a finite dimensional irreducible module for $\mathfrak{sl}_{n}$,
where $\mu$ is a dominant integral weight. Let $I$ act by a scalar $c$ on
$V(\mu)$ and denote the resultant finite dimensional irreducible
$\mathfrak{gl}_{n}$ module by $V(\mu,c).$ For $\alpha\in\mathbb{C}^{n},$ we
denote $F^{\alpha}(\mu,c)=V(\mu,c)\otimes A.$ We first recall $A\rtimes DerA$
modules which have been studied in [1].

######  Theorem 2.1.

$($Theorem 2.9; [1]$)$ Let $n\geq 2$ and $V$ be an irreducible module for
$A\rtimes DerA$ which is also a weight module for $H$ with finite dimensional
weight spaces. Also assume
$(1)$. $V$ is an $A$-module as an associative algebra and Lie module structure
of $A$ comes from associative algebra.
$(2)$. $1.v=v$ for all $v\in V$.
Then $V\simeq F^{\alpha}(\mu,c)$ for some $\alpha,c,\mu$ and actions of
$A\rtimes DerA$ on $F^{\alpha}(\mu,c)$ are given by

(2.5) $\displaystyle D(u,r)v\otimes t^{m}={(u,m+\alpha)v\otimes
t^{m+r}+\displaystyle{\sum_{i,j}(u_{i}r_{j}E_{ji}v)\otimes t^{m+r}}},$ (2.6)
$\displaystyle t^{r}v\otimes t^{m}=v\otimes t^{m+r}.$

## 3\.

######  Definition 3.1.

A module $V$ for $\tau$ is said to be weight module if the action of $H$ on
$V$ is diagonalizable, i.e $V$ can be decomposed as
$V=\displaystyle{\bigoplus_{\lambda\in H^{*}}V_{\lambda}}$, where
$H^{*}=Hom_{\mathbb{C}}(H,\mathbb{C})$ and $V_{\lambda}=\\{v\in
V:hv=\lambda(h)v$ for all $h\in H\\}.$ The space $V_{\lambda}$ is called the
weight space with respect to the weight $\lambda$ of $V$.

Let $\eta:B\to\mathbb{C}$ be an algebra homomorphism such that $\eta(1)=1$.
Define a $\tau$-module action on $F^{\alpha}(\mu,c)$ by

(3.1) $\displaystyle D(u,r)bv\otimes t^{m}=\eta(b)\\{(u,m+\alpha)v\otimes
t^{m+r}+\displaystyle{\sum_{i,j}(u_{i}r_{j}E_{ji}v)\otimes t^{m+r}}\\},$ (3.2)
$\displaystyle t^{r}b.v\otimes t^{m}=\eta(b)v\otimes t^{m+r}.$

It is easy to see that $F^{\alpha}(\mu,c)$ is a weight module for $\tau$ with
finite dimensional weight spaces $V(\mu)\otimes t^{m}$. By Theorem 2.1,
$F^{\alpha}(\mu,c)$ is an irreducible module for $\tau$.

The main purpose of this paper is to prove that under some natural conditions
$F^{\alpha}(\mu,c)$ exhausts all irreducible weight modules for $\tau$ with
finite dimensional weight spaces.

Let $V$ be an irreducible weight module for $\tau$ with finite dimensional
weight spaces. Also assume that the action of $A\otimes B$ on $V$ is
associative and $1$ acts as identity on $V$. Throughout this section we
consider $V$ as a $\tau$-module with these properties.

Let us assume that there exists a weight $\lambda\in H^{*}$ such that
$V_{\lambda}\neq 0$. Then due to irreducibility of $V$ we have
$V=\displaystyle{\bigoplus_{m\in\mathbb{Z}^{n}}}V_{\alpha+m},$ where
$V_{\alpha+m}=\\{v\in V:D(u,0)v=(u,m+\alpha)v$ for all $u\in\mathbb{C}^{n}\\}$
and
$\alpha=(\lambda(D^{1}(0),\lambda(D^{2}(0),....,\lambda(D^{n}(0))\in\mathbb{C}^{n}.$
Note that $V_{\lambda}=V_{\alpha}$ and $t^{k}$ is injective, hence
$t^{k}V_{\alpha}=V_{\alpha+k}$ for all $k\in\mathbb{Z}^{n}$. In particular,
all $V_{\alpha+m}$ have equal dimension. Therefore we can identify $V$ with
$V_{\alpha}\otimes A$ as vector spaces. Since $1\otimes B$ is central in
$\tau$ there exists an algebra homomorphism $\psi:B\to\mathbb{C}$ such that
$1\otimes b.v=\psi(b)v$ for all $v\in V$. Let $M$ be the kernel of this
homomorphism. Then $M$ is a maximal ideal in $B$, since $\psi(1)=1.$

Let $U$ be the universal enveloping algebra of $\tau$. Then
$U=\displaystyle{\bigoplus_{m\in\mathbb{Z}^{n}}}U_{m},$ where $U_{m}=\\{X\in
U:[D(u,0),X]=(u,m)X$ for all $u\in\mathbb{C}^{n}\\}$. Let $L$ be the two sided
ideal in $U$ generated by $t^{r}b.t^{s}b^{\prime}-t^{r+s}bb^{\prime}$ and
$t^{0}\otimes 1-1.$ Since $A\otimes B$ acts associatively on $V$ so $L$ acts
trivially on $V$ and hence $V$ is an irreducible $U/L$-module.

Let us consider the elements $T(u,r,b_{1},b_{2})=t^{-r}b_{1}D(u,r)b_{2}\in
U/L$ for all $r\in\mathbb{Z}^{n},u\in\mathbb{C}^{n},b_{1},b_{2}\in B$ and
define

$D=span\\{T(u,r,b_{1},b_{2}):r\in\mathbb{Z}^{n},u\in\mathbb{C}^{n},b_{1},b_{2}\in
B\\}.$

######  Lemma 3.1.

1\. $D$ is a Lie algebra with the Lie bracket

$[T(u,r,b_{1},b_{2}),T(v,s,b_{3},b_{4})]=T(w,r+s,b_{1}b_{3},b_{2}b_{4})-(u,s)T(v,s,b_{1}b_{2}b_{3},b_{4})$
$+(v,r)T(u,r,b_{1}b_{3}b_{4},b_{2}),$

where $w=(u,s)v-(v,r)u.$

2\. $V_{\alpha}$ is an irreducible $D$-module.

###### Proof.

1\. $[t^{-r}b_{1}D(u,r)b_{2},t^{-s}b_{3}D(v,s)b_{4}]$

$=[t^{-r}b_{1}D(u,r)b_{2},t^{-s}b_{3}]D(v,s)b_{4}+t^{-s}b_{3}[t^{-r}b_{1}D(u,r)b_{2},D(v,s)b_{4}]$

$=-(u,s)t^{-s}b_{1}b_{2}b_{3}D(v,s)b_{4}+(v,r)t^{-r}b_{1}b_{4}b_{3}D(u,r)b_{2}+t^{-r-s}b_{1}b_{3}D(w,r+s)b_{2}b_{4}$

$=T(w,r+s,b_{1}b_{3},b_{2}b_{4})-(u,s)T(v,s,b_{1}b_{2}b_{3},b_{4})+(v,r)T(u,r,b_{1}b_{3}b_{4},b_{2}),$

where $w=(u,s)v-(v,r)u.$

2\. Check that $D(w,0)$ commutes with $D$ for all $w\in\mathbb{C}^{n}$,
consequently $V_{\alpha}$ is a $D$-module. Let $v,w\in V_{\alpha}$ be two
arbitrary non-zero vectors. Then by weight arguments and the irreducibility of
$V$ implies that there exists $X\in U_{0}$ such that $X.v=w$. By the PBW
theorem we have $U(D)=U_{0}$ and hence $V_{\alpha}$ is an irreducible
$D$-module. ∎

######  Lemma 3.2.

$($ [3]$)$ 1\. Let $L\subset\mathfrak{gl}(V)$ $(V$ is finite dimensional $)$
be a non-zero Lie algebra and acting irreducibly on $V$. Then $L$ is reductive
and its center is at most one-dimensional.
2\. Let $L$ be a finite dimensional reductive Lie algebra. Then $L=[L,L]\oplus
Z(L)$ and $[L,L]$ is semi-simple.

Let us consider the elements
$T_{1}(u,r,b_{1},b_{2})=t^{-r}b_{1}D(u,r)b_{2}-D(u,0)b_{1}b_{2}$ and define

$D_{1}=span\\{T_{1}(u,r,b_{1},b_{2}):r\in\mathbb{Z}^{n},u\in\mathbb{C}^{n},b_{1},b_{2}\in
B\\}.$

Also let $I(u,r,b_{1},b_{2})=\psi(b_{1})D(u,r)b_{2}-D(u,0)b_{1}b_{2}$ and
define

$D_{2}=span\\{I(u,r,b_{1},b_{2}):r\in\mathbb{Z}^{n},u\in\mathbb{C}^{n},b_{1},b_{2}\in
B\\}.$

Moreover consider the subspace $W$ of $V,$ defined by

$W=span\\{t^{r}v-v:v\in V_{\alpha},r\in\mathbb{Z}^{n}\\}=span\\{t^{r}v-v:v\in
V,r\in\mathbb{Z}^{n}\\}$

Note that $dim(V/W)=dim(V_{\alpha})$.

######  Proposition 3.1.

1\. $D_{1}$ is a Lie algebra with the lie bracket

$[T_{1}(u,r,b_{1},b_{2}),T_{1}(v,s,b_{3},b_{4})]=T_{1}(w,r+s,b_{1}b_{3},b_{2}b_{4})-(u,s)T_{1}(v,s,b_{3},b_{1}b_{2}b_{4})$
$+(v,r)T_{1}(u,r,b_{1},b_{2}b_{3}b_{4}),$

where $w=(u,s)v-(v,r)u$. Moreover $[D,D]\subseteq D_{1}.$

2\. $D_{2}$ is a Lie sub-algebra of $DerA\otimes B$ with the Lie bracket

$[I(u,r,b_{1},b_{2}),I(v,s,b_{3},b_{4})]=I(w,r+s,b_{1}b_{3},b_{2}b_{4})-(u,s)I(v,s,b_{3},b_{1}b_{2}b_{4})$
$+(v,r)I(u,r,b_{1},b_{2}b_{3}b_{4}),$

where $w=(u,s)v-(v,r)u$.
3\. $\pi:D_{1}\to D_{2}$ defined by
$\pi(T_{1}(u,r,b_{1},b_{2})=I(u,r,b_{1},b_{2})$ is a surjective Lie algebra
homomorphism.
$4$. $V_{\alpha}$ is an irreducible $D_{1}$-module.
5\. $W$ is a $D_{2}$-sub-module. In particular $V/W$ is a $D_{2}$-module and a
$D_{1}$-module via $\pi$.
6\. $V_{\alpha}\simeq V/W$ as $D_{1}$-module.

###### Proof.

1\. From Lemma 3.1 we have

$[T(u,r,b_{1},b_{2}),T(v,s,b_{3},b_{4})]$

$=T(w,r+s,b_{1}b_{3},b_{2}b_{4})-(u,s)T(v,s,b_{1}b_{2}b_{3},b_{4})+(v,r)T(u,r,b_{1}b_{3}b_{4},b_{2})$

$=\\{T(w,r+s,b_{1}b_{3},b_{2}b_{4})-D(w,0)b_{1}b_{2}b_{3}b_{4}\\}-(u,s)\\{T(v,s,b_{1}b_{2}b_{3},b_{4})-D(v,0)b_{1}b_{2}b_{3}b_{4}\\}$

$+(v,r)\\{T(u,r,b_{1}b_{3}b_{4},b_{2})-D(u,0)b_{1}b_{2}b_{3}b_{4}\\}$

$=T_{1}(w,r+s,b_{1}b_{3},b_{2}b_{4})-(u,s)T_{1}(v,s,b_{1}b_{2}b_{3},b_{4})+(v,r)T_{1}(u,r,b_{1}b_{3}b_{4},b_{2})$.

From above relation we have $[D,D]\subseteq D_{1}.$

Now $[T_{1}(u,r,b_{1},b_{2}),T_{1}(v,s,b_{3},b_{4})]$

$=[t^{-r}b_{1}D(u,r)b_{2},t^{-s}b_{3}D(v,s)b_{4}]-[D(u,0)b_{1}b_{2},t^{-s}b_{3}D(v,s)b_{4}]+[D(v,0)b_{3}b_{4},t^{-r}b_{1}D(u,r)b_{2}]$

$=[T(u,r,b_{1}b_{2}),T(v,s,b_{3}b_{4})]-\\{[D(u,0)b_{1}b_{2},t^{-s}b_{3}]D(v,s)b_{4}+t^{-s}b_{3}[D(u,0)b_{1}b_{2},D(v,s)b_{4}]\\}$

$+\\{[D(v,0)b_{3}b_{4},t^{-r}b_{1}]D(u,r)b_{2}+t^{-r}b_{1}[D(v,0)b_{3}b_{4},D(u,r)b_{2}]\\}$

$=[T(u,r,b_{1}b_{2}),T(v,s,b_{3}b_{4})]+(u,s)T(v,s,b_{1}b_{2}b_{3},b_{4})-(u,s)T(v,s,b_{3},b_{1}b_{2}b_{4})$

$-(v,r)T(u,r,b_{1}b_{3}b_{4},b_{2})+(v,r)T(u,r,b_{1},b_{2}b_{3}b_{4})$

$=T(w,r+s,b_{1}b_{3},b_{2}b_{4})-(u,s)T(v,s,b_{3},b_{1}b_{2}b_{4})+(v,r)T(u,r,b_{1},b_{2}b_{3}b_{4})$
(using Lemma 3.1(1))

$=T_{1}(w,r+s,b_{1}b_{3},b_{2}b_{4})-(u,s)T_{1}(v,s,b_{3},b_{1}b_{2}b_{4})+(v,r)T_{1}(u,r,b_{1},b_{2}b_{3}b_{4}).$

2\. $[I(u,r,b_{1},b_{2}),I(v,s,b_{3},b_{4})]$

$=[\psi(b_{1})D(u,r)b_{2}-D(u,0)b_{1}b_{2},\psi(b_{3})D(v,0)b_{4}-D(v,0)b_{3}b_{4}]$

$=\psi(b_{1}b_{3})[D(u,r)b_{2},D(v,s)b_{4}]+(v,r)\psi(b_{1})D(u,r)b_{2}b_{3}b_{4}-(u,s)\psi(b_{3})D(v,s)b_{1}b_{2}b_{4}$

$=\psi(b_{1}b_{3})D(w,r+s)b_{2}b_{4}+(v,r)\psi(b_{1})D(u,r)b_{2}b_{3}b_{4}-(u,s)\psi(b_{3})D(v,s)b_{1}b_{2}b_{4}$

$=I(w,r+s,b_{1}b_{3},b_{2}b_{4})+(v,r)I(u,r,b_{1},b_{2}b_{3}b_{4})-(u,s)I(v,s,b_{3},b_{1}b_{2}b_{4})$.

3\. Follows from 1 and 2.

$4$. Check that $D(w,0)$ commutes with $D_{1}$ for all $w\in\mathbb{C}^{n}$.
Therefore $V_{\alpha}$ is a $D_{1}$-module. By Lemma 3.1, we have $V_{\alpha}$
is an irreducible $D$-module. Let $\phi:D\to End(V_{\alpha})$ be the
representation. Then by Lemma 3.2, we have $\phi(D)$ is reductive and
$\phi(D)=Z(\phi(D))\oplus\phi([D,D]).$ Since $V_{\alpha}$ is irreducible,
$Z(\phi(D))$ acts as scalar on $V_{\alpha}$. Consequently $V_{\alpha}$ is an
irreducible module for $[D,D]\subset D_{1}.$

5\. Since the action of $A\otimes B$ is associative, so $t^{s}b.v=t^{s}\otimes
1.1\otimes b.v=\psi(b)t^{s}v$, for all $s\in\mathbb{Z}^{n},b\in B$ and $v\in
V.$

Now, $(\psi(b_{1})D(u,r)b_{2}-D(u,0)b_{1}b_{2}).(t^{s}v-v)$

$=\psi(b_{1})[t^{s}(D(u,r)b_{2}.v+(u,s)t^{s+r}b_{2}.v-D(u,r)b_{2}.v]-[t^{s}D(u,0)b_{1}b_{2}.v+(u.s)t^{s}b_{1}b_{2}.v-D(u,0)b_{1}b_{2}.v]$

$=\psi(b_{1})[t^{s}(D(u,r)b_{2}.v-D(u,r)b_{2}.v]+(u,s)\psi(b_{1})[t^{s+r}b_{2}.v-t^{s}b_{2}.v]-[t^{s}D(u,0)b_{1}b_{2}.v-D(u,0)b_{1}b_{2}.v]\in
W$.

6\. Define $\phi:V\to V/W$ by $\phi(v)=\overline{v}$ for all $v\in V$. Now
restrict $\phi$ to $V_{\alpha}$ and denote by $\phi|_{V_{\alpha}}$. Note that
$\overline{t^{s}v}=\overline{v}$ for all $s\in\mathbb{Z}^{n},v\in V.$

Claim:
$\phi(T_{1}(u,r,b_{1},b_{2}).v)=\pi(T_{1}(u,r,b_{1},b_{2})).\overline{v},$ for
all $v\in V_{\alpha},u\in\mathbb{C}^{n},r\in\mathbb{Z}^{n},b_{1},b_{2}\in B$.
Now, $\phi(T_{1}(u,r,b_{1},b_{2}).v)$

$=\overline{t^{-r}b_{1}D(u,r)b_{2}.v-D(u,0)b_{1}b_{2}.v}$

$=\overline{\psi(b_{1})t^{-r}D(u,r)b_{2}.v}-\overline{D(u,0)b_{1}b_{2}.v}$

$=\overline{\psi(b_{1})D(u,r)b_{2}.v}-\overline{D(u,0)b_{1}b_{2}.v}$ (Since
$\overline{t^{s}v}=\overline{v}$)

$=\overline{\psi(b_{1})D(u,r)b_{2}.v-D(u,0)b_{1}b_{2}.v}$

$=\pi(T_{1}(u,r,b_{1},b_{2})).\overline{v}$.

Thus $\phi|_{V_{\alpha}}$ is a $D_{1}$-module map. Now $\phi|_{V_{\alpha}}$ is
non-zero and $V_{\alpha}$ is an irreducible $D_{1}$-module, so
$\phi|_{V_{\alpha}}$ is injective. $\phi|_{V_{\alpha}}$ is surjective because
of equal dimension and hence $V_{\alpha}\simeq V/W$ as $D_{1}$-module. ∎

Let us consider the space

$\widetilde{D}=span\\{\psi(b_{1})D(u,r)b_{2}-D(u,r)b_{1}b_{2}:r\in\mathbb{Z}^{n},u\in\mathbb{C}^{n},b_{1},b_{2}\in
B\\}.$

Observe that $\widetilde{D}\subset D_{2}$ and $\widetilde{D}=DerA\otimes M$,
where $M$ is the maximal ideal $ker\psi$. Therefore $\widetilde{D}$ is an
ideal in $DerA\otimes B$ and hence an ideal in $D_{2}$.

######  Lemma 3.3.

$D(u,0)b$ acts as scalar on $V_{\alpha}$ for all $u\in\mathbb{C}^{n},b\in B.$

###### Proof.

By Proposition 3.1, $D_{2}$ acts on $V/W$ irreducibly. Let $\phi:D_{2}\to
End(V/W)$ be the finite dimensional irreducible representation and
$\rho=\phi|_{\widetilde{D}}.$ We prove this lemma in two cases.
Case 1: Let $M^{k}=0$ for some $k\in\mathbb{N}$.
We have $[DerA\otimes M^{i},DerA\otimes M^{j}]=DerA\otimes M^{i+j}$ for all
$i,j\in\mathbb{N}$, since $DerA$ is a simple Lie algebra. This implies that
$DerA\otimes M$ is a solvable ideal in $D_{2}$, thus $\phi(\widetilde{D})$ is
a solvable ideal in the Lie algebra $\phi(D_{2})$. By Lemma 3.2, $\phi(D_{2})$
is a reductive Lie algebra and hence $\phi(\widetilde{D})$ acts as a scalar on
$V/W$. In particular, $\psi(b)D(u,0)-D(u,0)b$ acts as scalar on $V/W.$ Now by
Proposition 3.1(6,3), we have $T_{1}(u,0,b,1)$ acts as scalar on $V_{\alpha}$.
This gives us $D(u,0)b$ acts as scalar on $V_{\alpha}$.

Case 2: Let $M^{k}\neq 0$ for all $k\in\mathbb{N}.$ In this case, it is
sufficient to show that $DerA\otimes M^{k}$ acts trivially on $V/W$ for some
$k\in\mathbb{N}$. Because $[\frac{DerA\otimes
M^{i}}{ker\phi},\frac{DerA\otimes M^{j}}{ker\phi}]=\frac{DerA\otimes
M^{i+j}}{ker\phi}$ for all $i,j\in\mathbb{N}$ implies that $\frac{DerA\otimes
M^{2}}{ker\phi}$ is a solvable ideal contained in the semi-simple Lie algebra
$[\frac{D_{2}}{ker\phi},\frac{D_{2}}{ker\phi}].$ Thus $DerA\otimes M^{2}$ acts
trivially on $V/W.$ Therefore $\phi(DerA\otimes M)$ is an abelian ideal in
$\phi(D_{2})$. Now the result follows by same argument as case 1.

Claim: $DerA\otimes M^{2N+6}.V/W=0$, where $N=$ dim $(DerA\otimes M/J)$,
$J=ker\rho$.

Sub-claim 1: $D(v,0)\otimes M^{2N+5}\subset J$ for all $v\in\mathbb{C}^{n},$
for all $n\geq 2.$

Let $b=b_{1}b_{2}..,b_{2N+5}$ be a non-zero element of $M^{2N+5}$. Consider
the set of vectors

$\\{D(u_{i},r_{i})b_{1}+J:u_{i}\in\mathbb{C}^{n}-\\{0\\},r_{i}\in\mathbb{Z}^{n}-\\{0\\},r_{i}\neq
r_{j},\forall i\neq j,1\leq i\leq N+1\\}.$

Since dim $(DerA\otimes M/ker\rho)=N,$ there exists a non-zero vector
$X=\displaystyle{\sum_{i=1}^{k}D(u_{i},r_{i})b_{1}}\in J$ for some $k\leq
N+1.$
Now,

$[X,D(v,-r_{1})b_{2}]=D(w_{1},0)b_{1}b_{2}+\displaystyle{\sum_{i=2}^{k}}D(w_{i},r_{i}-r_{1})b_{1}b_{2},$

where $w_{1}=-(u_{1},r_{1})v-(v,r_{1})u_{1}$ and
$w_{i}=-(u_{i},r_{1})v-(v,r_{i})u_{i}$ for all $i\neq 1.$

Thus if $(u_{1},r_{1})=0$ choose $v$ such that $(v,r_{1})\neq 0$ and if
$(u_{1},r_{1})\neq 0$ choose $v$ such that $(v,r_{1})=0$, in both cases
$w_{1}\neq 0.$ If all $w_{i}=0$ for all $i\neq 1,$ then we have
$D(w_{1},0)b_{1}b_{2}\in J.$ If $w_{i}\neq 0$ for some $i\neq 1$, say
$w_{j}\neq 0$. Then choosing any $v^{\prime}$ such that
$(v^{\prime},r_{j}-r_{1})\neq 0$ and considering
$[[X,D(v,-r_{1})b_{2}],D(v^{\prime},0)b_{3}]$ we get the sum of at most $k-1$
terms. Continuing the above process we can conclude that
$D(u,0)b_{1}b_{2}..b_{2k+1}\in J$ for some non-zero $u\in\mathbb{C}^{n}.$

For any non-zero $v\in\mathbb{C}^{n}$ choose $s\in\mathbb{Z}^{n}$ such that
$(u,s)\neq 0$. Then we have

$[D(v,s)b_{2k+2},D(u,0)b_{1}b_{2}..b_{2k+1}]=(u,s)D(v,s)b_{1}b_{2}...b_{2k+2}\in
J.$

Again,

$[D(w,-s)b_{2k+3},D(v,s)b_{1}b_{2}...b_{2k+2}]=D((w,s)v+(v,s)w,0)b_{1}...b_{2k+3}\in
J.$

Now, if $(v,s)=0$ choose $(w,s)\neq 0$ and if $(v,s)\neq 0$ choose $w=v$, then
with this choice we have for all $v(\neq 0)\in\mathbb{C}^{n}$,
$D(v,0)b_{1}b_{2}....b_{2k+3}\in J$. Since $k\leq N+1$ we have $D(v,0)\otimes
M^{2N+5}\subset J$ for all $v\in\mathbb{C}^{n}.$

Sub-claim 2: $d_{0}\otimes M^{2N+5}\subset J$ for $n=1.$

Let $b=b_{1}b_{2}...b_{2N+5}\in M^{2N+5}$. Let us consider the set of vectors

$\\{d_{r_{i}}b_{1}+J:r_{i}\in\mathbb{N},r_{i}>r_{j}$ for $i>j$ and $2r_{i}\neq
r_{j}+r_{k}$ for distinct $i,j,k,1\leq i\leq N+1\\}$

( In particular $2^{i},i\in\mathbb{N}$ can be chosen as $r_{i})$. Since
dim$(DerA\otimes M)/J=N$, there exists a non-zero vector
$X=\displaystyle{\sum_{i=1}^{k}\lambda_{i}d_{r_{i}}b_{1}}\in J$ with some
$\lambda_{i}(\neq 0)\in\mathbb{C}$, $k\leq N+1$. Let us assume
$\lambda_{1}\neq 0$ and consider

$[d_{-r_{1}}b_{2},X]=2r_{1}\lambda_{1}d_{0}b_{1}b_{2}+\displaystyle{\sum_{i=2}^{k}\lambda_{i}(r_{i}+r_{1})d_{r_{i}-r_{1}}b_{1}b_{2}}\in
J.$

If $\lambda_{i}=0$ for all $i\geq 2$ we are done. If not, let $\lambda_{2}\neq
0$ and consider

$[d_{r_{1}-r_{2}}b_{4},[d_{0}b_{3},[d_{-r_{1}}b_{2},X]]]=2(r_{2}-r_{1})^{2}(r_{2}+r_{1})\lambda_{2}d_{0}b_{1}b_{2}b_{3}b_{4}$
$+\displaystyle{\sum_{i=3}^{k}\lambda_{i}(r_{i}^{2}-r_{1}^{2})(r_{i}+r_{2}-2r_{1})d_{r_{i}-r_{2}}b_{1}b_{2}b_{3}b_{4}}\in
J.$

Continuing this process we can conclude that $d_{0}\otimes M^{2N+5}\subset J$,
since $k\leq N+1.$ Therefore sub-claim 1 is true for all $n\geq 1.$

Now for any $b\in M$, $b^{\prime}\in M^{2N+5}$, $u(\neq 0)\in\mathbb{C}^{n}$
and $s(\neq 0)\in\mathbb{Z}^{n}$, choose $v$ such that $(v,s)\neq 0$ and then
consider $[D(u,s)b,D(v,0)b^{\prime}]$ which gives $D(u,s)bb^{\prime}\in J.$
Hence the claim follows. ∎

Let $T(u,r)b=T(u,r,1,b)$ for all $u,r,b$ and consider $T_{B}$ be the space
spanned by $T(u,r)b.$ It is easy to see that $T_{B}$ is a sub-algebra of
$D_{1}.$ In fact one can check that

$[T(v,s)b,T(u,r)b^{\prime}]=T(w,r+s)bb^{\prime}+(u,s)T(v,s)bb^{\prime}-(v,r)T(u,r)bb^{\prime},$

where $w=(v,r)u-(u,s)v.$

######  Theorem 3.1.

Let $V$ be an irreducible weight module for $\tau$ with finite dimensional
weight spaces. Also let the action of $A\otimes B$ on $V$ be associative and
$1.v=v$ for all $v\in V.$ Then $V$ is an irreducible module for $A\rtimes
DerA.$

###### Proof.

By Lemma 3.3, $D(u,0)b$ acts on $V_{\alpha}$ as a scalar, say $\lambda(u,b)$
for all $u,b$. Then we have,

(3.3) $\displaystyle D(u,0)b(v_{\alpha}\otimes
t^{m})=(\lambda(u,b)+(u,m)\psi(b))v_{\alpha}\otimes t^{m}$ (3.4)
$\displaystyle t^{m}b.(v_{\alpha}\otimes t^{k})=\psi(b)v_{\alpha}\otimes
t^{m+k}=\psi(b)t^{m}.(v_{\alpha}\otimes t^{k}).$

for all $m,k\in\mathbb{Z}^{n}$ and $v_{\alpha}\in V_{\alpha}.$
Consider the actions of $T(u,r)b$ on $V$, for all $r\neq 0,u,b.$

$T(u,r)b.v\otimes t^{k}=(t^{-r}D(u,r)b-D(u,0)b)v\otimes t^{k}.$

Now using (3.3) and the fact that $t^{r}$ commutes with $T(u,r)b$ we have,

(3.5) $\displaystyle D(u,r)b.v\otimes t^{k}=T(u,r)b.v\otimes
t^{k+r}+(\lambda(u,b)+(u,k)\psi(b))v\otimes t^{k+r}$

for all $v\in V_{\alpha}$ and $k\in\mathbb{Z}^{n}$.
In particular for $b=1$, (3.5) gives

(3.6) $\displaystyle D(u,r).v\otimes t^{k}=T(u,r).v\otimes
t^{k+r}+(u,k+\alpha)\psi(b)v\otimes t^{k+r},$

since $\lambda(u,1)=(u,\alpha)$.
For any $r\neq 0$ choose $v$ such that $(v,r)\neq 0$ and consider

(3.7) $\displaystyle[D(v,0)b,D(u,r)]=(v,r)D(u,r)b.$

Then using the actions of (3.3), (3.5) and (3.6) on the both sides of (3.7) we
have the following action of $T(u,r)b$ on $V$,

(3.8) $\displaystyle T(u,r)b=\psi(b)T(u,r)+\psi(b)(u,\alpha)-\lambda(u,b)$

for all $r\neq 0.$
Now using (3.5), (3.6) and (3.8) it is immediate to check that

(3.9) $\displaystyle D(u,r)b=\psi(b)D(u,r)$

on $V$ for all $r\neq 0.$
For all non-zero $u,v,r,s$ and $b$ we have

(3.10) $\displaystyle[D(v,s)b,D(u,r)]w_{0}\otimes
t^{k}=D((v,r)u-(u,s)v,r+s)bw_{0}\otimes t^{k},$

for all $w_{0}\in V_{\alpha}$ and $k\in\mathbb{Z}^{n}$.
Using (3.9) in the left hand side of (3.10) we have

(3.11) $\displaystyle\psi(b)D((v,r)u-(u,s)v,r+s)w_{0}\otimes
t^{k}=D((v,r)u-(u,s)v,r+s)b.w_{0}\otimes t^{k}.$

In (3.11), put $u=v$ and find $r$ such that $(u,r)\neq 0$. Then put $s=-r$ and
conclude that $D(u,0)b=\psi(b)D(u,0)$ on $V$. Hence
$\lambda(u,b)=\psi(b)(u,\alpha)$ and $D(u,r)b=\psi(b)D(u,r)$ on $V$ for all
$u,r$. This completes the proof.

∎

######  Remark 3.1.

Theorem 3.1 shows that our module for $\tau$ is actually an evaluation module
at a single point. For more details on evaluation module see [2, 5, 19] and
references therein.

## 4\.

In [1], Theorem 2.1 was proved for $A=\mathbb{C}[t_{1}^{\pm 1},t_{2}^{\pm
1},....,t_{n}^{\pm 1}]$ with the assumption $n\geq 2$. In this section we show
that the Theorem 2.1 holds for $A=\mathbb{C}[t^{\pm 1}]$ too.

Let $V$ be a module for $A\rtimes DerA$ satisfying all properties of Theorem
2.1. Let us assume that there exists $\lambda\in H^{*}$ such that
$V_{\lambda}\neq 0$. Then
$V=\displaystyle{\bigoplus_{m\in\mathbb{Z}}}V_{m+\alpha}$,
where $V_{m+\alpha}=\\{v\in V:d_{0}v=(m+\alpha)v\\}$,
$\alpha=\lambda(d)\in\mathbb{C}.$ Note that $V_{\lambda}=V_{\alpha}$ and
$t^{k}$ is injective, hence $t^{k}V_{\alpha}=V_{k}$ for all $k\in\mathbb{Z}$.
Therefore we can identify $V$ with $V_{\alpha}\otimes A$ as vector space.

Let $I(r)=(t^{r}-1)d$, for all $r\in\mathbb{Z}$ and denote
$T^{\prime}=span\\{I(r):r\in\mathbb{Z}\\}$. Let  m be the maximal ideal in $A$
generated by $(t-1)$. It is clear that $\textbf{m}d=T^{\prime}$.

######  Lemma 4.1.

$[I(r),I(s)]=(s-r)I(r+s)+rI(r)-sI(s)$, for all $s,r\in\mathbb{Z}.$ In
particular, $T^{\prime}$ is a sub-algebra of $A\rtimes DerA$.

Let $W=span\\{t^{r}.v-v:v\in V_{\alpha}$ and $r\in\mathbb{Z}\\}$. It is easy
to see that $W$ is a $T^{\prime}$ sub-module of $V$ and hence
$\overline{V}=V/W$ is a $T^{\prime}$ module. Now we define an action of
$A\rtimes DerA$ on $\overline{V}\otimes A$ by

(4.1) $\displaystyle d_{r}.v\otimes t^{s}=(I(r).v)\otimes
t^{r+s}+(s+\alpha)v\otimes t^{r+s},$ (4.2) $\displaystyle t^{r}.v\otimes
t^{s}=v\otimes t^{r+s},$

for all $r,s\in\mathbb{Z}$, $v\in\overline{V}$ and for some
$\alpha\in\mathbb{C}$. It is easy to see that this action define a module
structure on $\overline{V}\otimes A.$ We will denote this module by
$\overline{V}^{\alpha}\otimes A$.

######  Proposition 4.1.

$V\simeq\overline{V}^{\alpha}\otimes A$ as $A\rtimes DerA$-module and
$\overline{V}^{\alpha}$ is a finite dimensional irreducible
$T^{\prime}$-module.

###### Proof.

Define $\phi:V\to\overline{V}^{\alpha}\otimes A$ by
$\phi(t^{m}v)=\overline{t^{m}v}\otimes t^{m}$ for all $v\in V_{\alpha}$ and
$m\in\mathbb{Z}$. Then $\phi$ is a $A\rtimes DerA$ module homomorphism. $\phi$
is clearly surjective. Since $\phi(V_{\alpha})=\overline{V}^{\alpha}$, hence
$\phi$ is non-zero. So by irreducibility of $V$ we have $\phi$ is injective.
Suppose $\overline{W}$ be a $T^{\prime}$ sub-module of
$\overline{V}^{\alpha}$, then $\overline{W}\otimes A$ becomes a $A\rtimes
DerA$ sub-module of $\overline{V}^{\alpha}\otimes A$, a contradiction. ∎

######  Lemma 4.2.

1.

$\left[(t-1)^{k}d_{i},(t-1)^{l}d_{j}\right]=(l-k+j-i)(t-1)^{k+l}d_{i+j}+(l-k)(t-1)^{k+l-1}d_{i+j},$

for all $k,l\in\mathbb{N},i,j\in\mathbb{Z}.$

2\. For any polynomial $f(\lambda)$,

$f((1-t^{-1})d_{0}-k).(t-1)^{k+1}d_{-1}.v=(t-1)^{k+1}d_{-1}.f((1-t^{-1})d_{0}).v,$

for all $v\in V$ and $k\in\mathbb{N}$.

###### Proof.

1\. Follows from Lemma 2.4 of [4].

2\. Note that, $((1-t^{-1})d_{0}-k).(t-1)^{k+1}d_{-1}.v$

$=\left[(1-t^{-1})d_{0},(t-1)^{k+1}d_{-1}\right].v+(t-1)^{k+1}d_{-1}.(1-t^{-1})d_{0}.v-k(t-1)^{k+1}d_{-1}.v$

$=\left[(t-1)d_{-1},(t-1)^{k+1}d_{-1}\right].v+(t-1)^{k+1}d_{-1}.(1-t^{-1})d_{0}.v-k(t-1)^{k+1}d_{-1}.v$

$=k(t-1)^{k+2}d_{-2}.v+k(t-1)^{k+1}d_{-2}.v+(t-1)^{k+1}d_{-1}.(1-t^{-1})d_{0}.v-k(t-1)^{k+1}d_{-1}.v$

$=k(t-1)^{k+1}[(t-1)t^{-1}+t^{-1}]d_{-1}.v+(t-1)^{k+1}d_{-1}.(1-t^{-1})d_{0}.v-k(t-1)^{k+1}d_{-1}.v$

$=(t-1)^{k+1}d_{-1}.(1-t^{-1})d_{0}.v$

Now assume that,

(4.3)
$\displaystyle((1-t^{-1})d_{0}-k)^{n}.(t-1)^{k+1}d_{-1}.v=(t-1)^{k+1}d_{-1}.((1-t^{-1})d_{0})^{n}.v,$

for some $n\in\mathbb{N}.$ Then we have,

$((1-t^{-1})d_{0}-k)^{n+1}.(t-1)^{k+1}d_{-1}.v$

$=((1-t^{-1})d_{0}-k).[(t-1)^{k+1}d_{-1}.((1-t^{-1})d_{0})^{n}.v],$ by (4.3)

$=(t-1)^{k+1}d_{-1}.(1-t^{-1})d_{0}.((1-t^{-1})d_{0})^{n}.v$

$=(t-1)^{k+1}d_{-1}.((1-t^{-1})d_{0})^{n+1}.v$

Hence by induction we have the result for any polynomial $f(\lambda)$. ∎

######  Proposition 4.2.

1\. $\textbf{m}^{2}d\overline{V}^{\alpha}=0$.
2\. $V$ is an irreducible module for $\mathbb{C}$ and satisfy the actions
given in Theorem 2.1.

###### Proof.

1\. It is sufficient to prove that $\textbf{m}^{k}d\overline{V}^{\alpha}=0$
for some $k\in\mathbb{N}.$ Because then using same argument as Lemma 3.3 (Case
2) we have the result. Consider the operator
$(1-t^{-1})d_{0}:\overline{V}^{\alpha}\to\overline{V}^{\alpha}$. Let
$f(\lambda)$ be the characteristic polynomial of this operator. Since we are
over the field $\mathbb{C}$, there exists a $k\in\mathbb{N}$ such that
$gcd(f(\lambda),f(\lambda-l))=1$ for all $l>k$. Now using Lemma 4.2(2), we get
$(t-1)^{l+1}d_{-1}.v=0$ for all $l>k$ and $v\in\overline{V}^{\alpha}.$ By
Lemma 4.2(1), we have the relation

$[(t-1)^{l+2}d_{-1},(t-1)d_{i}]-[(t-1)^{l+1}d_{-1},(t-1)^{2}d_{i}]=-2(t-1)^{l+2}d_{i}$

for all $i\in\mathbb{Z},l\in\mathbb{N}.$ This implies that
$(t-1)^{l+2}t^{i}d.v=0$ for all $v\in\overline{V}^{\alpha}$, $i\in\mathbb{Z},$
$l>k.$ Therefore there exists $k_{0}\in\mathbb{N}$ such that
$\textbf{m}^{k_{0}}d\overline{V}^{\alpha}=0.$

2\. Let us define a map $\phi:{\textbf{m}d}/{\textbf{m}^{2}d}\to\mathbb{C}$ by
$\phi((t-1)f(t)d)=f(1)$. It is easy to see that $\phi$ is a Lie algebra
isomorphism and image of $I(r)$ under this map is $r$ for all
$r\in\mathbb{Z}.$ Now by Proposition 4.1 and the action defined by (4.1), we
have

$d_{r}.v\otimes t^{s}=r(1.v)\otimes t^{r+s}+(s+\alpha)v\otimes t^{r+s}$

∎

######  Theorem 4.1.

Let $V$ be an irreducible weight module for $\tau$ with finite dimensional
weight spaces. Also let the action of $A\otimes B$ on $V$ be associative and
$1.v=v$ for all $v\in V.$ Then $V\simeq F^{\alpha}(\mu,c)$ for some dominant
integral weight $\mu$, $c\in\mathbb{C}$ and $\alpha\in\mathbb{C}^{n}$. Actions
of the elements of $\tau$ on $F^{\alpha}(\mu,c)$ are given by

$D(u,r)bv\otimes t^{m}=\psi(b)\\{(u,m+\alpha)v\otimes
t^{m+r}+\displaystyle{\sum_{i,j}(u_{i}r_{j}E_{ji}v)\otimes t^{m+r}}\\},$

$t^{r}b.v\otimes t^{m}=\psi(b)v\otimes t^{r+m},$

for all $u\in\mathbb{C}^{n},r\in\mathbb{Z}^{n},b\in B$ and for some algebra
homomorphism $\psi:B\to\mathbb{C}.$

###### Proof.

Follows from Theorem 2.1, Theorem 3.1, Proposition 4.1 and Proposition 4.2. ∎

Acknowledgments: The first author would like to thank Prof. Rencai Lu for some
helpful discussion regarding one variable case of $(A,DerA)$.

## References

  * [1] S. Esawara Rao, Partial classification of modules for Lie-algebra of diffeomorphisms of $d-$dimensional torus.
  * [2] Xiangqian Guo, Rencai Lu, Kaiming Zhao, Simple Harish-chandra modules and Verma modules over loop-virasoro algebras, Forum Math 23(5), 1029-1052, 2011
  * [3] J. E. Humphreys, Introduction to Lie algebras and Representation theory. Second printing, revised. Graduate Texts in Mathematics, 9. Springer-Verlag, New York-Berlin, 1978
  * [4] Yan-an Cai, Rencai Lu, Yan Wang; Weight modules for map (super) algebra related to Virasoro algebra ; arxiv: 2007.02582v1
  * [5] A savage, Classification of irreducible quasifinite modules over map Virasoro algebras, Transformation groups 17(2), 547-570, 2012
  * [6] S. Eswara Rao, Modules for Loop Affine- Virasoro algebras, Journal of Algebra and Its Applications, Vol.20, No.4,(2021),2150055(13 pages)
  * [7] E. Neher , A Savage., P. Senesi, Irreducible finite dimensional representations of equivariant map algebras, Trans. AMS, 364(5) 2617-2646(2012).
  * [8] V.Chari, G. Fourier, T. Khandai, A Categorical approch to Weyl modules, Transfromation Groups,15(3), 517-549 (2010).
  * [9] S. Eswara Rao, Sachin S. Sharma, Sudipta Mukherjee, Integrable modules for the loop Affine-Virasoro algebras, arXiv:2005.08175v1 [math.RT] 17 May 2020
  * [10] Yuly Billig, Vyacheslav Futorny, Classification of irreducible representations of Lie algebra of vector fields on a torus. J. Renie Angew. Math 720(2016) 199-216.
  * [11] Xiangqian Guo and Kaiming Zhao, Irrefucible weight modules over Witt algebras, Proceedings of ams, vol 139, no 7, july 2011, 2367-2373.
  * [12] Ramos, E., Sah, C. H., and Shrock, R. E., Algebra of diffeomorphisms of $N$-torus, J. Math phys, 31, 1805-1816,1990.
  * [13] Larsson, T. A., Conformal fields: A class of representations of Vect (N), Int. J. Mod. Phys. A 7, 6493-6508(1992).
  * [14] P Chakraborty, P Batra, Classification of irreducible integrable representations for loop toroidal Lie algebras, arXiv:2007.06415
  * [15] O. Mathieu, Classification of Harish-Chandra modules over the Virasoro Algebras, Invent. Math. 107, 225-234, 1992
  * [16] Mazorchuk, Volodymyr; Zhao, Kaiming Classification of simple weight Virasoro modules with a finite-dimensional weight space. J. Algebra, 307, 2007, 209-214
  * [17] Vyjayanthi Chari and Andrew Pressley. Unitary representations of the Virasoro algebra and a conjecture of Kac. Compositio Math., 67(3):315-342, 1988
  * [18] S.Eswara Rao, Irreducible representations for Toroidal Lie algebras, Journal of Pure and Applied Algebra, 202(1-3), 2005, 102-117
  * [19] D. Britten, M. Lau, F. Lemire, Weight modules for current algebras, J. Algebra, 440(2015), 245-263.


Further author information: (Send correspondence to J.T.)
J.T.: E-mail<EMAIL_ADDRESS>
# A Near-Infrared Pyramid Wavefront Sensor for the MMT

Jacob Taylor  David A. Dunlap Department of Astronomy & Astrophysics,
University of Toronto  Dunlap Institute for Astronomy & Astrophysics  Suresh
Sivanandam  David A. Dunlap Department of Astronomy & Astrophysics, University
of Toronto  Dunlap Institute for Astronomy & Astrophysics  Narsireddy Anugu
Center for High Angular Resolution Astronomy  Adam Butko  David A. Dunlap
Department of Astronomy & Astrophysics, University of Toronto  Dunlap
Institute for Astronomy & Astrophysics  Shaojie Chen  Dunlap Institute for
Astronomy & Astrophysics  Olivier Durney  Steward Observatory, University of
Arizona  Tim Hardy  NRC Herzberg Astronomy and Astrophysics Research Centre
Masen Lamb  Dunlap Institute for Astronomy & Astrophysics  Manny Montoya
Steward Observatory, University of Arizona  Katie Morzinski  Steward
Observatory, University of Arizona  Robin Swanson  Dunlap Institute for
Astronomy & Astrophysics  Department of Computer Science, University of
Toronto

###### Abstract

The MMTO Adaptive optics exoPlanet characterization System (MAPS) is an
ongoing upgrade to the 6.5-meter MMT Observatory on Mount Hopkins in Arizona.
MAPS includes an upgraded adaptive secondary mirror (ASM), upgrades to the
ARIES spectrograph, and a new AO system containing both an optical and near-
infrared (NIR; 0.9-1.8 µm) pyramid wavefront sensor (PyWFS). The NIR PyWFS
will utilize an IR-optimized double pyramid coupled with a SAPHIRA detector: a
low-read noise electron Avalanche Photodiode (eAPD) array. This NIR PyWFS will
improve MAPS’s sky coverage by an order of magnitude by allowing redder guide
stars (e.g. K & M-dwarfs or highly obscured stars in the Galactic plane) to be
used. To date, the custom designed cryogenic SAPHIRA camera has been fully
characterized and can reach sub-electron read noise at high avalanche gain. In
order to test the performance of the camera in a closed-loop environment prior
to delivery to the observatory, an AO testbed was designed and constructed. In
addition to testing the SAPHIRA’s performance, the testbed will be used to
test and further develop the proposed on-sky calibration procedure for MMTO’s
ASM. We will report on the anticipated performance improvements from our NIR
PyWFS, the SAPHIRA’s closed-loop performance on our testbed, and the status of
our ASM calibration procedure.

###### keywords:

Adaptive optics, pyramid wavefront sensor, near infrared, MMT Observatory,
MAPS, SAPHIRA, avalanche photodiode array

##  1 INTRODUCTION

As pristine starlight passes through the atmosphere, its wavefront becomes
undulated and distorted which ultimately limits the resolution of large
ground-based telescopes. Adaptive optics (AO) is a method of improving the
quality of ground-based astronomical observations by rapidly measuring and
correcting the wavefront error introduced by turbulence in the Earth’s
atmosphere. In recent decades, AO systems have become more sensitive, robust,
and prevalent; moreover, the next-generation of extremely large telescopes
will be reliant on AO to achieve many of their key scientific goals. While
there is substantial variety across AO systems, they all have three key
components: a wavefront sensor (WFS), a deformable mirror (DM), and a real-
time controller (RTC). These components typically work together in a ‘closed-
loop’ configuration to compute and correct atmospheric distortions in real-
time.thesis

The MMT observatory, situated on Mount Hopkins in Arizona, hosts a 6.5m
primary mirror and has had an operational AO system since 2002 [1]; however,
the system has reached the end of its lifetime and a new system has been
commissioned. The upgraded AO system is part of the MMTO Adaptive optics
exoPlanet characterization System (MAPS) which consists of a new AO system, a
refurbished high-resolution IR spectrograph (ARIES, 1-5$\mu$m) and a new AO-
optimized upgrade to the MMT-sensitive polarimeter (MMT-Pol) [2]. The
project’s main science goal is to characterize exoplanet atmospheres through
the marriage of a state-of-the-art AO system and broad wavelength, high-
resolution IR spectroscopy (HRS). By cross-correlating spectra with molecular
templates, MAPS will be able to characterize atmospheric properties of
exoplanets. Please see proceedings by Montoya, M et al. at this conference for
more details on the project as a whole. MMT’s new AO system comes with some
significant upgrades over its predecessor. All 336 voice coil actuators of the
MMT adaptive secondary mirror (ASM) have been replaced with a new design of
actuators and control electronics, and the legacy 12x12 Shack-Hartmann
wavefront sensor has been replaced by two high-sensitivity pyramid wavefront
sensors (one optimized for visible wavelengths and the other infrared). The
near-infrared pyramid wavefront sensor (NIR PyWFS) is of particular importance
for MAPS since it will drastically increase the instrument’s sky coverage.

The atmosphere’s rapid time-evolution, combined with the faintness of most
stars, limits natural guide-star (NGS) AO systems to the fraction of the sky
containing sufficiently bright sources whose wavefront can be easily measured.
For this reason, while NGS adaptive optics continues to be a powerful tool for
astronomers, its applicability is not universal; however, recently there have
been some key advancements in infrared detectors which have the potential to
significantly increase NGS AO sky-coverage. Astronomers now have access to
high-speed IR electron avalanche photodiode (eAPD) arrays that can reach sub-
electron read noise (e.g. Leonardo’s SAPHIRA [3]). eAPD arrays are able to
reduce the CMOS read noise barrier of near infrared sensors by applying a
reverse-bias voltage across each pixel’s P-N junction. This voltage
accelerates photoelectrons along the electric field, knocking off additional
electrons. By increasing the number of electrons prior to readout, the eAPD
arrays are able to achieve sub-electron effective read noise at millisecond
readout rates. These detectors have paved the way for fast and efficient
wavefront sensing in the near-infrared which, in turn, significantly improves
sky coverage by utilizing guide stars which are normally obscured behind
interstellar dust as well as an abundance of red stars (type K and M) whose
brightness peaks in the infrared [4]. The MAPS NIR PyWFS will operate
predominately in the J-band, operate at speeds up to 1Khz, and is anticipated
to use guide stars down to 18th magnitude (R-band). This paper will provide an
overview of the MAPS NIR PyWFS and an update on it’s current commissioning
status.

##  2 INSTRUMENTATION

The MAPS AO system consists of three main components: a 336 voice coil
actuator adaptive secondary mirror; a visible pyramid wavefront sensor ($\sim$
0.6-0.8 um); and a near infrared pyramid wavefront sensor ($\sim$ 0.9-1.8 um).
Both WFSs are housed together in the MAPS WFS unit with a flip mirror that
will select which will be active. Additionally, both WFSs will be modulated
using the same Physik Instrumente (PI) S-330 Tip Tilt Platform and E-727
Digital Multi-Channel Piezo Controller. Figure 1 shows a diagram of the MAPS
WFS unit. See Montoya, M et al. at this conference for updates on the MAPS
adaptive secondary and MAPS visible wavefront sensor.

---
Figure 1: A labelled diagram of the MAPS WFS optical design. The design is
divided into 5 channels: periscope channel (blue); baseline channel (green);
acquisition channel (purple); visible PyWFS channel (red); NIR PyWFS channel
(yellow). The majority of the MAPS WFS unit has been fabricated and installed
and the system is currently undergoing optical alignment.

###  2.1 SAPHIRA DETECTOR AND CRYOSTAT

Butko et al. 2018[5] reported on the specifics of our SAPHIRA control system
implementation and cryostat chamber which we summarize here. The chosen
detector for the MAPS NIR PyWFS is Leonardo’s SAPHIRA Mark 13, a 320 x 256
pixel format HgCdTe CMOS detector integrated with electron avalanche
photodiodes (eAPD). The SAPHIRA is controlled by a custom Astronomical
Research Camera (ARC) controller. The ARC controller utilizes two ARC-46 IR
video processing board links to read out the detector in 16 channels (an
additional one was installed since Butko et al. 2018[5]). Both boards are
required to achieve the on-sky frame rate goal of 1 kHz using an 80x80 region
of interest (ROI) while employing correlated double sampling (CDS) ($\sim
0.48ms$ per read). Additionally, our controller consists of several other ARC
boards: an ARC-64 PCIe board and ARC-22 timing board with three fiber links
for fast data transfer; an ARC-32 CCD & IR clock driver board which provides
clock signals and voltage supplies for the detector; and an ARC-33 bias board
capable of providing a 0 to 14 V bias for avalanche gain. The detector is
installed on a preamplifier board with differential preamplifier channels,
bias decoupling and over-voltage protection which is placed inside a cryogenic
Dewar. The assembly is shown installed inside the MAPS WFS unit in Figure 2.

---
Figure 2:  A photograph of the SAPHIRA cryostat assembly installed into the
MAPS WFS unit.

The SAPHIRA sensor is housed within a cryostat Dewar, designed by the National
Research Council (NRC) Hertzberg, that cools the detector to its operating
temperature of 85K. The cryostat utilizes a Stirling cooler (Sunpower’s
CryoTel MT Cryocooler) with an Active Vibration Cancellation (AVC) system.
Butko et al. 2018[5] reports on the Dewar design, cryocooler performance and
vibration testing.

###  2.2 AO TESTBED

The Renovated Adaptive Z-band Optic Relay system (RAZOR) is an adaptive optics
testbed located at the University of Toronto’s Spectroscopic High Angular
Resolution/Photonics (SHARP) imaging lab. RAZOR was constructed to test novel
AO techniques and technologies in a controlled environment prior to their
deployment at an observatory; in particular, the closed-loop performance
testing of the SAPHIRA sensor. RAZOR has been designed to test emerging
techniques and technologies from various AO sub-fields: the testbed hosts
three independent wavefront sensing arms; operates at both optical and near-
infrared wavelengths; and employs a custom real-time control software
developed in house. These specifications provide the flexibility and
capability required to test and develop a range of cutting edge AO techniques.

---
Figure 3:  A photo of our adaptive optics testbed (RAZOR) from above. The
light path for the experiments reported in this proceedings is shown in red
and the relevant components are labelled. The orange box shows a zoomed in
view of the pyramid.

RAZOR uses two co-aligned monochromatic point sources to facilitate operation
in both the visible and near-infrared: one 650nm red laser and a 1.31um
infrared laser that can be swapped for an 890nm superluminescent diode. The
testbed makes use of achromatic lenses so that both wavelengths can be used
simultaneously with minimal chromatic error. Both sources pass through
simulated atmosphere — one of four possible rotating phase screens with
$r_{0}$ between 0.5 and 1.5 millimeters — before illuminating the surface of a
97 actuator ALPAO deformable mirror (DM97-15). After reflecting off of the
DM97, the light is separated into three independent wavefront sensing arms: a
HASO4 First Shack-Hartmann wavefront sensor (SHWFS) from Imagine Optic, a
custom-built SHWFS arm, and a PyWFS arm. The PyWFS arm is composed of a
PBL25Y/S-BAM12 glass Pyramid (spare copy of MAPS pyramid with manufacturing
defects), a tip-tilt modulator (PI S-330 Tip Tilt and E-727 Controller), and
the SAPHIRA camera. To quantify the quality of the correction, RAZOR’s
‘science’ arm utilizes a XIMEA CMOS camera (MQ013MG-ON) or a Raptor Photonics
InGaAs CMOS camera (Owl 320 HS) depending on the operating wavelength.

RAZOR is controlled through an in-house real-time control software called
SharpRTC. The core components of SharpRTC are written in C/C++ with an
emphasis on object-oriented design. Each hardware component is implemented as
a stand alone ‘object class’ and interactions between hardware components are
implemented as a ‘process class’. At runtime, each object class is
instantiated with parameters extracted from a user defined configuration file
(YAML format). The configuration file also determines which calibration or
pre-processing features will be executed before running the closed AO loop.
This design gives the user a large amount of customizability surrounding the
many parameters of an AO loop as well as providing a clear path for adding
additional functionality. In addition to the core components, SharpRTC also
includes a variety of analysis scripts written in Python. The analysis scripts
contain tools for probing performance parameters such as loop latency, slope
diagnostics, modulation shape, and Strehl-ratios. Together, all of the
SharpRTC components offer the user a significant amount of flexibility in
their operation and caters to a wide variety of AO testing and development.

##  3 CHARACTERIZATION & RESULTS

###  3.1 SAPHIRA Characterization

In early 2020, a suite of characterization tests were performed on our SAPHIRA
detector in order to determine each pixel’s avalanche gain, conversion gain,
nominal read noise, effective read noise, and excess noise. Since we only
require a small region of interest on the detector for wavefront sensing, we
purchased an engineering grade SAPHIRA that is known to have regions with sub-
optimal performance; therefore, the purpose of the characterization was to
verify performance and to determine the optimal region of interest.

The first characterization parameter of interest is avalanche gain ($M$) — the
multiplicative factor of photoelectrons resulting from impact ionization — as
a function of the reverse-bias voltage, $V_{bias}$. Light frames were acquired
using a near-infrared LED; 128 exposures were acquired with 100 Samples Up the
Ramp (SUTR) each. To account for dark current, background frames were captured
using the same exposure time as light frames but with the NIR LED off. The
second read from each exposure was used to perform CDS and calibrate out each
pixel’s pedestal voltage. The avalanche gain for each pixel was obtained by
dividing the linear slope (in ADU/s) by the calibrated slope that yields unity
avalanche gain ($V_{bias}=2V$). The results agree with previous studies and
are summarized in Figure 4.

---
Figure 4: Median avalanche gain across the array as function of reverse-bias
voltage. The blue curve represents the reported avalanche gain for SAPHIRA
Mark 3 & 10 arrays.

Conversion gain and read noise for the array were determined using the Photon
Transfer Curve (PTC) method. The same data used to calculate the avalanche
gain were also used to compute the conversion gain. Each pixel’s PTC was
estimated by plotting the pixel variance as a function of mean pixel value
across the 128 exposures. A line was fit to the linear regime of the PTC and
the conversion gain was taken to be the inverse slope of the fit. To measure
the read noise, a similar process was conducted using dark frames to better
populate the read noise dominated regime. The dark frames were acquired with a
cooled aluminum blank filter installed in front of the array to reduce the
amount of NIR background light illuminating the detector. From the resulting
PTC, the y-intercept of the linear-fit was converted to electrons RMS using
the pixel’s conversion gain (an additional factor of $\sqrt{2}$ is needed to
account for the CDS). The median conversion gain of the detector was found to
be $11\frac{e^{-}}{ADU}$ and the median read noise was found to be $58e^{-}$.
The full distributions of both parameters are shown in Figure 5.

---
Figure 5: Histograms showing each pixel’s measured conversion gain (left) and
read noise (right) for our SAPHIRA detector. The median conversion gain and
read noise of the detector were found to be $11\frac{e^{-}}{ADU}$ and
$58e^{-}$ respectively.

Using the nominal read noise, conversion gain, and avalanche gain we are able
to estimate the effective read noise using the following relation:

$ERN_{e^{-}}(V_{bias})=G\frac{\sigma(V_{bias})}{M(V_{bias})}$ (1)

Where $G$ is the pixels conversion gain, $\sigma$ is the pixel’s read noise in
ADU, and $M$ is the pixel’s avalanche gain. We found that a median effective
read noise of $0.85e^{-}$ is achieved with a bias voltage of 14V.

The impact ionization responsible for avalanche gain is a stochastic process
that introduces some excess noise. This excess noise degrades the signal-to-
noise ratio and can be quantified as the ratio, $F$, of the observed photon
noise to what we would expect if there was no excess noise. The excess noise
can be measured as the quotient of the slopes of PTCs in the Poisson-noise
regime for $V_{bias}>2V$ and the product of each pixel’s conversion and
avalanche gains for that bias. The median excess noise factors across the
array are summarized in Figure 6.

---
Figure 6: The median excess noise factor across the array as a function of
applied bias voltage. There is a more scatter in the excess noise measurement
after a bias voltage of 8V. It is possible this is a result of having fewer
points within the Poisson-dominated regime of the PTCs for these trials, and
therefore, less accuracy in our slope measurements.

###  3.2 Closed-loop Performance

Prior to shipping the SAPHIRA and cryostat assembly to be integrated into the
MAPS wavefront sensor unit at the University of Arizona, we tested the camera
in closed-loop on our AO testbed. The camera was illuminated using the 1.31um
laser source and the PSF was monitored using the Owl 320 HS infrared camera
from Raptor Photonics. In order to best replicate MMT atmospheric conditions,
the tests were conducted mimicking our expected aperture size to atmosphere
coherence length ratio at the MMT ($\frac{D}{r_{0}}\approx 13.5$), and phase
plate rotation speeds were chosen to be analogous to typical wind speeds
($10mm/s$ on the phase plate $\approx 10m/s$ at MMTO). We were able to test
for minimum required functionality and did not test the PyWFS with modulation.
Additional testing will be performed once the camera is fully integrated into
the MAPS WFS unit.

Our tests employed a standard AO control loop. An empirical interaction matrix
(IM) was computed using the poke/pull method whereby each actuator is
repeatedly perturbed and the slope response is recorded. The command matrix
(CM) was computed via a Singular Value Decomposition (SVD) of the IM. To
improve stability, up to 30 modes were dropped during the SVD; we believe the
lack of modulation and imperfections on the phase screen were the main causes
of the instabilities, both of which should not be a problem under nominal
operation at MMTO. The loop was run 1.1Khz, limited by the readout speed of
the detector (ROI of 96x96 pixels with dark subtraction and deinterlacing).

In order to quantify the performance of our system, we compute the Strehl
ratio relative to a best-fit airy model (see Figure 7). During closed-loop
operations, photos of the corrected PSF were captured by a high speed infrared
camera. The Strehl ratio was estimated through a multi-step process: the
images were stacked together to produce a ‘science’ PSF approximating a long
exposure; an airy disk model was assumed based on the best fit to the
unabberrated testbed PSF; the empirical PSF was shifted to get a more accurate
peak estimate; the Airy model was down-sampled and shifted to match the
empirical PSF; both images were normalized and the Strehl ratio was estimated
to be the ratio of their central values. This process was repeated as a
function of phase screen speed, producing Figure 8. We observe a Strehl ratio
of about 65% with a slight degradation as wind speed increased. This level of
performance meets our expectation for our experimental set-up and turbulence
parameters. This shows that the SAPHIRA assembly has met minimum requirements
for integration into the MAPS WFS.

---
Figure 7: A comparison between the observed closed loop PSF (1000 stacked PSF
frames taken over one phase screen rotation, estimate Strehl $\approx$ 65%)
and the down-sampled best-fit airy model for the underrated testbed PSF.
---
Figure 8:  The estimated Strehl ratio across a range of phase screen speeds.
The laser source approximated a 3 mag J-band guide star. The SAPHIRA avalanche
gain was set to unity and no modulation was used. The non-aberrated Strehl
ratio of our testbed is about 85% from alignment and optical errors.

##  4 FUTURE WORK

###  4.1 Status

The NIR PyWFS has officially completed its commissioning phase and has entered
its integration phase. In June 2022, the camera was shipped to the Steward
Observatory at the University of Arizona where it is currently being
integrated into the MAPS WFS unit. Once the camera is installed into the WFS
unit and its operation has been verified, we will begin the process of
coordinating with the other WFS components and the MAPS control software. The
current timeline projects that the MAPS WFS and ASM will begin on-sky testing
at MMTO in the Fall of 2022.

###  4.2 Calibration

Typically, AO systems operate as a self-contained instrument which is fed
directly from the focal plane of a telescope. MAPS’s adaptive secondary,
however, is an example of a ‘pre-focal’ deformable mirror with no intermediate
focus. While adaptive secondary mirrors have several key advantages over their
‘post-focal’ counterparts, this configuration substantially increases the
complexity of calibration. Systems with concave ASMs such as the Large
Binocular Telescope (LBT) have overcome this issue by installing calibration
sources at their intermediate focus; however, MMT uses a convex ASM with no
access to a usable focal plane. This is a well-known problem in the field for
which several solutions have been theorized, investigated, and implemented.
Without access to a calibration source, there are two options to calibrate the
system: use on-sky data; or generate a synthetic calibration matrix based on
detailed simulations of the system. [6][7][8] Synthetic calibration is a good
option for systems which are only able to correct low spatial frequency
aberrations since simulations tend to perform well at those spatial
frequencies. Since MMTO’s former AO system was not sensitive to higher order
aberrations ($\sim 50$ modes), it was able to use a synthetic calibration.
Synthetic calibration for higher order systems such as the new MAPS AO system
have been previously implemented[7], but require a substantial amount of
empirical data about the adaptive secondary and wavefront sensor. Obtaining
enough empirical data to verify a high-order simulation is much more difficult
for convex adaptive secondaries for the same reason as calibration.

Using on-sky data is non-trivial since the wavefront sensor will
simultaneously measure both the atmospheric and induced aberrations, making it
difficult to recover the ASM’s influence on the wavefront error during
calibration. Pinna et al. 2012[7] reported on the first high order calibration
of an adaptive optics system using on-sky data. Their methodology employed a
sinusoidal modulation technique whereby the magnitude a given basis ASM shape
is varied over time with a known frequency. After measuring several cycles of
this modulated signal, the AO system telemetry will now have a distinct
feature at the modulation frequency from which the calibration matrix can be
determined. The authors were able to compare system performance using a
traditional calibration to both a synthetic and on-sky calibration. They
concluded that a combination of synthetic and on-sky methods is likely to
produce the best results. Later, hybrid methods such as the SPRINT calibration
strategy [9] defined feedback loops that adjust simulated calibration matrices
using on-sky data, potentially eliminating the requirement for sophisticated
measurements of the ASM. We are currently investigating the implementation of
a hybrid calibration strategy for MAPS, but may additionally implement purely
empirical methods such as the recent DO-CRIME algorithm [10].

###  4.3 Conclusion

This proceedings reported on the status of the near-infrared pyramid wavefront
(NIR PyWFS) sensor currently being commissioned as part of the MAPS upgrade to
the MMT observatory. Advancements in avalanche photodiode arrays, such as
Leonardo’s SAPHIRA, have made NIR wavefront sensing a powerful tool for
increasing NGS AO sky coverage. To date, readout electronics, firmware, and a
cryostat Dewar have been developed for the wavefront sensor. Additionally, our
SAPHIRA detector has undergone a suite of characterization testing and been
operated in closed-loop on the AO testbed. Recently, the camera and cryostat
assembly have been installed into the MAPS wavefront sensor unit at the
University of Arizona with first light scheduled for the Fall of 2022.

###### Acknowledgements.

The MAPS project is primarily funded through the NSF Mid-Scale Innovations
Program, programs AST-1636647 and AST-1836008. The NIR PyWFS contributions
have been funded by the Canada Foundation of Innovation and the Ontario
Research Funds.

## References

  * [1] Brusa-Zappellini, G., Riccardi, A., Biliotti, V., Vecchio, C. D., Salinari, P., Stefanini, P., Mantegazza, P., Biasi, R., Andrighettoni, M., Franchini, C., and Gallieni, D., “Adaptive secondary mirror for the 6.5-m conversion of the Multiple Mirror Telescope: first laboratory testing results,” in [Adaptive Optics Systems and Technology ], Tyson, R. K. and Fugate, R. Q., eds., 3762, 38 – 49, International Society for Optics and Photonics, SPIE (1999).
  * [2] Morzinski, K. M., Montoya, M., Fellows, C., Durney, O., Ford, J., West, G., Gardner, A., Vaz, A., Anugu, N., Mailhot, E., Carlson, J., Harrison, L., Gacon, F., Downey, E., Hinz, P., Jones, T., Patience, J., Sivanandam, S., Chen, S., Lamb, M., Butko, A., Liu, S., Hardy, T., and Jannuzi, B., “Development and status of MAPS, the MMT AO exoPlanet characterization system,” in [Adaptive Optics Systems VII ], Schreiber, L., Schmidt, D., and Vernet, E., eds., 11448, 360 – 370, International Society for Optics and Photonics, SPIE (2020).
  * [3] Finger, G., Baker, I., Alvarez, D., Dupuy, C., Ives, D., Meyer, M., Mehrgan, L., Stegmeier, J., and Weller, H. J., “Sub-electron read noise and millisecond full-frame readout with the near infrared eAPD array SAPHIRA,” in [Adaptive Optics Systems V ], Marchetti, E., Close, L. M., and Véran, J.-P., eds., 9909, 354 – 373, International Society for Optics and Photonics, SPIE (2016).
  * [4] Wizinowich, P., Chun, M., Mawet, D., Agapito, G., Dekany, R., Esposito, S., Fusco, T., Guyon, O., Hall, D., Plantet, C., and Rigaut, F., “Near-infrared wavefront sensing,” in [Adaptive Optics Systems V ], Marchetti, E., Close, L. M., and Véran, J.-P., eds., 9909, 395 – 407, International Society for Optics and Photonics, SPIE (2016).
  * [5] Butko, A., Sivanandam, S., Hardy, T., Liu, S., Chen, S., Veran, J., and Hinz, P., “Developing an infrared apd array camera for near-infrared wavefront sensing,” in [Ground-based and Airborne Instrumentation for Astronomy VII ], Simard, L., Simard, L., Evans, C., and Takami, H., eds., 10702, SPIE (1 2018).
  * [6] Riccardi, A., Briguglio, R., Pinna, E., Agapito, G., Quiros-Pacheco, F., and Esposito, S., “Calibration strategy of the pyramid wavefront sensor module of ERIS with the VLT deformable secondary mirror,” in [Adaptive Optics Systems III ], Ellerbroek, B. L., Marchetti, E., and Véran, J.-P., eds., 8447, 1842 – 1853, International Society for Optics and Photonics, SPIE (2012).
  * [7] Pinna, E., Quirós-Pacheco, F., Riccardi, A., Briguglio, R., Puglisi, A., Busoni, L., Arcidiacono, C., Argomedo, J., Xompero, M., Marchetti, E., and Esposito, S., “First on-sky calibration of a high order adaptive optics system,” in [Adaptive Optics Systems III ], Ellerbroek, B. L., Marchetti, E., and Véran, J.-P., eds., 8447, 869 – 880, International Society for Optics and Photonics, SPIE (2012).
  * [8] Riccardi, A., Briguglio, R., Pinna, E., Agapito, G., Quiros-Pacheco, F., and Esposito, S., “Calibration strategy of the pyramid wavefront sensor module of ERIS with the VLT deformable secondary mirror,” in [Adaptive Optics Systems III ], Ellerbroek, B. L., Marchetti, E., and Véran, J.-P., eds., 8447, 1842 – 1853, International Society for Optics and Photonics, SPIE (2012).
  * [9] Heritier, C. T., Fusco, T., Oberti, S., Neichel, B., Esposito, S., and Madec, P.-Y., “Sprint, system parameters recurrent invasive tracking: a fast and least-cost online calibration strategy for adaptive optics,” Monthly Notices of the Royal Astronomical Society 504, 4274–4290 (May 2021).
  * [10] Lai, O., Chun, M., Dungee, R., Lu, J., and Carbillet, M., “DO-CRIME: dynamic on-sky covariance random interaction matrix evaluation, a novel method for calibrating adaptive optics systems,” Monthly Notices of the Royal Astronomical Society 501, 3443–3456 (dec 2020).