$R^{\beta}$ has IIA level $l$.

By construction $R^{\beta}$ belongs to $E_{9}$ which is associated with nodes
3-10 and node 11 and it has, by assumption, IIA level $l$., However, it does
not contain the root $\alpha_{2}$ and so it has level zero for this node. In
contrast we see from equation (5.4.3) that $R$ has level two with respect to
node 2 but it has IIA level zero, Consequently $R^{s}$ has level $2s$ and 0
with respect to nodes 2 and 10. As such the root of equation (5.4.4), which
corresponds to $R^{s}\wedge R^{\beta}$, has levels $2s$ and $l$ with respect
to nodes 2 and 10.

Following, for example, the review of chapter 16 of [19] we can analyse the
$A_{1}\otimes D_{8}$ representations that occur in the roots of the form of
equation (5.4.4). The first step is to write the roots $\alpha_{2}$ and
$\alpha_{10}$ in the form

$\alpha_{2}=y-\nu_{1}-\mu\ \ \ \ \,\ \ \ \ \alpha_{10}=x-\nu_{7}$ $None$

where $\mu$ is the fundamental weight of $A_{1}$ and $\nu_{1}$ and $\nu_{7}$
are the fundamental weights of $D_{8}$ corresponding to nodes 3 and 9 in the
above Dynkin diagram respectively. The quantities $x$ and $y$ are orthogonal
to all the roots, and so the weights, of $A_{1}$ and $D_{8}$. These roots must
have length squared two and so

$\alpha_{2}^{2}=y^{2}+\nu_{1}^{2}+\mu^{2}=y^{2}+1+{1\over 2}=2\ \ \ {\rm and\
so}\ \ \ y^{2}={1\over 2}$ $None$

and

$\alpha_{10}^{2}=x^{2}+\nu_{7}^{2}=x^{2}+2=2\ \ \ {\rm and\ so}\ \ \ x^{2}=0$
$None$

In evaluating these equations we used the scalar products of the fundamental
weights which can be found, for example, in appendix D of [19]. It is straight
forward to check that they have the correct scalar products with the other
$E_{11}$ roots with one exception, namely $\alpha_{2}\cdot\alpha_{9}$ which
requires more care. We have that

$\alpha_{2}\cdot\alpha_{9}=\nu_{1}\cdot\nu_{7}+x\cdot y={1\over 2}+x\cdot y=0\
\ \ {\rm and\ so}\ \ \ x\cdot y=-{1\over 2}$ $None$

This novel feature arises as we have deleted two nodes.

Substituting these expressions for the roots $\alpha_{2}$ and $\alpha_{10}$ in
the root $\alpha$ of equation (5.4.4) is given by

$\alpha=\epsilon-2s\mu+2sy+\gamma-2s\nu_{1}-l\nu_{7}+lx$ $None$

We can analyse the $A_{1}\otimes D_{8}$ representations that occur in the
roots of the form of equation (5.4.4). The part of the root in equation
(5.4.4) in the space of the $A_{1}$ algebra is $\epsilon-s\mu$ and this must,
for suitable $\epsilon$, be a highest weight of $A_{1}$. We therefore find the
equation

$q\mu=2s\mu-\epsilon=2s\mu-n\hat{\alpha}_{1},\ \ {\rm and\ so}\ \ {q\over
2}=s-n,\ \ {\rm for}\ \ n\geq 0$ $None$

where $q$ is the Dynkin index of the highest representation and
$\hat{\alpha}_{1}$ is the simple root of $A_{1}$. Proceeding in a similar way
for the $D_{8}$ part of the algebra we find that

$\sum_{j\in\\{3,..,9,11\\}}p_{j}\lambda_{j}=l\nu_{7}+2s\nu_{1}-\gamma=l\nu_{7}+2s\nu_{1}-\sum_{j\in\\{3,..,9,11\\}}m_{j}\hat{\alpha}_{j},\
\ {\rm for}\ \ m_{j}\geq 0$ $None$

We have written $\gamma=\sum_{j\in\\{3,..,9,11\\}}m_{j}\hat{\alpha}_{j}$ where
$\hat{\alpha}_{j}$, $j=3,..,9,11$ are the simple roots of $D_{8}$ and
$\lambda_{j}$ its fundamental weights. Taking the dot product of equation
(5.4.11) with $\lambda_{i}$ and using the definition of the inverse $D_{8}$
Cartan matrix $(A_{ij}^{D_{8}})^{-1}=\lambda_{i}\cdot\lambda_{j}$, we find

$\sum_{j\in\\{3,..,9,11\\}}p_{j}(A_{ji}^{D_{8}})^{-1}=l\nu_{7}\cdot\lambda_{i}+2s\nu_{1}\cdot\lambda_{i}-m_{i}$
$None$

By analysing equations (4.4.11) and (4.4.12) one can find at low levels the
possible representations of $A_{1}\otimes D_{8}$ that can occur.

In any Kac-Moody algebra the lengths of the roots are bounded by $2$. Using
equations (4.4.10) and (4.4.11) we find that

$\alpha^{2}={q^{2}\over
2}+\sum_{i,j}p_{i}(A_{ij}^{D_{8}})^{-1}p_{j}+{2s^{2}}-2sl\leq 2.$ $None$

This will only be less than or equal to 2 if $s\leq l$ holds. This indeed
proves the identity of equation (5.4.1) since the root corresponding to
$R^{l+1}\wedge R^{\beta}$ does not exist in the $E_{11}$ algebra.

6 String states and an irreducible representation of the string little algebra

We begin this section by clarifying how the string little algebra sits in
$E_{11}$ in relation to the well known subalgebras $E_{9}$, $E_{8}$ and its
Cartan involution subalgebra SO(16).

6.1 Relationship of the string little algebra to $E_{9}$

In section five we found the subalgebra ${\cal H}$ of
$I_{c}(E_{11})\otimes_{s}l_{1}$ that preserves the string charges. We referred
to this as the string little algebra. It contained the generators $J_{ab}$ and
$S_{ab}$, $a,b=0,1$, which generate $SO(2)\otimes SO(2)$, and the generator
$L_{ai}^{(1)}$. It also contained the generators

${\cal H}_{r}=\\{J_{ij},S_{ij};L_{\alpha}\ ;\ \alpha\in{\rm positive\ roots\
of\ }E_{9}\\}.$ $None$

where $L_{\alpha}$ is defined in equation (5.1.1). As explained in section 5.2
the generators in equation (6.1.1) obey an algebra that has the algebra
$SO(8)\otimes SO(8)$ at IIA level zero but at higher levels it has commutators
that have the same algebra as the Borel subalgebra of $E_{9}$.

In this paper we wish to construct the string analogue of the irreducible
particle representations first constructed by Wigner [13]. To do this we must
choose a representation of the string little algebra and then carry out the
corresponding boost. As discussed in the previous section we may choose the
generators $J_{ab}$, $S_{ab}$ and $L_{ai}^{(1)}$ of ${\cal H}$ to be trivially
realised. As such we just have to choose a representation of the algebra
obeyed by the algebra ${\cal H}_{r}$ of equation (6.1.1).

To better understand the algebra ${\cal H}_{r}$ it will be useful to
understand in detail its relation to $E_{9}$. The Dynkin diagram of $E_{9}$ is
found by deleting the first two nodes, labelled one and two and indicated by a
$\oplus$ , from the Dynkin diagram of $E_{11}$:

$\matrix{&&&&&&&&&&&&&&\bullet&11&\bullet&10\cr&&&&&&&&&&&&&&|&&|\cr\oplus&-&\oplus&-&\bullet&-&\bullet&-&\bullet&-&\bullet&-&\bullet&-&\bullet&-&\bullet&&\cr
1&&2&&3&&4&&5&&6&&7&&8&&9&&\cr}$

Since we are considering the ten dimensional IIA theory we delete node ten,
called the IIA node, in the above Dynkin diagram. We will decompose the
$E_{9}$ algebra with respect to the remaining $D_{8}={\rm SO}(8,8)$ subalgebra
as shown by the diagram

$\matrix{&&&&&&&&&&&&&&\bullet&11&\oplus&10\cr&&&&&&&&&&&&&&|&&|\cr\oplus&-&\oplus&-&\bullet&-&\bullet&-&\bullet&-&\bullet&-&\bullet&-&\bullet&-&\bullet&&\cr
1&&2&&3&&4&&5&&6&&7&&8&&9&&\cr}$

The low level generators in such a decomposition can be found by using Simplie
[16]. Deleting node 10 we find that $E_{9}$ decomposes into the 120
dimensional adjoint representation of $D_{8}$ at level zero and the 128
dimensional spinor representation of $D_{8}$ at level one. This pattern
repeats itself, at all even levels and odd levels we find the adjoint and
spinor representations of $D_{8}$ respectively. In order to recognise this
result in terms of generators we are familiar with we will not only delete the
IIA node ten but also node eleven to result in a decomposition to the algebra
SL(8) which corresponds in the Dynkin diagram consisting of nodes 3 to 9
inclusive. At IIA level zero, the 120 generators which belong to the adjoint
of $D_{8}=S0(8,8)$ is given by

$K_{(0)}{}^{i}{}_{j}=K^{i}{}_{j}\ ,\ R^{ij}_{(0)}=R^{ij}\ ,\
R_{(0)}{}_{ij}=R_{ij},\ \ \ i,j,\ldots=3,4,\ldots,10$ $None$

Here the $K^{i}{}_{j}$ generate GL(8) and together with $R^{ij}$ and $R_{ij}$
they generate SO(8,8). The subscripts here, and in the equations that follow,
denote the IIA level of the generators. At level zero only one finds an
additional generator with no SL(8) indices which we have omitted to write. It
is the analogue of the dilaton generator in the IIA theory.

At IIA level one we find the generators

$R^{i}_{(1)}=R^{i}\ ,\ R^{i_{1}i_{2}i_{3}}_{(1)}=R^{i_{1}i_{2}i_{3}}\ ,\
R^{i_{1}\ldots i_{5}}_{(1)}=R^{i_{1}\ldots i_{5}}\ ,\ R^{i_{1}\ldots
i_{7}}_{(1)}=R^{i_{1}\ldots i_{7}}\ ,$ $None$

These 128 generators belong to the Majorana-Weyl spinor representation of
$D_{8}$. Similarly at level minus we find another 128 dimensional spinor
representation that is given by

$R_{(-1)}{}_{i}=R_{i}\ ,\ R_{(-1)}{}_{i_{1}i_{2}i_{3}}=R_{i_{1}i_{2}i_{3}}\ ,\
R_{(-1)}{}_{i_{1}\ldots i_{5}}=R_{i_{1}\ldots i_{5}}\ ,\
R_{(-1)}{}_{i_{1}\ldots i_{7}}=R_{i_{1}\ldots i_{7}}\ ,$ $None$

At level two we find the generators

$K_{(2)}{}^{j}{}_{i}={1\over 7!}\varepsilon_{ik_{1}\ldots k_{7}}R^{k_{1}\ldots
k_{7},j}\ ,\ \tilde{R}_{(2)}={1\over 8!}\varepsilon_{k_{1}\ldots
k_{8}}R^{k_{1}\ldots k_{8}}\ ,$ $R^{ij}_{(2)}={1\over
6!}\varepsilon^{ij}{}_{k_{1}\ldots k_{6}}R^{k_{1}\ldots k_{6}}\ ,\
R_{(2)}{}_{ij}={1\over 8!}\varepsilon_{k_{1}\ldots k_{8}}R^{k_{1}\ldots
k_{8},ij}\ ,$ $None$

which belong to another copy of the 120 dimensional adjoint representation of
$D_{8}$. The first and second generator generate GL(8). While at level minus
we have the same representation

$K_{(-2)}{}^{i}{}_{j}={1\over 7!}\varepsilon^{ik_{1}\ldots
k_{7}}R_{k_{1}\ldots k_{7},j}\ ,\ \tilde{R}_{(-2)}={1\over
8!}\varepsilon^{k_{1}\ldots k_{8}}R_{k_{1}\ldots k_{8}}\ ,$
$R^{ij}_{(-2)}={1\over 6!}\varepsilon^{ijk_{1}\ldots k_{6}}R_{k_{1}\ldots
k_{6}}\ ,\ R_{(-2)}{}_{ij}={1\over 8!}\varepsilon^{k_{1}\ldots
k_{8}}R_{k_{1}\ldots k_{8},ij}\ ,$ $None$

At level three we find that the generators belong to the 128 dimensional
spinor representation of $D_{8}$. They are given by

$R_{(3)}^{i}={1\over 8!}\varepsilon_{k_{1}\ldots k_{8}}R^{k_{1}\ldots
k_{8},i}\ ,\ R^{i_{1}i_{2}i_{3}}_{(3)}={1\over 8!}\varepsilon_{k_{1}\ldots
k_{8}}R^{k_{1}\ldots k_{8},i_{1}i_{2}i_{3}}\ ,$ $R^{i_{1}\ldots
i_{5}}_{(3)}={1\over 8!}\varepsilon_{k_{1}\ldots k_{8}}R^{k_{1}\ldots
k_{8},i_{1}\ldots i_{5}}\ ,\ R^{i_{1}\ldots i_{7}}_{(3)}={1\over
8!}\varepsilon_{k_{1}\ldots k_{8}}R^{k_{1}\ldots k_{8},i_{1}\ldots i_{7}}\ ,$
$None$

Similarly at level minus three we find another spinor of $D_{8}$,

$R_{(-3)}{}_{i}={1\over 8!}\varepsilon^{k_{1}\ldots k_{8}}R_{k_{1}\ldots
k_{8},i}\ ,\ R_{(-3)}{}_{i_{1}i_{2}i_{3}}={1\over 8!}\varepsilon^{k_{1}\ldots
k_{8}}R_{k_{1}\ldots k_{8},i_{1}i_{2}i_{3}}\ ,$ $R_{(-3)}{}_{i_{1}\ldots
i_{5}}={1\over 8!}\varepsilon^{k_{1}\ldots k_{8}}R_{k_{1}\ldots
k_{8},i_{1}\ldots i_{5}}\ ,\ R_{(-3)}{}_{i_{1}\ldots i_{7}}={1\over
8!}\varepsilon^{k_{1}\ldots k_{8}}R_{k_{1}\ldots k_{8},i_{1}\ldots i_{7}}\ ,$
$None$

In section three we did not list the correspondence between the $E_{11}$
generators that appear in the eleven dimensional theory at IIA level three and
the generators as they appear in the IIA theory, that is, when the node ten is
deleted. We now give this correspondence

$R^{\underline{a}_{1}\ldots\underline{a}_{8},\underline{b}}=\hat{R}^{\underline{a}_{1}\ldots\underline{a}_{8},\underline{b}}\
,\
R^{\underline{a}_{1}\ldots\underline{a}_{8},\underline{b}_{1}\underline{b}_{2}\underline{b}_{3}}=\hat{R}^{\underline{a}_{1}\ldots\underline{a}_{8}11,\underline{b}_{1}\underline{b}_{2}\underline{b}_{3}}\
,$
$R^{\underline{a}_{1}\ldots\underline{a}_{8},\underline{b}_{1}\ldots\underline{b}_{5}}=\hat{R}^{\underline{a}_{1}\ldots\underline{a}_{8}11,\underline{b}_{1}\ldots\underline{b}_{5}11}\
,\
R^{\underline{a}_{1}\ldots\underline{a}_{8},\underline{b}_{1}\ldots\underline{b}_{7}}=\hat{R}^{\underline{a}_{1}\ldots\underline{a}_{8}11,\underline{b}_{1}\ldots\underline{b}_{7}11,11}\
.$ $None$

The identifications at level minus three are given by

$R_{\underline{a}_{1}\ldots\underline{a}_{8},\underline{b}}=\hat{R}_{\underline{a}_{1}\ldots\underline{a}_{8},\underline{b}}\
,\
R_{\underline{a}_{1}\ldots\underline{a}_{8},\underline{b}_{1}\underline{b}_{2}\underline{b}_{3}}=\hat{R}_{\underline{a}_{1}\ldots\underline{a}_{8}11,\underline{b}_{1}\underline{b}_{2}\underline{b}_{3}}\
,\ $
$R_{\underline{a}_{1}\ldots\underline{a}_{8},\underline{b}_{1}\ldots\underline{b}_{5}}=\hat{R}_{\underline{a}_{1}\ldots\underline{a}_{8}11,\underline{b}_{1}\ldots\underline{b}_{5}11}\
,\
R^{\underline{a}_{1}\ldots\underline{a}_{8},\underline{b}_{1}\ldots\underline{b}_{5}}=\hat{R}_{\underline{a}_{1}\ldots\underline{a}_{8}11,\underline{b}_{1}\ldots\underline{b}_{7}11,11}\
.$ $None$

The pattern that emerges is obvious, at even IIA levels $2n=0,\pm 2,\pm
4,\ldots$ the $E_{9}$ algebra contains the adjoint representation of ${\rm
SO}(8,8)$, which we denote by ${\rm SO}(8,8)_{2n}$, with generators

$K_{(2n)}{}^{i}{}_{j}\ \ ,\ \ R_{(2n)}^{ij}\ \ ,\ \ R_{(2n)}{}_{ij}\ \ ,$
$None$

While at each odd $E_{9}$ level, $2n+1=1,3,\ldots$ , we find a Majorana-Weyl
spinor representation of $D_{8}$, which we denote by ${\bf 128}_{2n+1}$, with
generators

$R_{(2n+1)}^{i}\ ,\ R_{(2n+1)}^{i_{1}i_{2}i_{3}}\ ,\ R_{(2n+1)}^{i_{1}\ldots
i_{5}}\ ,\ R_{(2n+1)}^{i_{1}\ldots i_{7}}\ \ ,\ \ n=1,\pm 3,\pm 5,\ldots\ .$
$None$

We can thus write $E_{9}$ as

$E_{9}=\sum_{n\in Z}{\rm SO}(8,8)_{2n}\oplus{\bf 128}_{2n+1}\ .$ $None$

The algebra of $E_{9}$ reflects its affine character and can be written as

$[{\bf 120}_{2n},{\bf 120}_{2m}]={\bf 120}_{2(n+m)}\ ,\ [{\bf 120}_{2n},{\bf
128}_{2m+1}]={\bf 128}_{2(n+m)+1}\ ,$ $[{\bf 128}_{2n+1},{\bf
128}_{2m+1}]={\bf 120}_{2(n+m+1)}$ $None$

The Cartan involution invariant algebra of $E_{9}$ has generators that are
given by $R^{\alpha}+I_{c}(R^{\alpha})$ where $R^{\alpha}$ is a positive root
generator of $E_{9}$. Using the action of the Cartan involution given in
equation (2.7) we find at IIA level zero that the Cartan involution invariant
generators are given by

$K^{i}{}_{j}-K^{j}{}_{i},\ \ R^{i_{1}i_{2}}-R_{i_{1}i_{2}}$ $None$

which generate $SO(8)\otimes SO(8)$. While examining equation (2.8) the Cartan
involution invariant generators arising from IIA levels $\pm 1$ are given by

$R^{i}-R_{i}\ ,\ R^{i_{1}i_{2}i_{3}}-R_{i_{1}i_{2}i_{3}}\ ,R^{i_{1}\ldots
i_{5}}+R_{i_{1}\ldots i_{5}}\ ,\ R^{i_{1}\ldots i_{7}}-R_{i_{1}\ldots i_{7}}$
$None$

The pattern at higher levels is obvious.

We can now discuss the string little algebra in more detail and in particular
the generators given in equation (6.1.1). The generator $L_{\alpha}$ has the
form

$L_{\alpha}=R^{\alpha}+I_{c}(R^{\alpha})+\ {\rm terms\ involving\ commutators\
with\ R\ or\ \overline{R}}$ $None$

We recognise the two terms shown explicitly in $L_{\alpha}$ in equation
(6.1.17) as an element of $I_{c}(E_{9})$ but , as indicated, it also contains
terms involving the commutators with $R$ or $\overline{R}$. Thus the
generators in the string little algebra ${\cal H}_{r}$ are in one to one
correspondence with those of $I_{c}(E_{9})$. However, $L_{\alpha}$ belongs to
$I_{c}(E_{11})$ rather than $I_{c}(E_{9})$ as the generators $R$ and
$\overline{R}$. are not in $E_{9}$.

At IIA level zero there are no $R$ and $\overline{R}$ commutators as these
vanish and the generators in ${\cal H}_{r}$ are those of equation (6.1.15).
Thus, at level zero, the algebra of ${\cal H}_{r}$ and $I_{c}(E_{9})$ are the
same, namely they just contain the generators of $SO(8)\otimes SO(8)$.
However, for all higher levels the generators are very different. The
commutators of the generators in ${\cal H}_{r}$ with IIA level greater than
zero have the same algebra as $R^{\alpha}$, where $\alpha$ is a positive root
of $E_{9}$, that is, their algebra is isomorphic to the part of the Borel
subalgebra of $E_{9}$ that contains generators corresponding to positive
roots. In particular the generators in ${\cal H}_{r}$ that contain the
positive roots of $D_{8}^{+}$ obey the same algebra as the commutators of the
Borel algebra of $D_{8}^{+}$.

As is well known $E_{9}$ is nothing but affine $E_{8}$, that is, $E_{8}^{+}$.
However, so far we have hardly mentioned $E_{8}$, or it’s Cartan involution
invariant subalgebra $I_{c}(E_{8})=SO(16)$ which both play such an important
role in the eleven dimensional $E_{11}$ theory. In this theory one deletes
node 11 to leave a GL(11) algebra. This contrasts with the IIA theory being
studied in this paper in which one deletes the IIA node, that is, node ten
which leaves the algebra $D_{10}$. To be relevant to our current context we
will first delete the nodes 1 and 2 as shown in the Dynkin diagram in figure
above to leave $E_{9}$ rather than consider the full $E_{11}$ algebra. Further
deleting node eleven we find the algebra $A_{8}=SL(9)$ however, this is not
the deletion that corresponds to the IIA theory which is of interest to us
here.

The algebra $E_{9}$ is affine $E_{8}$ but this $E_{8}$ arises in the $E_{11}$
Dynkin diagram consisting of nodes 4-10 and node 11 and is found by deleting
node three. It is useful to first analyse how the generators of $E_{8}$ appear
from the eleven dimensional theory by deleting node eleven and the resulting
SL(8) algebra which corresponds to nodes 4-10. The resulting Dynkin diagram is
given by

$\matrix{&&&&&&&&&&&&&&\oplus&11&\bullet&10\cr&&&&&&&&&&&&&&|&&|\cr\oplus&-&\oplus&-&\oplus&-&\bullet&-&\bullet&-&\bullet&-&\bullet&-&\bullet&-&\bullet&&\cr
1&&2&&3&&4&&5&&6&&7&&8&&9&&\cr}$

The generators of $E_{8}$ are given by

$\hat{K}^{\underline{i}}{}_{\underline{j}},\
\hat{R}^{\underline{i}_{1}\underline{i}_{2}\underline{i}_{3}},\
\epsilon_{\underline{k}_{1}\underline{k}_{2}\underline{i}_{1}\ldots\underline{i}_{6}},\hat{R}^{\underline{i}_{1}\ldots\underline{i}_{6}},\epsilon_{\underline{i}_{1}\ldots\underline{i}_{8},}\hat{R}^{\underline{i}_{1}\ldots\underline{i}_{8},\underline{j}},\
\ $ $None$

where the indices take the values
$\underline{i},\underline{j},\underline{i}_{1},\underline{i}_{2},\ldots=4,\ldots,11$.
These generators have levels zero, one, two and three respectively with
respect to the level corresponding to node eleven. Their negative level
analogues are given by

$\hat{R}_{\underline{i}_{1}\underline{i}_{2}\underline{i}_{3}},\
\epsilon^{\underline{k}_{1}\underline{k}_{2}\underline{i}_{1}\ldots\underline{i}_{6}}\hat{R}_{\underline{i}_{1}\ldots\underline{i}_{6}},\
\epsilon^{\underline{i}_{1}\ldots\underline{i}_{8},}\hat{R}_{\underline{i}_{1}\ldots\underline{i}_{8},\underline{j}}$
$None$

The Cartan involution acts on these generators as specified in equation (2.7)
and the invariant algebra is generated by

$\hat{K}^{\underline{i}}{}_{\underline{j}}-\hat{K}^{\underline{j}}{}_{\underline{i}},\
\hat{R}^{\underline{i}_{1}\underline{i}_{2}\underline{i}_{3}}-\hat{R}_{\underline{i}_{1}\underline{i}_{2}\underline{i}_{3}},\
\hat{R}^{\underline{i}_{1}\ldots\underline{i}_{6}}+\hat{R}_{\underline{i}_{1}\ldots\underline{i}_{6}},\
\hat{R}^{\underline{i}_{1}\ldots\underline{i}_{8},\underline{j}}-\hat{R}_{\underline{i}_{1}\ldots\underline{i}_{8},\underline{j}}$
$None$

The second and third generators obey the algebra of $SO(8)\otimes SO(8)$ while
the second and fourth generators belong to the $(8_{s},8_{c})$ representations
of this algebra where $8_{s}$ and $8_{c}$ are the spinor representations of
$SO\otimes SO(8)$. Taken together these 120 generators obey the algebra of
SO(16). A detailed discussion can be found in section four of reference [15].
The 128 Cartan involution odd generators in $E_{8}$ have the opposite signs to
those in equation (6.1.18) and they belong to the Majorana-Weyl spinor
representation of this SO(16).

The $E_{9}$ is then given by the affinisation of the $E_{8}$ algebra. It
generators are

$E_{9}=\sum_{n}248_{n}=\sum_{n}(SO(16)_{n}\oplus 128_{n})\ ,\ \ n=0,\pm
1,\ldots$ $None$

where 248 is the adjoint representation of $E_{8}$ and the second line
contains the decomposition of $E_{9}$ that we just discussed into its Cartan
involution invariant sublagebra SO(16) of $E_{8}$.

From the IIA perspective studied in this paper we have the same $E_{9}$ but we
deleted the IIA node, node 10, to find $D_{8}=SO(8,8)$ and we decomposed the
$E_{9}$ algebra in terms of this algebra. To aid our analysis we also deleted
node eleven and decomposed to the remaining $A_{8}$ algebra associated with
nodes 3-9. While the algebra $D_{8}$ appears in both the eleven dimensional
and IIA viewpoints they are not the same algebra. This is easily seen once we
realise that in the IIA viewpoint node ten is deleted but from the eleven
dimensional viewpoint this node it is part of $E_{8}$ Dynkin diagram. Thus in
the IIA and eleven dimensional viewpoints different nodes are deleted. Hence
the $D_{8}=SO(8,8)$ we discussed in the IIA theory is not subalgebra of
$E_{8}$ as it involves node three which is not a node of the $E_{8}$ that
appears from the elven dimensional viewpoint. In contrast the $E_{8}$ of the
eleven dimensional theory requires node ten to be active but this is deleted
in the IIA theory. It is intriguing that despite this $E_{9}$ has a very
similar decomposition into the two different $D_{8}$’s. In the IIA case we can
write $E_{9}$ as in equation (6.1.13) while from the eleven dimensional
viewpoint we write it as in equation (6.1.19). One can think of it as a kind
of duality.

6.2 The relation of the string little algebra to the string states.  As we
explained in the previous section the string little algebra involves an
algebra that is the same as the Borel subalgebra of $E_{9}$ at higher IIA
levels and $SO(8)\otimes SO(8)$ at IIA level zero. Such a little algebra is
unexpected and is quite different to the little algebras for the point
particles. At first sight it is not clear why it has such a simple form.

In principle we should now choose an irreducible representation of the string
little algebra and then carry out the boost to find the full irreducible
representation of $I_{c}(E_{11})\otimes_{s}l_{1}$. We will not be able to show
that the well known string states do carry a representation of the full string
little algebra, rather we will make progress towards this goal. In particular
the algebra $SO(8,8)^{+}$ is a subalgebra of $E_{9}$ and we will focus on
considering a representation of the Borel subalgebra of $SO(8,8)^{+}$ part of
the string little algebra.

We first consider the open superstring and in particular the NS sector. As
explained in section 4.3, from the $b_{r}^{i}$ oscillators one can construct a
bilinear quantity whose generators are those of $SO(8)^{+}$, see equation
(4.3.1). However, these generators do not act on physical states to give
physical states as they do not boost the momentum of the states to that
required by the physical state conditions. As explained at the end of section
four, one can replace them by their DDF analogues, see equation (4.3.3), or
use the boost of equation (4.3.5), so that they act on physical states. to
give physical states. Thus the open string states in the NS sector do carry a
representation of $SO(8)^{+}$ and so the Borel subalgebra of $SO(8)^{+}$.
Since the generators are bilinear in the oscillators they act on the physical
states so as to take states with an even (odd) number of oscillators to states
which also have an even (odd) number of oscillators. As such the physical
states do not carry an irreducible representation of $SO(8)^{+}$ but contain
two irreducible representations which have an odd and even in the number of
the oscillators. If we choose to have the irreducible representation that is
odd in oscillators then we make the GSO [25] projection in this sector. Thus
the GSO projection can be understood as a restriction to have an irreducible
representation of $SO(8)^{+}$.

Let us now consider the NS-NS sector of the closed superstring as discussed in
section 4.3. Now we have the oscillators
$b^{I}_{r}\equiv(b_{r}^{i},\bar{b}_{r}^{i})$ where the indices $I,J,\ldots$
take $8+8=16$ values and the generators ${\cal
M}^{c}{}_{n}{}^{IJ}=\sum_{r}:b_{n-r}^{[I}b_{r}^{J]}:$ which obey the algebra
$SO(8,8)^{+}$. As for the open string these do not take physical states to
physical states, however, we can use their DDF analogues or do a boost similar
to that in equation (4.3.5). At first sight the string states in the NS-NS
sector do carry a representation of $SO(8,8)^{+}$ and so the Borel subalgebra
of $SO(8,8)^{+}$ at IIA levels greater than one and $SO(8)\otimes SO(8)$ at
level zero which a large part of the string little algebra.

However, this statement does not take account of the GSO projection which
ensures that one has odd number of $b_{r}^{i}$ oscillators and also odd
numbers of $\bar{b}_{r}^{i}$. This choice is preserved by $SO(8)^{+}\otimes
SO(8)^{+}$ but not the generators ${\cal M}^{c}{}_{0}{}^{IJ}$ for
$I=1,\ldots,8$ and $j=9,\ldots,16$. However, it is preserved by the action of
bilinears of the latter generators. There is also the level matching condition
to take account of. This would only be preserved if we took suitable
combinations of generators. However, one could also allow windings, or achieve
matching by also acting with the bosonic oscillators. The string states come
so close to carrying a representation of $SO(8,8)^{+}$ that it is tempting to
think that in a more complete treatment would work better.

One could also consider generators that transform the NS-NS into the R-R
sectors. Such generators could take the generic form

$-2D_{k}|R><N|b_{{1\over 2}}{}^{(i}\bar{b}_{{1\over
2}}^{k)}+D^{i}{}_{j_{1}j_{2}}|R><N|b_{{1\over 2}}^{j_{1}}\bar{b}_{{1\over
2}}^{j_{2}}+\ldots$ $9D^{[i_{1}}|R><N|b_{{1\over 2}}{}^{i_{2}}\bar{b}_{{1\over
2}}^{i_{3}]}-6D^{k[i_{1}i_{2}}|R><N|b_{{1\over 2}}^{(i_{3}]}\bar{b}_{{1\over
2}}^{k)}+\ldots$

where $|R>$ and $|N>$ are the vacuum states in the R-R and NS-NS sectors,
$D^{k}=d_{0}^{k}+i\bar{d}_{0}^{k}$ and $+\ldots$ are higher level terms. In
fact these generators are the Borel part of the transformations of the
massless states that appear when one considers the irreducible representation
of $I_{c}(E_{11})\otimes l_{1}$ suitable to the IIA theory. This can be
deduced from the eleven dimensional case given in references [14,15] by
carrying out a dimensional reduction.

In the R and R-R sectors of the open and closed sectors of the superstring we
have the $d_{n}^{i}$ and the $d_{n}^{I}=(d_{n}^{i},\bar{d}_{n}^{i})$
oscillators respectively. From these one can form bilinear expressions that
generate the algebras $SO(8)^{+}$ and $SO(8,8)^{+}$ respectively. It would be
interesting to find their replacement by the analogue DDF operators, or
include momentum boosts, which take physical states to physical sates.

We now study the situation from the view point of the Green-Schwarz
formulation of the IIA superstring. In the light-cone gauge one has a residual
SO(8) symmetry of the Lorentz algebra. In addition to the bosonic oscillators
the theory contains two fermionic spinor oscillators $S_{n}^{\alpha}$ and
$\bar{S}_{n}^{\alpha}$ with $\alpha,\beta,\ldots=1,\ldots,8$. These are
Majorana-Weyl spinors of Spin(8) of opposite chirality. They obey the anti-
commutators

$\\{S_{n}^{\alpha},S_{m}^{\beta}\\}=\delta^{\alpha,\beta}\delta_{n+m,0},\
\\{\bar{S}_{n}^{\alpha},\bar{S}_{m}^{\beta}\\}=\delta^{\alpha,\beta}\delta_{n+m,0},\
\\{S_{n}^{\alpha},\bar{S}_{m}^{\beta}\\}=0$ $None$

We will need the properties of such spinors. Looking at section 5.6 of the
book [19] on finds for this Euclidean case that we may choose $B=C=I$ and
$\rho=1$. Then $\epsilon=1$ and $(\gamma^{i})^{*}=\gamma^{i}$ and
$(\gamma^{i})^{T}=\gamma^{i}$. It follows that the matrices

$\gamma^{i},\ \gamma^{i_{1}\ldots i_{4}},\ \gamma^{i_{1}\ldots i_{5}},\
\gamma^{i_{1}\ldots i_{8}},\ $ $None$

are symmetric and the others are antisymmetric. We also find that if
$\gamma_{9}\chi=\chi$, where $\gamma_{9}=\gamma_{1}\ldots\gamma_{8}$, then
$\bar{\chi}\gamma_{9}=\bar{\chi}$ for a Majorana-Weyl spinor and as such only
the expressions $\bar{\chi}\gamma^{i_{1}i_{2}}$ and
$\bar{\chi}\gamma^{i_{1}\ldots i_{6}}\chi$ are non zero. However, these two
expressions are related by a duality transformation.

Taking into account the above properties we can form the operators

$M_{n}^{i_{1}i_{2}}=\sum_{p}S_{n-p}\gamma^{i_{1}i_{2}}S_{p},\ \
\bar{M}_{n}^{i_{1}i_{2}}=\sum_{p}\bar{S}_{n-p}\gamma^{i_{1}i_{2}}\bar{S}_{p}$
$None$

In this equation
$S_{n-p}\gamma^{i_{1}i_{2}}S_{p}=S_{n-p}^{\alpha}(\gamma^{i_{1}i_{2}})_{\alpha\beta}S_{p}^{\beta}$.
The first expression is the only bilinear object in $S_{n}^{\alpha}$ alone
that we can construct. The same applies for $\bar{S}_{n}^{\alpha}$. Using the
relations of equation (6.2.1) we find that the generators of equation (6.2.3)
have the algebra $SO(8)^{+}\otimes SO(8)^{+}$ which we can think of acting on
the left and right moving parts of the string.

For the mixed bilinear expressions we have spinors of opposite Weyl character
and so we can only have

$L^{I}_{n}\equiv\sum_{p}S_{n-p}\gamma^{I}\bar{S}_{p}$ $None$

where $\gamma^{I}=(\gamma^{i},\gamma^{i_{1}i_{2}i_{3}})$. We have taken
account of the fact that the gamma matrices between the spinors obey duality
relations.

The commutators of the $L^{I}_{n}$ with
$M_{n}^{i_{1}i_{2}}+\bar{M}_{n}^{i_{1}i_{2}}$ is given by

$[M_{n}^{i_{1}i_{2}}+\bar{M}_{n}^{i_{1}i_{2}},L^{I}_{n}]=2\sum_{p}S_{n+m-p}[\gamma^{i_{1}i_{2}},\gamma^{I}]\bar{S}_{p}$
$None$

which is just the usual affine rotation. With
$M_{n}^{i_{1}i_{2}}-\bar{M}_{n}^{i_{1}i_{2}}$ we find that

$[M_{n}^{i_{1}i_{2}}-\bar{M}_{n}^{i_{1}i_{2}},L^{I}_{n}]=2\sum_{p}S_{n+m-p}\\{\gamma^{i_{1}i_{2}},\gamma^{I}\\}\bar{S}_{p}$
$None$

Thus the $L^{I}_{n}$ belong to a representation of $SO(8)^{+}\otimes
SO(8)^{+}$ and indeed they belong to the $(8_{s},8_{c})_{n}$ representation.

The commutators of the $L^{I}_{n}$ are given by

$[L^{I}_{n},L^{J}_{m}]=\sum_{p}S_{n+m-p}\gamma^{J}\gamma^{I}S_{p}\eta_{I}-\sum_{p}\bar{S}_{n+m-p}\gamma^{I}\gamma^{J}\bar{S}_{p}\eta_{I}$
$None$

where $\eta_{I}=-1$ if $\gamma^{I}$ equals $\gamma^{i}$ and $\eta_{I}=1$ if
$\gamma^{i_{1}\ldots i_{5}}$. In view of the above comments these commutators
can only belong to $SO(8)^{+}\otimes SO(8)^{+}$. As such the generators of
equations (6.2.3) and (6.2.4 ) close to give the algebra $SO(8,8)^{+}$.

In the open superstring, the states are formed by the oscillators
$S_{n}^{\alpha}$ and the bosonic oscillators acting on the 16 dimensional
$8_{v}\oplus 8_{c}$ states which belong to a representation of
$S_{0}^{\alpha}$ which itself belongs to the $8_{s}$ representation of
spin(8). This is part of the spacetime supersymmetry and it takes the 8 spin
one states into the 8 spin ”one” ”half” states and vice-versa. The
$8_{v}\oplus 8_{c}$ are the massless states of the open string. For a review
of states in the superstring in the Green-Schwarz formulation see section 5.3
of reference [26].

The $SO(8)^{+}$ generators $M_{n}^{i_{1}i_{2}}$ clearly act on the states of
the open superstring to give other states which will be physical states
provided we also inject a corresponding momenta. The latter could either be
achieved by the analogue of a DDF construction or the boosts we considered at
the end of section four. Assuming this to be possible the open string states
belong to a representation of $SO(8)^{+}$ in agreement with our result in the
Neveu-Schwarz formulation. We note that this is a representation of the bosons
and also the fermion as the generator $M_{n}^{i_{1}i_{2}}$ does not mix these
two.

The closed string physical states result from the left moving oscillators
$S_{n}^{\alpha}$ and the right moving oscillators $\bar{S}_{n}^{\alpha}$ and
also the bosonic oscillators acting on the $256=16\otimes 16$ dimensional
formed from the tensor product of the left and right moving states
$(8_{v}+8_{c})\otimes(8_{v}+8_{s})$. The states $8_{v}\otimes 8_{v}$ belong to
the $1+28+35_{v}$ dimensional representations of spin(8) which are the
massless states in the NS-NS sector. While the $8_{c}\otimes 8_{s}$ belong to
the $8_{v}+56_{v}$ representation of spin(8) which are the massless states of
the R-R sector. The generators of equation (6.2.3) and (6.3.4) act on the
physical states. The left moving generators $M_{n}^{i_{1}i_{2}}$ and
$\bar{M}_{n}^{i_{1}i_{2}}$ give affine Lorentz rotations of the left moving
states and the right moving sectors. The generators $L^{I}$ act on the left
and right movers so as to transform the NS-NS states into the R-R states into
each other. Assuming we can provide the required boost of the momentum when
these generators act on the physical states they will carry a representation
of the algebra $SO(8,8)^{+}$ and also take account of the level matching. This
latter constraint can perhaps be achieved by taking certain combinations of
generators, or involving the bosonic operators, or allowing winding,

So far we did not discuss the role of the bosonic oscillators $\alpha_{n}^{i}$
, $\bar{\alpha}_{n}^{i}$, $i=2,\ldots,9$. They should also play an important
and one would expect the above symmetries to also act on these oscillators.
However, as we explained at the end of section four, the natural
transformations of $SO(8,8)^{+}$ of the oscillators, see equations (4.3.7) and
(4.3.8), do not respect their commutation relations of equation (4.3.6).
However, the string little algebra for the superstring is not $I_{c}(E_{9})$
but is essentially the Borel subalgebra of $E_{9}$. A subalgebra of this
algebra is the Borel subalgebra of $SO(8,8)^{+}$ which we can take to have the
generators $M_{n}^{ij}$, $\bar{M}_{n}^{ij}$, $V_{n}^{ij}$ and
$\bar{V}_{n}^{ij}$ for $n\leq 0$. These act on the negatively moded
oscillators as in equations (4.3.7) and (4.3.8) so as to take the negatively
modeed into themselves. The physical states contain negatively moded bosonic
oscillators acting on the vacuum and so, with a suitable momentum boost, these
generators take physical states to physical states. While somewhat
unconventional the action of these generators on the positively modded
oscillators can be chosen to preserve the commutation relations. Thus if would
appear that one can have the action of a significant part of the string little
algebra. One could also apply this strategy to the fermionic oscillators.

In this section we have made a fragmented and partial attempt to understand if
a representation of the string little algebra is carried by the well known
physical states of the superstring. While there are many encouraging features
there are also some unresolved issues. We have only investigated the Borel
part of $SO(8,8)^{+}$ and not the 128 dimensional spinor representation that
occurs at even levels. We have also not taken a unified approach in that we
did not consider the bosonic and fermionic oscillators in a unified way. It
would be good to persue these matters further.

7 Discussion  E theory provides, for the first time, a unified approach to
branes within which one can hope to treat point particles, string and branes
in the same way. Indeed one can take the attitude that they are all
irreducible representations of $I_{c}(E_{11})\otimes_{s}l_{1}$ in much the
same way as particles, in our familiar quantum field theories, are irreducible
representations of the Poincare algebra [14]. In this paper we have applied
this strategy to strings and in particular the IIA string. Taking the
corresponding charges in the vector representation to be non zero we found the
little algebra for the string. It has an unusual form in that it is
essentially the Borel subalgebra of $E_{9}$. However, this $E_{9}$ is not the
$E_{9}$ that is obviously apparent form the Dynkin diagram of $E_{11}$ by
deleting node two, but only emerged after a rather involved calculation
involving generators in $E_{11}$ which are outside this obvious $E_{9}$. The
Borel character of the little algebra has its origins in the fact that the
string charges involve the momenta and the string charge which satisfy a
relation rather than each have specific values. This is a novel feature of
finding the irreducible representations for extended objects. This contrasts
with the massless case where it is $I_{c}(E_{9})$.

The next step in the procedure would be to take a representation of the little
algebra. However in section six we investigated if the known states of the
superstring did indeed carry a representation of essentially the Borel
subalgebra of $E_{9}$. While we were not able to show that this was the case
we did find quite a bit of evidence that there are additional symmetries which
are in this algebra. Starting from the fact that bilinears in the known
oscillators can generate the algebra $SO(8,8)^{+}$, whose Borel subalgebra, is
a substantial subalgebra of the string little algebra we investigated if the
physical states carried a representation of this algebra. If one allowed
winding or some other way of relaxing the level matching requirements, and one
could carry out the required momentum boosts, then it would appear that this
was true. Our treatment was also provisional in that it did not consider the
symmetry to act in a unified way on all the oscillators. We note that it is
only required to show that the physical states carry a representation of this
algebra and this may not have a simple expression in terms of the known
oscillators.

Assuming that we had a representation of the string little algebra the next
step would be to boost it up to a representation of
$I_{c}(E_{11})\otimes_{s}l_{1}$. Such a step was discussed in reference [14]
and it would be very interesting to do this for the string. The net result
should be a wavefunction that depends on the string coordinates and obeys the
physical state conditions. These conditions also follow from the string
dynamics which can also be constructed from and $E_{11}$ viewpoint as a non-
linear realisation of $I_{c}(E_{11})\otimes_{s}l_{1}$ with a suitable local
subalgebra [7,27]. This local subalgebra has been found at low levels and it
essentially agrees with the string little algebra found in this paper even
though the two derivations are very different. It would be interesting to join
up these two different approaches and this may help illuminate the unknown
parts of each approach.

It would be very interesting to extend the approach of this paper to the
branes and in particular the M2 and M5 branes. The first step would be to find
their little algebras. We would expect the calculation to have many points in
common with the derivation of the string little algebra found in this paper.

String theory on a torus has additional symmetries which can, for a suitable
torus, be any of the Lie algebras in the Killing-Cartan list. It is believed
that these are a quirk of taking the string on the torus and not a sign of
some underlying symmetry of string theory. A similar attitude was taken to the
Cremmer-Julia symmetries in the maximal supergravity theories; they were
thought just to be a consequence of the dimensional reduction on a torus.
However, we now know that they are symmetries of the underlying theory, namely
E theory, although this theory was not discovered by thinking in this way [1].
Perhaps we should think that the torus probes aspects of string theory not
present in our usual spacetime and this allows us to see some hidden
symmetries.

In E theory space time has an infinite number of coordinates [2]. Taking
string theory on a torus introduces the winding coordinates that are just a
few of these coordinates. It is useful to think of the infinite coordinates as
a kind of effective spacetime reflecting some of the properties of the
underlying structure. The idea that taking particular situations may display
hidden symmetries has a precedent in particle physics. For example symmetries
that are hidden at low energy in particle physics, as they are spontaneously
broken, will show up at high energy. Another example is the discovery of
asymptotic charges in general relativity when we consider spacetime that are
not flat but asymptotically flat.

Acknowledgement Peter West has benefitted from grant numbers ST/P000258/1 and
ST/T000759/1 from STFC in the UK. Keith Glennon would like to thank Kings
College for support during his PhD studies.

Appendix A: The IIA Commutators

In this appendix we will construct at low levels the commutators of the
$E_{11}$ algebra in its decomposition that leads to the IIA theory. As
described in section two of this paper this is in the decomposition to
SO(10,10) which is for ease of recognition further decomposed into
representation of GL(10). Given the generators at a low IIA levesl this
algebra can be determined directly by considering all possible expressions
consistent with the ${\rm SL}(10)$ character of the generators involved in a
given commutator, the fact that the commutators preserve the level and then
requiring that these expressions satisfy the Jacobi identities. Equivalently
the algebra can be determined from the algebra of the $E_{11}$ algebra in its
eleven dimensional decomposition as given, for example, in [19]. We give the
IIA algebra to level one, but only include those generators which involve the
underlying $E_{11}$ generators to $E_{11}$ level three for simplicity. This
means for example that we will ignore
$R^{{\underline{a}}_{1}\ldots{\underline{a}}_{9}}=R^{{\underline{a}}_{1}\ldots{\underline{a}}_{9}11,1111}$
in the algebra in this approximation as it has level four with respect to the
deletion of node eleven. The algebra of the generators at IIA level zero is

$[K^{\underline{a}}{}_{\underline{b}},K^{\underline{c}}{}_{\underline{d}}]=\delta^{\underline{c}}{}_{\underline{b}}K^{\underline{a}}{}_{\underline{d}}-\delta^{\underline{a}}{}_{\underline{d}}K^{\underline{c}}{}_{\underline{b}}\
\ ,\ \ [K^{\underline{a}}{}_{\underline{b}},\tilde{R}]=0\ \ ,\ \
[K^{\underline{a}}{}_{\underline{b}},R^{\underline{c}\underline{d}}]=2\delta^{[\underline{c}}{}_{\underline{b}}R^{|\underline{a}|\underline{d}]}\
\ ,$ $\ \
[K^{\underline{a}}{}_{\underline{b}},R_{\underline{c}\underline{d}}]=-2\delta^{\underline{a}}{}_{[\underline{c}}R_{|\underline{b}|\underline{d}]}\
\ ,\ \ [\tilde{R},\tilde{R}]=0\ \ ,\ \
[\tilde{R},R^{\underline{a}\underline{b}}]=0\ \ ,\ \
[\tilde{R},R_{\underline{a}\underline{b}}]=0\ \ ,$
$[R^{\underline{a}\underline{b}},R^{\underline{c}\underline{d}}]=0\ \ ,\ \
[R^{\underline{a}\underline{b}},R_{\underline{c}\underline{d}}]=4\delta^{[\underline{a}}{}_{[\underline{c}}K^{\underline{b}]}{}_{\underline{d}]}\
\ ,\ \ [R_{\underline{a}\underline{b}},R_{\underline{c}\underline{d}}]=0\ \ .$
$None$

These generators obey the algebra of $D_{10}\otimes GL(1)=SO(10,10)\otimes
GL(1)$ as they must.

The commutators between the IIA level one generators are given by

$[R^{\underline{a}},R^{\underline{b}}]=0\ ,\
[R^{\underline{a}},R^{\underline{b}_{1}\underline{b}_{2}\underline{b}_{3}}]=0\
,\
[R^{\underline{a}},R^{\underline{b}_{1}\ldots\underline{b}_{5}}]=-R^{\underline{a}\,\underline{b}_{1}\ldots\underline{b}_{5}}\
,\ $
$[R^{\underline{a}},R^{\underline{b}_{1}\ldots\underline{b}_{7}}]=-2R^{\underline{a}\,\underline{b}_{1}\ldots\underline{b}_{7}}-7R^{\underline{a}[\underline{b}_{1}\ldots\underline{b}_{6},\underline{b}_{7}]}\
,\
[R^{\underline{a}_{1}\underline{a}_{2}\underline{a}_{3}},R^{\underline{b}_{1}\underline{b}_{2}\underline{b}_{3}}]=2R^{\underline{a}_{1}\underline{a}_{2}\underline{a}_{3}\underline{b}_{1}\underline{b}_{2}\underline{b}_{3}}\
,$
$[R^{\underline{a}_{1}\underline{a}_{2}\underline{a}_{3}},R^{\underline{b}_{1}\ldots\underline{b}_{5}}]=-3R^{\underline{b}_{1}\ldots\underline{b}_{5}[\underline{a}_{1}\underline{a}_{2},\underline{a}_{3}]}\
,\
[R^{\underline{a}_{1}\underline{a}_{2}\underline{a}_{3}},R^{\underline{b}_{1}\ldots\underline{b}_{7}}]=0\
,$
$[R^{\underline{a}_{1}\ldots\underline{a}_{5}},R^{\underline{b}_{1}\ldots\underline{b}_{5}}]=0\
,\
[R^{\underline{a}_{1}\ldots\underline{a}_{5}},R^{\underline{b}_{1}\ldots\underline{b}_{7}}]=0\
,\
[R^{\underline{a}_{1}\ldots\underline{a}_{7}},R^{\underline{b}_{1}\ldots\underline{b}_{7}}]=0\
,$ $None$

and for those with IIA level minus one by

$[R_{\underline{a}},R_{\underline{b}}]=0\ ,\
[R_{\underline{a}},R_{\underline{b}_{1}\underline{b}_{2}\underline{b}_{3}}]=0\
,\
[R_{\underline{a}},R_{\underline{b}_{1}\ldots\underline{b}_{5}}]=+R_{\underline{a}\,\underline{b}_{1}\ldots\underline{b}_{5}}\
,\ $
$[R_{\underline{a}},R_{\underline{b}_{1}\ldots\underline{b}_{7}}]=+2R_{\underline{a}\,\underline{b}_{1}\ldots\underline{b}_{7}}+7R_{\underline{a}[\underline{b}_{1}\ldots\underline{b}_{6},\underline{b}_{7}]}\
,\
[R_{\underline{a}_{1}\underline{a}_{2}\underline{a}_{3}},R_{\underline{b}_{1}\underline{b}_{2}\underline{b}_{3}}]=2R_{\underline{a}_{1}\underline{a}_{2}\underline{a}_{3}\underline{b}_{1}\underline{b}_{2}\underline{b}_{3}}\
,$
$[R_{\underline{a}_{1}\underline{a}_{2}\underline{a}_{3}},R_{\underline{b}_{1}\ldots\underline{b}_{5}}]=-3R_{\underline{b}_{1}\ldots\underline{b}_{5}[\underline{a}_{1}\underline{a}_{2},\underline{a}_{3}]}\
,\
[R_{\underline{a}_{1}\underline{a}_{2}\underline{a}_{3}},R_{\underline{b}_{1}\ldots\underline{b}_{7}}]=0\
,$
$[R_{\underline{a}_{1}\ldots\underline{a}_{5}},R_{\underline{b}_{1}\ldots\underline{b}_{5}}]=0\
,\
[R_{\underline{a}_{1}\ldots\underline{a}_{5}},R_{\underline{b}_{1}\ldots\underline{b}_{7}}]=0\
,\
[R_{\underline{a}_{1}\ldots\underline{a}_{7}},R_{\underline{b}_{1}\ldots\underline{b}_{7}}]=0\
.$ $None$

The algebra of IIA level zero generators with IIA level one generators are
given by

$[K^{\underline{a}}{}_{\underline{b}},R^{\underline{c}}]=R^{\underline{a}}\delta^{\underline{c}}{}_{\underline{b}}-{1\over
2}\delta^{\underline{a}}{}_{\underline{b}}R^{\underline{c}}\ ,\
[K^{\underline{a}}{}_{\underline{b}},R^{\underline{c}_{1}\underline{c}_{2}\underline{c}_{3}}]=3\delta^{[\underline{c}_{1}}{}_{\underline{b}}R^{|\underline{a}|\underline{c}_{2}\underline{c}_{3}]}-{1\over
2}\delta^{\underline{a}}{}_{\underline{b}}R^{\underline{c}_{1}\underline{c}_{2}\underline{c}_{3}}\
,$
$[K^{\underline{a}}{}_{\underline{b}},R^{\underline{c}_{1}\ldots\underline{c}_{5}}]=5\delta^{[\underline{c}_{1}}{}_{\underline{b}}R^{|\underline{a}|\underline{c}_{2}\ldots\underline{c}_{5}]}-{1\over
2}\delta^{\underline{a}}{}_{\underline{b}}R^{\underline{c}_{1}\ldots\underline{c}_{5}}\
,$
$[K^{\underline{a}}{}_{\underline{b}},R^{\underline{c}_{1}\ldots\underline{c}_{7}}]=7\delta^{[\underline{c}_{1}}{}_{\underline{b}}R^{|\underline{a}|\underline{c}_{2}\ldots\underline{c}_{7}]}-{1\over
2}\delta^{\underline{a}}{}_{\underline{b}}R^{\underline{c}_{1}\ldots\underline{c}_{7}}\
\ ,$ $[\tilde{R},R^{\underline{a}}]=-3R^{\underline{a}}\ \ ,\ \
[\tilde{R},R^{\underline{a}_{1}\underline{a}_{2}\underline{a}_{3}}]=-3R^{\underline{a}_{1}\underline{a}_{2}\underline{a}_{3}}\
\ ,\ \
[\tilde{R},R^{\underline{a}_{1}\ldots\underline{a}_{5}}]=-3R^{\underline{a}_{1}\ldots\underline{a}_{5}}\
\ ,$
$[\tilde{R},R^{\underline{a}_{1}\ldots\underline{a}_{7}}]=-3R^{\underline{a}_{1}\ldots\underline{a}_{7}}\
\ ,\ \
[R^{\underline{a}_{1}\underline{a}_{2}},R^{\underline{b}}]=-R^{\underline{a}_{1}\underline{a}_{2}\underline{b}}\
,$
$[R^{\underline{a}_{1}\underline{a}_{2}},R^{\underline{b}_{1}\underline{b}_{2}\underline{b}_{3}}]=-2R^{\underline{a}_{1}\underline{a}_{2}\underline{b}_{1}\underline{b}_{2}\underline{b}_{3}}\
,\
[R^{\underline{a}_{1}\underline{a}_{2}},R^{\underline{b}_{1}..\underline{b}_{5}}]=-R^{\underline{a}_{1}\underline{a}_{2}\underline{b}_{1}..\underline{b}_{5}}$
$[R^{\underline{a}_{1}\underline{a}_{2}},R^{\underline{b}_{1}..\underline{b}_{7}}]=0\
,\ [R_{\underline{a}_{1}\underline{a}_{2}},R^{\underline{b}}]=0\ ,\
[R_{\underline{a}_{1}\underline{a}_{2}},R^{\underline{b}_{1}\underline{b}_{2}\underline{b}_{3}}]=-6\delta^{[\underline{b}_{1}\underline{b}_{2}}_{\underline{a}_{1}\underline{a}_{2}}R^{\underline{b}_{3}]}\
,$
$[R_{\underline{a}_{1}\underline{a}_{2}},R^{\underline{b}_{1}..\underline{b}_{5}}]=-10R^{[\underline{b}_{1}\underline{b}_{2}\underline{b}_{3}}\delta^{\underline{b}_{4}\underline{b}_{5}]}_{\underline{a}_{1}\underline{a}_{2}}\
,\
[R_{\underline{a}_{1}\underline{a}_{2}},R^{\underline{b}_{1}..\underline{b}_{7}}]=-42R^{[\underline{b}_{1}..\underline{b}_{5}}\delta^{\underline{b}_{6}\underline{b}_{7}]}_{\underline{a}_{1}\underline{a}_{2}}\
,$ $None$

The commutators of the IIA level zero with the IIA level minus one are given
by

$[K^{\underline{a}}{}_{\underline{b}},R_{\underline{c}}]=-R_{\underline{b}}\delta^{\underline{a}}{}_{\underline{c}}+{1\over
2}\delta^{\underline{a}}{}_{\underline{b}}R_{\underline{c}}\ ,\
[K^{\underline{a}}{}_{\underline{b}},R_{\underline{c}_{1}\underline{c}_{2}\underline{c}_{3}}]=-3\delta^{[\underline{a}}{}_{\underline{c}_{1}}R_{|\underline{b}|\underline{c}_{2}\underline{c}_{3}]}+{1\over
2}\delta^{\underline{a}}{}_{\underline{b}}R^{\underline{c}_{1}\underline{c}_{2}\underline{c}_{3}}\
,$
$[K^{\underline{a}}{}_{\underline{b}},R_{\underline{c}_{1}\ldots\underline{c}_{5}}]=-5\delta^{[\underline{a}}{}_{\underline{c}_{1}}R_{|\underline{b}|\underline{c}_{2}\ldots\underline{c}_{5}]}+{1\over
2}\delta^{\underline{a}}{}_{\underline{b}}R_{\underline{c}_{1}\ldots\underline{c}_{5}}\
\ ,$
$[K^{\underline{a}}{}_{\underline{b}},R_{\underline{c}_{1}..\underline{c}_{7}}]=-7\delta^{\underline{b}}{}_{[\underline{c}_{1}}R_{|\underline{b}|\underline{c}_{2}..\underline{c}_{7}]}+{1\over
2}\delta^{\underline{a}}{}_{\underline{b}}R_{\underline{c}_{1}..\underline{c}_{7}},$
$[\tilde{R},R_{\underline{a}}]=3R_{\underline{a}}\ \ ,\ \
[\tilde{R},R_{\underline{a}_{1}\underline{a}_{2}\underline{a}_{3}}]=3R_{\underline{a}_{1}\underline{a}_{2}\underline{a}_{3}}\
\ ,\ \
[\tilde{R},R_{\underline{a}_{1}\ldots\underline{a}_{5}}]=3R_{\underline{a}_{1}\ldots\underline{a}_{5}}\
\ ,$
$[\tilde{R},R_{\underline{a}_{1}\ldots\underline{a}_{7}}]=3R_{\underline{a}_{1}\ldots\underline{a}_{7}}\
\ ,\ \ [R^{\underline{a}_{1}\underline{a}_{2}},R_{\underline{b}}]=0\ \ ,\ \
[R^{\underline{a}_{1}\underline{a}_{2}},R_{\underline{b}_{1}\underline{b}_{2}\underline{b}_{3}}]=6\delta^{\underline{a}_{1}\underline{a}_{2}}_{[\underline{b}_{1}\underline{b}_{2}}R_{\underline{b}_{3}]}\
\ ,$
$[R^{\underline{a}_{1}\underline{a}_{2}},R_{\underline{b}_{1}..\underline{b}_{5}}]=-10R_{[\underline{b}_{1}\underline{b}_{2}\underline{b}_{3}}\delta^{\underline{a}_{1}\underline{a}_{2}}_{\underline{b}_{4}\underline{b}_{5}]}\
\ ,\ \
[R^{\underline{a}_{1}\underline{a}_{2}},R_{\underline{b}_{1}..\underline{b}_{7}}]=-42R_{[\underline{b}_{1}..\underline{b}_{5}}\delta^{\underline{a}_{1}\underline{a}_{2}}_{\underline{b}_{6}\underline{b}_{7}]}\
\ ,$
$[R_{\underline{a}_{1}\underline{a}_{2}},R_{\underline{b}}]=R_{\underline{a}_{1}\underline{a}_{2}\underline{b}}\
\ ,\ \
[R_{\underline{a}_{1}\underline{a}_{2}},R_{\underline{b}_{1}\underline{b}_{2}\underline{b}_{3}}]=-2R_{\underline{a}_{1}\underline{a}_{2}\underline{b}_{1}\underline{b}_{2}\underline{b}_{3}}\
\ ,$
$[R_{\underline{a}_{1}\underline{a}_{2}},R_{\underline{b}_{1}..\underline{b}_{5}}]=-2R_{\underline{a}_{1}\underline{a}_{2}\underline{b}_{1}..\underline{b}_{5}}\
\ ,\ \
[R_{\underline{a}_{1}\underline{a}_{2}},R_{\underline{b}_{1}..\underline{b}_{7}}]=0\
\ .$ $None$

The IIA level one generators with the IIA level minus generators are giv en by

$[R^{\underline{a}},R_{\underline{b}}]=K^{\underline{a}}{}_{\underline{b}}-{1\over
6}\delta^{\underline{a}}{}_{\underline{b}}(3D-\tilde{R})\ \ ,\ \
[R^{\underline{a}},R_{\underline{b}_{!}\underline{b}_{2}\underline{b}_{3}}]=-3\delta^{\underline{a}}{}_{[\underline{b}_{1}}R_{\underline{b}_{2}\underline{b}_{3}]}\
,$ $[R^{\underline{a}},R_{\underline{b}_{1}\ldots\underline{b}_{5}}]=0\ \ ,\ \
[R^{\underline{a}},R_{\underline{b}_{1}\ldots\underline{b}_{7}}]=0\ ,\
[R^{\underline{a}_{1}\underline{a}_{2}\underline{a}_{3}},R_{\underline{b}}]=-3\delta^{[\underline{a}_{1}}{}_{\underline{b}}R^{\underline{a}_{2}\underline{a}_{3}]}\
\ ,$
$[R^{\underline{a}_{1}\underline{a}_{2}\underline{a}_{3}},R_{\underline{b}_{1}\underline{b}_{2}\underline{b}_{3}}]=18\delta^{[\underline{a}_{1}\underline{a}_{2}}_{[\underline{b}_{1}\underline{b}_{2}}K^{\underline{a}_{3}]}{}_{\underline{b}_{3}]}-\delta^{\underline{a}_{1}\underline{a}_{2}\underline{a}_{3}}_{\underline{b}_{1}\underline{b}_{2}\underline{b}_{3}}(3D-\tilde{R})\
\ ,$
$[R^{\underline{a}_{1}\underline{a}_{2}\underline{a}_{3}},R_{\underline{b}_{1}..\underline{b}_{5}}]=30\delta^{\underline{a}_{1}\underline{a}_{2}\underline{a}_{3}}_{[\underline{b}_{1}\underline{b}_{2}\underline{b}_{3}}R_{\underline{b}_{4}\underline{b}_{5}]}\
,\
[R^{\underline{a}_{1}\underline{a}_{2}\underline{a}_{3}},R_{\underline{b}_{1}..\underline{b}_{7}}]=0\
,$ $[R^{\underline{a}_{1}..\underline{a}_{5}},R_{\underline{b}}]=0\ ,\
[R^{\underline{a}_{1}..\underline{a}_{5}},R_{\underline{b}_{1}\underline{b}_{2}\underline{b}_{3}}]=-30\delta^{[\underline{a}_{1}\underline{a}_{2}\underline{a}_{3}}_{[\underline{b}_{1}\underline{b}_{2}\underline{b}_{3}}R^{\underline{a}_{4}\underline{a}_{5}]}\
,$
$[R^{\underline{a}_{1}..\underline{a}_{5}},R_{\underline{b}_{1}\ldots\underline{b}_{5}}]=-5\cdot
5\cdot
6\delta^{[\underline{a}_{1}..\underline{a}_{4}}_{[\underline{b}_{1}\ldots\underline{b}_{4}}K^{\underline{a}_{5}]}{}_{\underline{b}_{5}]}+5\delta^{\underline{a}_{1}..\underline{a}_{5}}_{\underline{b}_{1}\ldots\underline{b}_{5}}(3D-\tilde{R})$
$[R^{\underline{a}_{1}..\underline{a}_{5}},R_{\underline{b}_{1}\ldots\underline{b}_{7}}]=-9\cdot
70\delta^{\underline{a}_{1}..\underline{a}_{5}}_{[\underline{b}_{1}\ldots\underline{b}_{5}}R_{\underline{b}_{6}\underline{b}_{7}]}\
,\ [R^{\underline{a}_{1}..\underline{a}_{7}},R_{\underline{b}}]=0\ ,$
$[R^{\underline{a}_{1}..\underline{a}_{7}},R_{\underline{b}_{1}\underline{b}_{2}\underline{b}_{3}}]=0\
,\
[R^{\underline{a}_{1}..\underline{a}_{7}},R_{\underline{b}_{1}\ldots\underline{b}_{5}}]=9\cdot
70R^{[\underline{a}_{1}\underline{a}_{2}}\delta^{\underline{a}_{3}\ldots\underline{a}_{7}]}_{\underline{b}_{1}\ldots\underline{b}_{5}}\
,$
$[R^{\underline{a}_{1}..\underline{a}_{7}},R_{\underline{b}_{1}..\underline{b}_{7}}]=7\cdot
7\cdot
180\delta^{[\underline{a}_{1}\ldots\underline{a}_{6}}_{[\underline{b}_{1}\ldots\underline{b}_{6}}K^{\underline{a}_{7}]}{}_{\underline{b}_{7}]}-7\cdot
6\cdot
5\delta^{\underline{a}_{1}..\underline{a}_{7}}_{\underline{b}_{1}..\underline{b}_{7}}(3D-\tilde{R})\
\ ,$ $None$

where $D=\sum_{\underline{a}=0}^{9}K^{\underline{a}}{}_{\underline{a}}$.

We now give the commutators of the IIA generators with the $l_{1}$ generators.
Again these results can be derived directly or equivalently from the
commutators of the $E_{11}\otimes_{s}l_{1}$ algebra in its eleven dimensional
formulation [19]. Here we consider the algebra to level one in the IIA and
$l_{1}$ representations and again only to $E_{11}$ level three. The IIA level
zero generators with the IIA level zero $l_{1}$ generators have the
commutators

$[K^{\underline{a}}{}_{\underline{b}},P_{\underline{c}}]=-\delta^{\underline{a}}{}_{\underline{c}}P_{\underline{b}}\
\ ,\ \ [\tilde{R},P_{\underline{c}}]=-3P_{\underline{c}}\ \ ,\ \
[R^{\underline{a}_{1}\underline{a}_{2}},P_{\underline{b}}]=-2\delta^{[\underline{a}_{1}}{}_{\underline{b}}Q^{\underline{a}_{2}]}\
\ ,$ $[R_{\underline{a}_{1}\underline{a}_{2}},P_{\underline{b}}]=0\ \ ,\ \
[K^{\underline{a}}{}_{\underline{b}},Q^{\underline{c}}]=+\delta^{\underline{c}}{}_{\underline{b}}Q^{\underline{a}}\
\ ,\ \ [\tilde{R},Q^{\underline{c}}]=-3Q^{\underline{c}}\ \ ,$
$[R^{\underline{a}_{1}\underline{a}_{2}},Q^{\underline{c}}]=0\ \ ,\ \
[R_{\underline{a}_{1}\underline{a}_{2}},Q^{\underline{c}}]=+2\delta^{\underline{c}}{}_{[\underline{a}_{1}}P_{\underline{a}_{2}]}\
\ ,$ $None$

The IIA level zero $IIA$ generators with the IIA level one $l_{1}$ generators
have the commutators

$[K^{\underline{a}}{}_{\underline{b}},Z]=-{1\over
2}\delta^{\underline{a}}{}_{\underline{b}}Z\ ,\
[K^{\underline{a}}{}_{\underline{b}},Z^{\underline{c}_{1}\underline{c}_{2}}]=2\delta^{[\underline{c}_{1}}{}_{\underline{b}}Z^{|\underline{a}|\underline{c}_{2}]}-{1\over
2}\delta^{\underline{a}}{}_{\underline{b}}Z^{\underline{c}_{1}\underline{c}_{2}}\
,\ $
$[K^{\underline{a}}{}_{\underline{b}},Z^{\underline{c}_{1}\ldots\underline{c}_{4}}]=4\delta^{[\underline{c}_{1}}{}_{\underline{b}}Z^{|\underline{a}|\underline{c}_{2}\underline{c}_{3}\underline{c}_{4}]}-{1\over
2}\delta^{\underline{a}}{}_{\underline{b}}Z^{\underline{c}_{1}\ldots\underline{c}_{4}}\
,$
$[K^{\underline{a}}{}_{\underline{b}},Z^{\underline{c}_{1}\ldots\underline{c}_{6}}]=6\delta^{[\underline{c}_{1}}{}_{\underline{b}}Z^{|\underline{a}|\underline{c}_{2}\ldots\underline{c}_{6}]}-{1\over
2}\delta^{\underline{a}}{}_{\underline{b}}Z^{\underline{c}_{1}\ldots\underline{c}_{6}}\
\ ,$ $[\tilde{R},Z]=-6Z\ \ ,\ \
[\tilde{R},Z^{\underline{a}_{1}\underline{a}_{2}}]=-6Z^{\underline{a}_{1}\underline{a}_{2}}\
\ ,\ \
[\tilde{R},Z^{\underline{a}_{1}\ldots\underline{a}_{4}}]=-6Z^{\underline{a}_{1}\ldots\underline{a}_{4}}\
\ ,$
$[\tilde{R},Z^{\underline{a}_{1}\ldots\underline{a}_{6}}]=-6Z^{\underline{a}_{1}\ldots\underline{a}_{6}}\
\ ,\ \
[R^{\underline{a}_{1}\underline{a}_{2}},Z]=Z^{\underline{a}_{1}\underline{a}_{2}}\
\ ,\ \ [R_{\underline{a}_{1}\underline{a}_{2}},Z]=0\ \ ,$
$[R^{\underline{a}_{1}\underline{a}_{2}},Z^{\underline{b}_{1}\underline{b}_{2}}]=Z^{\underline{a}_{1}\underline{a}_{2}\underline{b}_{1}\underline{b}_{2}}\
,\
[R_{\underline{a}_{1}\underline{a}_{2}},Z^{\underline{b}_{1}\underline{b}_{2}}]=2\delta^{\underline{a}_{1}\underline{a}_{2}}_{\underline{b}_{1}\underline{b}_{2}}Z\
,$
$[R^{\underline{a}_{1}\underline{a}_{2}},Z^{\underline{b}_{1}..\underline{b}_{4}}]={1\over
3}Z^{\underline{a}_{1}\underline{a}_{2}\underline{b}_{1}..\underline{b}_{4}}\
,\
[R_{\underline{a}_{1}\underline{a}_{2}},Z^{\underline{b}_{1}..\underline{b}_{4}}]=12\delta^{[\underline{b}_{1}\underline{b}_{1}}_{\underline{a}_{1}\underline{a}_{2}}Z^{\underline{b}_{3}\underline{b}_{4}]}\
,$
$[R^{\underline{a}_{1}\underline{a}_{2}},Z^{\underline{b}_{1}..\underline{b}_{6}}]=0\
,\
[R^{\underline{a}_{1}\underline{a}_{2}},Z^{\underline{b}_{1}..\underline{b}_{6}}]=2\cdot
45\delta^{[\underline{b}_{1}\underline{b}_{2}}_{\underline{a}_{1}\underline{a}_{2}}Z^{\underline{b}_{3}..\underline{b}_{6}]}\
,$ $None$

The commutators of the IIA level one $IIA$ generators with level zero $l_{1}$
generators are given by

$[R^{\underline{a}},P_{\underline{b}}]=-\delta^{\underline{a}}{}_{\underline{b}}Z\
,\
[R^{\underline{a}_{1}\underline{a}_{2}\underline{a}_{3}},P_{\underline{b}}]=3\delta^{[\underline{a}_{1}}{}_{\underline{b}}Z^{\underline{a}_{2}\underline{a}_{3}]}\
,$ $[R^{\underline{a}_{1}\ldots\underline{a}_{5}},P_{\underline{b}}]=-{5\over
2}\delta^{[\underline{a}_{1}}{}_{\underline{b}}Z^{\underline{a}_{2}\ldots\underline{a}_{5}]}\
,\ [R^{\underline{a}_{1}\ldots\underline{a}_{7}},P_{\underline{b}}]={7\over
6}\delta^{[\underline{a}_{1}}{}_{\underline{b}}R^{\underline{a}_{2}\ldots\underline{a}_{7}]}\
,$ $[R^{\underline{a}},Q^{\underline{b}}]=Z^{\underline{a}\underline{b}}\ ,\
[R^{\underline{a}_{1}\underline{a}_{2}\underline{a}_{3}},Q^{\underline{b}}]=-Z^{\underline{a}_{1}\underline{a}_{2}\underline{a}_{3}\underline{b}}\
,$ $[R^{\underline{a}_{1}\ldots\underline{a}_{5}},Q^{\underline{b}}]={1\over
6}Z^{\underline{a}_{1}\ldots\underline{a}_{5}\underline{b}}\ ,\
[R^{\underline{a}_{1}\ldots\underline{a}_{7}},Q^{\underline{b}}]=0\ ,$ $None$

and the level minus one $IIA$ with level zero $l_{1}$ commutators are given by

$[R_{\underline{a}},P_{\underline{b}}]=0\ ,\
[R_{\underline{a}_{1}\underline{a}_{2}\underline{a}_{3}},P_{\underline{b}}]=0\
,\ [R_{\underline{a}_{1}\ldots\underline{a}_{5}},P_{\underline{b}}]=0\
,[R_{\underline{a}_{1}\ldots\underline{a}_{7}},P_{\underline{b}}]=0\ ,$
$[R_{\underline{a}},Q^{\underline{b}}]=0\ ,\
[R_{\underline{a}_{1}\underline{a}_{2}\underline{a}_{3}},Q^{\underline{b}}]=0\
,\ [R_{\underline{a}_{1}\ldots\underline{a}_{5}},Q^{\underline{b}}]=0\ ,\
[R_{\underline{a}_{1}\ldots\underline{a}_{7}},Q^{\underline{b}}]=0\ .$ $None$

The commutators of the level one $IIA$ generators with the level one $l_{1}$
commutators are given by

$[R^{\underline{a}},Z]=0\ ,\
[R^{\underline{a}},Z^{\underline{b}_{1}\underline{b}_{2}}]=0\ ,\
[R^{\underline{a}},Z^{\underline{b}_{1}..\underline{b}_{4}}]=Z^{\underline{a}\underline{b}_{1}..\underline{b}_{4}}\
,\ [R^{\underline{a}_{1}\underline{a}_{2}\underline{a}_{3}},Z]=0\ ,$
$[R^{\underline{a}},Z^{\underline{b}_{1}..\underline{b}_{6}}]=Z^{\underline{a}\underline{b}_{1}..\underline{b}_{6}}+Z^{\underline{b}_{1}..\underline{b}_{6},\underline{a}}\
,\
[R^{\underline{a}_{1}\underline{a}_{2}\underline{a}_{3}},Z^{\underline{b}_{1}\underline{b}_{2}}]=Z^{\underline{a}_{1}\underline{a}_{2}\underline{a}_{3}\underline{b}_{1}\underline{b}_{2}}$
$[R^{\underline{a}_{1}\underline{a}_{2}\underline{a}_{3}},Z^{\underline{b}_{1}..\underline{b}_{4}}]=Z^{\underline{b}_{1}..\underline{b}_{4}[\underline{a}_{1}\underline{a}_{2},\underline{a}_{3}]}-\overline{Z}^{\underline{a}_{1}\underline{a}_{2}\underline{a}_{3}\underline{b}_{1}..\underline{b}_{4}}\
\ ,\
[R^{\underline{a}_{1}\underline{a}_{2}\underline{a}_{3}},Z^{\underline{b}_{1}..\underline{b}_{6}}]=0\
,$ $[R^{\underline{a}_{1}..\underline{a}_{5}},Z]={1\over
2}R^{\underline{a}_{1}..\underline{a}_{5}}\ ,\
[R^{\underline{a}_{1}..\underline{a}_{5}},Z^{\underline{b}_{1}\underline{b}_{2}}]=-\overline{Z}^{\underline{a}_{1}..\underline{a}_{5}\underline{b}_{1}\underline{b}_{2}}-{1\over
3}Z^{\underline{a}_{1}..\underline{a}_{5}[\underline{b}_{1},\underline{b}_{2}]}\
,$
$[R^{\underline{a}_{1}..\underline{a}_{5}},Z^{\underline{b}_{1}..\underline{b}_{4}}]=0\
,\
[R^{\underline{a}_{1}..\underline{a}_{5}},Z^{\underline{b}_{1}..\underline{b}_{6}}]=0\
,$ $[R^{\underline{a}_{1}..\underline{a}_{7}},Z]=-{3\over
2}\overline{Z}^{\underline{a}_{1}..\underline{a}_{7}}-{1\over
6}Z^{\underline{a}_{1}..\underline{a}_{7}}\ ,\
[R^{\underline{a}_{1}..\underline{a}_{7}},Z^{\underline{b}_{1}\underline{b}_{2}}]=0\
,$
$[R^{\underline{a}_{1}..\underline{a}_{7}},Z^{\underline{b}_{1}..\underline{b}_{4}}]=0\
,\
[R^{\underline{a}_{1}..\underline{a}_{7}},Z^{\underline{b}_{1}..\underline{b}_{6}}]=0\
,$ $None$

and level minus one $IIA$ with the level one $l_{1}$ commutators are given by

$[R_{\underline{a}},Z]=-P_{\underline{a}}\ ,\
[R_{\underline{a}},Z^{\underline{b}_{1}\underline{b}_{2}}]=2\delta^{[\underline{b}_{1}}{}_{\underline{a}}Q^{\underline{b}_{2}]}\
,\ [R_{\underline{a}},Z^{\underline{b}_{1}..\underline{b}_{4}}]=0\ ,\
[R_{\underline{a}},Z^{\underline{b}_{1}..\underline{b}_{6}}]=0\ ,$ $\
[R_{\underline{a}_{1}\underline{a}_{2}\underline{a}_{3}},Z]=0\ ,\
[R_{\underline{a}_{1}\underline{a}_{2}\underline{a}_{3}},Z^{\underline{b}_{1}\underline{b}_{2}}]=6\delta^{\underline{b}_{1}\underline{b}_{2}}_{[\underline{a}_{1}\underline{a}_{2}}P_{\underline{a}_{3}]}\
,$
$[R_{\underline{a}_{1}\underline{a}_{2}\underline{a}_{3}},Z^{\underline{b}_{1}..\underline{b}_{4}}]=-24\delta^{[\underline{b}_{1}\underline{b}_{2}\underline{b}_{3}}_{\underline{a}_{1}\underline{a}_{2}\underline{a}_{3}}Q^{\underline{b}_{4}]}\
,\
[R_{\underline{a}_{1}\underline{a}_{2}\underline{a}_{3}},Z^{\underline{b}_{1}..\underline{b}_{6}}]=0\
,$ $[R_{\underline{a}_{1}..\underline{a}_{5}},Z]=0,\
[R_{\underline{a}_{1}..\underline{a}_{5}},Z^{\underline{b}_{1}\underline{b}_{2}}]=0\
,\
[R_{\underline{a}_{1}..\underline{a}_{5}},Z^{\underline{b}_{1}..\underline{b}_{4}}]=72\delta^{\underline{b}_{1}..\underline{b}_{4}}_{[\underline{a}_{1}..\underline{a}_{4}}P_{\underline{a}_{5}]}\
,$
$[R_{\underline{a}_{1}..\underline{a}_{5}},Z^{\underline{b}_{1}..\underline{b}_{6}}]=-135\cdot
8\delta_{\underline{a}_{1}..\underline{a}_{5}}^{[\underline{b}_{1}..\underline{b}_{5}}Q^{\underline{b}_{6}]}\
,\ [R_{\underline{a}_{1}..\underline{a}_{7}},Z]=0\ ,\
[R_{\underline{a}_{1}..\underline{a}_{7}},Z^{\underline{b}_{1}\underline{b}_{2}}]=0\
,$
$[R_{\underline{a}_{1}..\underline{a}_{7}},Z^{\underline{b}_{1}..\underline{b}_{4}}]=0\
,\
[R_{\underline{a}_{1}..\underline{a}_{7}},Z^{\underline{b}_{1}..\underline{b}_{6}}]=8\cdot
7\cdot
135P_{[\underline{a}_{1}}\delta_{\underline{a}_{1}..\underline{a}_{7}]}^{\underline{b}_{1}..\underline{b}_{6}}\
.$ $None$

Appendix B In this section we will give some of commutators of the algebra
$I_{c}(E_{11})\otimes_{s}l_{1}$ at level zero which will be useful in section
four of this paper. In appendix A of this paper the $E_{11}\otimes_{s}l_{1}$
in its decomposition corresponding to the IIA theory was found by deleting
node ten (the IIA node) from the $E_{11}$ Dynkin diagram, leaving the algebra
$SO(10,10)\otimes GL(1)$, followed by deleting node eleven to decompose the
result into representations of GL(10). The algebra of the generators at IIA
level zero was given in equation (A.1). In what follows we will discard the
GL(1) generator $\tilde{R}$.

At IIA level zero the vector representation contains the generators
$P_{a}=\hat{P}_{a}$ and $Q^{a}=-\hat{Z}^{a11}$ and their commutators with
$E_{11}$ was given in equation (A.13). The Cartan involution acts as in
equation (2.7)

Defining $J_{ab}=K^{a}{}_{b}-\eta^{ac}\eta_{bd}K^{d}{}_{c}$ ,
$S^{ab}=R^{ab}-\eta^{ac}\eta^{bd}R_{cd}$ we find the Cartan involution
invariant algebra, $I_{c}(SO(10,10))=I_{c}(D_{10})$ is given by

$[J^{a_{1}a_{2}},J_{b_{1}b_{2}}]=-4\delta^{[a_{1}}_{[b_{1}}J^{a_{2}]}{}_{b_{2}]},\
[J^{a_{1}a_{2}},S_{b_{1}b_{2}}]=-4\delta^{[a_{1}}_{[b_{1}}S^{a_{2}]}{}_{b_{2}]},\
$
$[S^{a_{1}a_{2}},S_{b_{1}b_{2}}]=-4\delta^{[a_{1}}_{[b_{1}}J^{a_{2}]}{}_{b_{2}]},\
$ $None$

Defining $M^{a_{1}a_{2}}={1\over 2}(J^{a_{1}a_{2}}+S^{a_{1}a_{2}})$,
$\bar{M}^{a_{1}a_{2}}={1\over 2}(J^{a_{1}a_{2}}-S^{a_{1}a_{2}})$ these
commutators become

$[M^{a_{1}a_{2}},M_{b_{1}b_{2}}]=-4\delta^{[a_{1}}_{[b_{1}}M^{a_{2}]}{}_{b_{2}]},[\bar{M}^{a_{1}a_{2}},\bar{M}_{b_{1}b_{2}}]=-4\delta^{[a_{1}}_{[b_{1}}\bar{M}^{a_{2}]}{}_{b_{2}]},[M^{a_{1}a_{2}},\bar{M}_{b_{1}b_{2}}]=0$
$None$

which we recognise as the algebra $I_{c}(SO(10,10))=SO(10)\otimes SO(10)$. In
this section all our equations apply equally well to $SO(D)\otimes SO(D)$ with
suitable adjustment of the range of the indices.

The Cartan involution odd generators are
$T_{ab}=\eta_{ac}K^{c}{}_{b}+\eta_{bc}K^{c}{}_{a}$ ,
$U^{ab}=R^{ab}+\eta^{ac}\eta^{bd}R_{cd}$ and they obey the algebra

$[J^{a_{1}a_{2}},T_{b_{1}b_{2}}]=-4\delta^{[a_{1}}_{(b_{1}}T^{a_{2}]}{}_{b_{2})},\
[J^{a_{1}a_{2}},U_{b_{1}b_{2}}]=-4\delta^{[a_{1}}_{[b_{1}}U^{a_{2}]}{}_{b_{2}]},\
$
$[S^{a_{1}a_{2}},T_{b_{1}b_{2}}]=4\delta^{[a_{1}}_{(b_{1}}U^{a_{2}]}{}_{b_{2})},\
[S^{a_{1}a_{2}},U_{b_{1}b_{2}}]=4\delta^{[a_{1}}_{[b_{1}}T^{a_{2}]}{}_{b_{2}]}$
$None$

and

$[T^{a_{1}a_{2}},T_{b_{1}b_{2}}]=4\delta^{(a_{1}}_{(b_{1}}J^{a_{2})}{}_{b_{2})},\
[U^{a_{1}a_{2}},U_{b_{1}b_{2}}]=4\delta^{[a_{1}}_{[b_{1}}J^{a_{2}]}{}_{b_{2}]},\
[T^{a_{1}a_{2}},U_{b_{1}b_{2}}]=4\delta^{(a_{1}}_{[b_{1}}S^{a_{2})}{}_{b_{2}]}$
$None$

Defining the generators

$V_{a_{1}a_{2}}=(T_{a_{1}a_{2}}-U_{a_{1}a_{2}}),\
\bar{V}_{a_{1}a_{2}}=(T_{a_{1}a_{2}}+U_{a_{1}a_{2}})$ $None$

the above commutators become

$[M^{a_{1}a_{2}},V_{b_{1}b_{2}}]=-4\delta^{[a_{1}}_{b_{1}}V^{a_{2}]}{}_{b_{2}},\
[M^{a_{1}a_{2}},\bar{V}_{b_{1}b_{2}}]=-4\delta^{[a_{1}}_{b_{2}}V^{a_{2}]}{}_{b_{1}},\
$
$[\bar{M}^{a_{1}a_{2}},\bar{V}_{b_{1}b_{2}}]=-4\delta^{[a_{1}}_{b_{1}}\bar{V}^{a_{2}]}{}_{b_{2}},\
[\bar{M}^{a_{1}a_{2}},V_{b_{1}b_{2}}]=-4\delta^{[a_{1}}_{b_{2}}\bar{V}^{a_{2}]}{}_{b_{1}}$
$None$

While the odd with odd generators obey

$[V^{a_{1}a_{2}},V_{b_{1}b_{2}}]=4\delta^{a_{1}}_{b_{1}}\bar{M}^{a_{2}}{}_{b_{2}}+4\delta^{a_{2}}_{b_{2}}M^{a_{1}}{}_{b_{1}},\
[\bar{V}^{a_{1}a_{2}},\bar{V}_{b_{1}b_{2}}]=4\delta^{a_{1}}_{b_{1}}M^{a_{2}}{}_{b_{2}}+4\delta^{a_{2}}_{b_{2}}\bar{M}^{a_{1}}{}_{b_{1}},\
$
$[V^{a_{1}a_{2}},\bar{V}_{b_{1}b_{2}}]=4\delta^{a_{2}}_{b_{1}}M^{a_{1}}{}_{b_{2}}+4\delta^{a_{1}}_{b_{2}}\bar{M}^{a_{2}}{}_{b_{1}},\
$ $None$

We will now consider the vector representation and it commutators with the odd
and even parts of $D_{10}$. The Cartan involution invariant algebra
$I_{c}(D_{10})$ acts on the vector representation as

$[J_{ab},P_{c}]=-2\eta_{c[a}P_{b]},\ [J_{ab},Q_{c}]=-2\eta_{c[a}Q_{b]},\
[S_{ab},P_{c}]=-2\eta_{c[a}Q_{b]},\ ,\ [S_{ab},Q_{c}]=-2\eta_{c[a}P_{b]}$
$None$

The above equations correct a number of typographical errors in reference
[17]. While with the Cartan involution odd generators we have

$[T_{ab},P_{c}]=-2\eta_{c(a}P_{b)},\ [T_{ab},Q_{c}]=+2\eta_{c(a}Q_{b)},\
[U_{ab},P_{c}]=-2\eta_{c[a}Q_{b]},\ ,\ [U_{ab},Q_{c}]=2\eta_{c[a}P_{b]}$
$None$

The commutators of the vector representation simplify if we write the
commutators in terms of

${\cal{\wp}}_{a}=P_{a}+Q_{a},\ \bar{\cal{\wp}}_{a}=P_{a}-Q_{a},\ $ $None$

Then

$[M_{a_{1}a_{2}},{\cal{\wp}}_{b}]=-2\eta_{c[a_{1}}{\cal{\wp}}_{a_{2}]},\
[\bar{M}^{a_{1}a_{2}},{\cal{\wp}}_{b}]=0,\
[\bar{M}_{a_{1}a_{2}},\bar{\cal{\wp}}_{b}]=-2\eta_{c[a_{1}}\bar{\cal{\wp}}_{a_{2}]},\
[M^{a_{1}a_{2}},\bar{\cal{\wp}}_{b}]=0,\ $ $None$

and

$[V_{a_{1}a_{2}},{\cal{\wp}}_{b}]=-2\eta_{ba_{1}}\bar{\cal{\wp}}_{a_{2}},\
[V_{a_{1}a_{2}},\bar{\cal{\wp}}_{b}]=-2\eta_{ba_{2}}{\cal{\wp}}_{a_{1}},\ $
$[\bar{V}_{a_{1}a_{2}},\bar{\cal{\wp}}_{b}]=-2\eta_{ba_{1}}{\cal{\wp}}_{a_{2}},\
[\bar{V}_{a_{1}a_{2}},{\cal{\wp}}_{b}]=-2\eta_{ba_{2}}\bar{\cal{\wp}}_{a_{1}},\
$ $None$

Thus the generators of $I_{c}(D_{10})\otimes_{s}l_{1}$ split into two
irreducible algebras which are

$M_{a_{1}a_{2}},\ {\cal{\wp}}_{a},\ \ \ {\rm and}\ \ \ \bar{M}_{a_{1}a_{2}},\
\bar{\cal{\wp}}_{b}$ $None$

Finally we give the explicit transformations of the vector representation
which were given by $\delta P_{a}=[\Lambda,P_{a}]$ and $\delta
Q_{a}=[\Lambda,Q_{a}]$ where we take the Lie algebra element of $D_{10}$ to be
given by

$\Lambda=\Lambda^{ab}J_{ab}+\tilde{\Lambda}^{ab}S_{ab}+\Omega^{ab}T_{ab}+\hat{\Omega}^{ab}U_{ab}=\lambda^{ab}M_{ab}+\tilde{\lambda}^{ab}\bar{M}_{ab}+\mu^{ab}V_{ab}+\tilde{\mu}^{ab}\bar{V}_{ab}$
$None$

We find that under $D_{10}$ transformations

$\delta{\cal{\wp}}_{a}=-2(\Lambda+\tilde{\Lambda})_{a}{}^{b}{\cal{\wp}}_{b}-2(\Omega-\hat{\Omega})_{a}{}^{b}\bar{\cal{\wp}}_{b}=-2\lambda_{a}{}^{b}{\cal{\wp}}_{b}-2(\mu_{a}{}^{b}+\bar{\mu}^{b}{}_{a})\bar{\cal{\wp}}_{b},$
$\delta\bar{\cal{\wp}}_{a}=-2(\Lambda-\tilde{\Lambda})_{a}{}^{b}\bar{\cal{\wp}}_{b}-2(\Omega+\hat{\Omega})_{a}{}^{b}{\cal{\wp}}_{b}=-2\tilde{\lambda}_{a}{}^{b}\bar{\cal{\wp}}_{b}-2(\tilde{\mu}_{a}{}^{b}+\mu^{b}{}_{a}){\cal{\wp}}_{b},$
$None$

It is straight forward to verify that

$P_{a}^{2}+Q_{a}^{2},\ \ \ {\rm and}\ \ \ P^{a}Q_{a}\ \ \ {\rm or\
equivalently}\ \ \ {\cal{\wp}}_{a}^{2},\ \ \ {\rm and}\ \ \
\bar{\cal{\wp}}_{a}^{2}$ $None$

are invariant under $I_{c}(D_{10})$ transformations.

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