#  $\mathbb{Z}_{2}-$Graded Lie Algebra of Quaternions and Superconformal
Algebra in $D=4$ dimensions

Bhupendra C. S. Chauhan1, Pawan Kumar Joshi2, B.C. Chanyal3

###### Abstract

In the present discussion, we have studied the $\mathbb{Z}_{2}-$$grading$ of
quaternion algebra $(\mathbb{H})$. We have made an attempt to extend the
quaternion Lie algebra to the graded Lie algebra by using the matrix
representations of quaternion units. The generalized Jacobi identities of
$\mathbb{Z}_{2}-graded$ algebra then result in symmetric graded partners
$(N_{1},N_{2},N_{3})$. The graded partner algebra $(\mathcal{F})$ of
quaternions $(\mathbb{H})$ thus has been constructed from this complete set of
graded partner units $(N_{1},N_{2},N_{3})$, and $N_{0}=C$. Keeping in view the
algebraic properties of the graded partner algebra $(\mathcal{F})$, the
$\mathbb{Z}_{2}-graded$ superspace $(S^{l,m})$ of quaternion algebra
$(\mathbb{H})$ has been constructed. It has been shown that the antiunitary
quaternionic supergroup $UU_{a}(l;m;\mathbb{H})$ describes the isometries of
$\mathbb{Z}_{2}-graded$ superspace $(S^{l,m})$. The Superconformal algebra in
$D=4$ dimensions is then established, where the bosonic sector of the
Superconformal algebra has been constructed from the quaternion algebra
$(\mathbb{H})$ and the fermionic sector from the graded partner algebra
$(\mathcal{F})$.

1,2Department of Physics, Kumaun University, S. S. J. Campus, Almora – 263601
(Uttarakhand), India.

3Department of Physics, G.B. Pant University of Agriculture and Technology,
Pantnagar-263145 (Uttarakhand ), India.

Email<EMAIL_ADDRESS>
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##  1 Introduction:

In order to unify the symmetry of the Poincaré group with some internal
groups, several attempts have been made. Coleman and Mandula [1] in $1967$ set
a restriction on the incorporation of Poincaré symmetry into the internal
symmetry group, up to which this unification is possible. But actually, this
doesn’t offer any unification of the Poincaré group with the internal symmetry
group. This theorem, which is called the no-go theorem of Coleman and Mandula,
is based on the extension of the symmetry of the $S$-matrix under the
assumption of physical conditions of locality, causality, positive energy, and
for finite numbers of particles. Weiss and Zumino [2, 3] realized that the
unification of the Poincaré group with the internal symmetry group is possible
by introducing anti-commutation relations of supersymmetric charges into the
theory, which relate the fermions to bosons. However, its proof has been
established by Haag, Lapuszanski, and Sohnius [3, 4].

Thus, supersymmetric field theories arise as the maximum symmetry of the
$S$-matrix that is possible. This is the largest extension of the Lie algebra
of the Poincaré group and the internal symmetry group, which has not only
commutators but also anti-commutators of supercharges that generate the
supersymmetric transformations [3, 4]. Since this theory can be a possible
answer for most of the hierarchy problems [4] of the standard model, the
unification of gravitation, dark matter, and dark energy, several attempts
have been made experimentally in search of this symmetry. But it has not been
confirmed yet. However, it has always been an interesting theory searched by
the theoretical physicists in an attempt to unify the fundamental forces of
nature.

In the theoretical literature on supersymmetric field theories, L. Brink et
al. [5] established the fact that supersymmetry is only possible for the cases
of dimensions $D=2,4,6,$ and $10,$ called critical dimensions. They showed
that the action is supersymmetric only for $D=4,6,$ and $10$ dimensions
without the inclusion of further fields. Also, it has been shown that [3, 4,
5] non-abelian Yang–Mills fields with minimal coupling to mass-less spinors
are supersymmetric if and only if the dimension of space-time is $3,4,6$, and
$10$.

On the other hand, according to the celebrated Hurwitz theorem [6]-[11], there
are four normed-division algebras consisting of $\mathbb{R}$ (real numbers),
$\mathbb{C}$ (complex numbers), $\mathbb{H}$ (quaternions)[6]-[14], and
$\mathbb{O}$ (octonions) [10]-[17]. All four algebras are alternative with
antisymmetric associators. Real numbers and complex numbers are limited to
only two dimensions; quaternions are extended to four dimensions; and
octonions represent eight dimensions. Keeping this in mind, many authors
[18]-[30] tried to establish a connection between the supersymmetric theories
in critical dimensions and the four normed-division algebras. One such
connection was studied by Kugo-Townsend [19], who established a relation
between the supersymmetric algebra in various dimensions and the four-division
algebras. This was further extended and generalized by Jerzy Lukierski et al.
[20, 21] and other authors [22]-[30] as well. It is summarized that the
algebras $\mathbb{R}$,$\mathbb{C}$,$\mathbb{H}$,$\mathbb{O}$ can be useful for
the description of supersymmetric field theories in higher dimensions.

Keeping in view the connection between the normed division algebras
($\mathbb{R},$$\mathbb{C}$,$\mathbb{H}$, $\mathbb{O}$) and supersymmetric
theories, in the present paper, we have made an attempt to study the
$\mathbb{Z}_{2}-grading$ [31] of quaternion algebra using matrix
representations of quaternion basis units $(e_{1},e_{2},e_{3})$.

The whole paper is arranged in seven sections, including the introduction.
Section $2$ contains a basic introduction to quaternion algebra and the matrix
representations of its basis units $(e_{1},e_{2},e_{3})$. In Section 3, we
have studied the graded Lie algebra of quaternions $(\mathbb{H})$, where we
have defined the graded partner matrices evaluated by the grading of
quaternion algebra $(\mathbb{H})$. In Section $4$, we have studied the
relations between quaternion algebra $(\mathbb{H})$ and the proposed graded
partner algebra $(\mathcal{F})$ composed by the graded partner matrices
($N_{1},N_{2},N_{3}$, and $N_{0}=C$). Section $5$ contains the dynamics of the
superspace constructed from the quaternion algebra $(\mathbb{H})$ and the
graded partner algebra $(\mathcal{F})$ under the quaternionic supergroups
[20]. In Section $6$, the Superconformal algebra in $D=4$ has been established
in terms of quaternions $(\mathbb{H})$ and its graded partner algebra
$(\mathcal{F})$. Section $7$ is for discussion and conclusions.

##  2 Quaternion Algebra $(\mathbb{H})$ (Definition):

According to the celebrated Hurwitz theorem [7, 11], there exist four normed
division algebras: $\mathbb{R}$,$\mathbb{C}$,$\mathbb{H}$, and $\mathbb{O}$,
respectively, named as the algebras of real numbers, complex numbers,
quaternions, and octonions. The quaternion algebra $(\mathbb{H})$ is the
second largest normed division algebra, which is non-commutative but
associative. Any element of this quaternion algebra $(\mathbb{H})$ is called a
quaternion, which is expressed over the field of real numbers as

$\displaystyle Q=$ $\displaystyle
q^{0}e_{0}+q^{1}e_{1}+q^{2}e_{2}+q^{3}e_{3}.\quad\quad\quad(Q\in\mathbb{H}).$
(1)

Where $q^{0}$ and $q^{j}\,\>(\forall j=1,2,3)$ are the real numbers and the
quaternion basis elements $e_{1},e_{2},e_{3}$ satisfy the following
multiplication rules:

$\displaystyle e_{i}e_{j}=$
$\displaystyle-\delta_{ij}+\in_{ijk}e_{k}\;\;\;\;(\forall i,j,k=1\,to\,3).$
(2)

The $\in_{ijk}$ is a Levi-Civita tensor, which is totally antisymmetric and
has a value of $+1$ for the permutations: $(ijk)=(123),(231),(312)$.

The quaternion conjugate is defined by
$\overline{Q}=q^{0}e_{0}-q^{1}e_{1}-q^{2}e_{2}-q^{3}e_{3}\>(q\in\mathbb{H}).$
It is to be noted that the quaternion conjugation operation satisfies the
following composition rule:

$\displaystyle\overline{(Q_{1}Q_{2})}=$
$\displaystyle\overline{Q}_{2}\overline{Q}_{1}\>\>\>\>\>\>\>\>\>\>(\forall
Q_{1},Q_{2}\in\mathbb{H}).$ (3)

The norm of a quaternion is positive-definite and computed as

$\displaystyle N(Q)=$ $\displaystyle
Q\overline{Q}=\overline{Q}Q=\left|Q\right|=(q^{0})^{2}+(q^{1})^{2}+(q^{2})^{2}+(q^{3})^{2}\geq
0.$ (4)

Quaternion is the second-highest normed division algebra; the norm of
quaternion satisfies the following multiplication rule:

$\displaystyle N(Q_{1})N(Q_{2})=$ $\displaystyle
N(Q_{1}Q_{2})\>\>\>\>\>\>\>\>\>\>(\forall Q_{1},Q_{2}\in\mathbb{H}).$ (5)

Following the eq. (4), the inverse of a quaternion is defined by
$Q^{-1}=\frac{\overline{Q}}{\left|Q\right|}$. Meanwhile, the inverse of the
multiplication of two quaternions has the following property:

$\displaystyle(Q_{1}Q_{2})^{-1}=$ $\displaystyle
Q_{1}^{-1}Q_{2}^{-1}\>\>\>\>\>\>\>\>\>\>(\forall Q_{1},Q_{2}\in\mathbb{H}).$
(6)

A $2\times 2$ quaternionic matrix is defined as

$\displaystyle H=$ $\displaystyle\left[\begin{array}[]{cc}h_{1}&h_{2}\\\
h_{3}&h_{4}\end{array}\right]\>\>\>\>\>\>\>\>\>\>(\forall
h_{1},h_{2},h_{3},h_{4}\in\mathbb{H}).$ (9)

Where $h_{1},h_{2},h_{3},$ and $h_{4}$ are the elements of quaternion algebra
$(\mathbb{H}).$ The Hermitian conjugate of this quaternionic matrix is defined
as

$\displaystyle H^{\dagger}=$
$\displaystyle\left[\begin{array}[]{cc}\overline{h}_{1}&\overline{h}_{3}\\\
\overline{h}_{2}&\overline{h}_{4}\end{array}\right],$ (12)

where $\overline{h}_{1},\overline{h}_{2},\overline{h}_{3},\overline{h}_{4}$
are the quaternion conjugates of $h_{1},h_{2},h_{3},h_{4}$, respectively. The
Hermitian conjugation operation in quaternionic matrices satisfies the
following composition rule:

$\displaystyle(H_{1}H_{2})^{\dagger}=$ $\displaystyle
H_{2}^{\dagger}H_{1}^{\dagger}.$ (13)

In a $4\times 4$ real matrix representation, the quaternion basis elements are
defined [20] as

$\displaystyle e_{1}=$ $\displaystyle\left[\begin{array}[]{cccc}0&1&0&0\\\
-1&0&0&0\\\ 0&0&0&-1\\\
0&0&1&0\end{array}\right],\>\>e_{2}=\left[\begin{array}[]{cccc}0&0&1&0\\\
0&0&0&1\\\ -1&0&0&0\\\
0&-1&0&0\end{array}\right],\,\,e_{3}=\left[\begin{array}[]{cccc}0&0&0&1\\\
0&0&-1&0\\\ 0&1&0&0\\\ -1&0&0&0\end{array}\right].$ (26)

These basis elements satisfy the following commutation algebra:

$\displaystyle[e_{i},e_{j}]=$ $\displaystyle
2\in_{ijk}e_{k}\quad\quad\quad(\forall i,j,k=1\,to\,3).$
$\displaystyle{}[e_{p},[e_{q},e_{r}]]+[e_{q},[e_{r},e_{p}]]+[e_{r},[e_{p},e_{q}]]=$
$\displaystyle 0\quad\quad\quad\quad\quad\quad\quad\>\>(\forall
p,q,r=1\,to\,3).$ (27)

The matrix representations of quaternion basis units defined in eq. (26) are
real and unitary, i.e., $e_{i}^{T}=-e_{i}=e_{i}^{-1}=e_{i}^{\dagger}$
$(\forall i=1,2,\,and\,3)$.

##  3 Graded Lie Algebra of Quaternions $(\mathbb{H})$:

The $Z_{2}$-graded algebra $L$ is the direct sum of two algebras [30, 31],
with $L_{0}\oplus L_{1}$ having the following properties:

(i) $L_{0}$ is an even Lie algebra with degree $g(L_{0})=0$ and $L_{0}\times
L_{0}\rightarrow L_{0}$

(ii) $L_{1}$ is an odd Lie algebra with degree $g(L_{1})=1$ and $L_{0}\times
L_{1}\rightarrow L_{1}$ , $L_{1}\times L_{1}\rightarrow L_{0}$

(iii) Representation of $L_{0}$ in $dim\,L_{1}\times dim\,L_{1}$

Now to construct $Z_{2}$-graded algebra for quaternions, we take the $4\times
4$ dimensional representation of quaternions defined in eq. (26). These
representations of quaternions satisfy the Lie algebra described in eq. (27).
This Lie algebra is closed, and hence it is an even algebra, $L_{0}$, that
constructs the bosonic part of the graded Lie algebra $(L)$. While the odd
algebra $L_{1}$ is not closed, as one can see in the second (ii) axiom of the
above-described properties of $Z_{2}-$graded algebra, In a mathematical way,
we now summarize these assertions for the $Z_{2}$-graded algebra $(L)$ of
quaternions $(\mathbb{H})$ as:

$\displaystyle 1.\>L_{0}=$ $\displaystyle\\{e_{i}\in
L_{0},\;\;[e_{i},e_{j}]=2\in_{ijk}e_{k},\>(\forall i,j,k=1\,to\,3)\\}$
$\displaystyle 2.\>L_{1}=$ $\displaystyle\\{Q_{a}\in L_{1},(\forall
a=1\,to\,4)\>\>[Q_{a},e_{i}]=\left(e_{i}\right)_{ab}Q_{b}\in L_{1},\,$
$\displaystyle and$
$\displaystyle\,\left\\{Q_{a},Q_{b}\right\\}=(N_{i})_{ab}e_{i}\in
L_{0}\;(\forall i=1\,to\,3,\forall a,b=1\,to\,4)\\},$ (28)

where the $N_{i}\,(\forall i=1\,to\,3)$ must be symmetric, i.e.,
$N_{i}=N_{i}^{T}\,(\forall i=1\,to\,3)$. The $Z_{2}$-graded algebra $(L$) of
quaternions is the direct sum of the two algebras as $L=L_{0}\oplus L_{1}.$
Now, to evaluate the representations of the symmetric $N_{i}$ matrices, we
consider the following generalized Jacobi identity [31]:

$\displaystyle[e_{l},\\{Q_{a},Q_{b}\\}]+\\{Q_{b},[e_{l},Q_{a}]\\}+\\{Q_{a},[Q_{b},e_{l}]\\}=$
$\displaystyle 0.$ (29)

Using the relations of graded Lie algebra defined in eq. (28), we get the
following simplified form of eq. (29) as

$\displaystyle
2(N_{m})_{ab}e_{n}\in_{lmn}+(e_{l})_{ac}(N_{m})_{bc}e_{m}+(e_{l})_{bc}(N_{m})_{ac}e_{m}=$
$\displaystyle 0,$ $\displaystyle e_{l}N_{m}+(e_{l}N_{m})^{T}=$ $\displaystyle
2\in_{lmn}N{}_{n}$ $\displaystyle e_{l}N_{m}-N_{m}e_{l}=$ $\displaystyle
2\in_{lmn}N{}_{n}\,\>\>\>\>\>\>\>\>(\forall
a,b,c=1\,to\,4\>\>\>and\,\,\,\forall l,m,n=1,2,3).$ (30)

The last line of the above eq. is obtained by using the symmetric property of
the $N_{i}$ matrix and noting the transpose properties of $e_{l}\>\>(\forall
l=1,2,3)$ from eq. (26). Now, we define the $N_{i}\,(\forall i=1,2,3)$
matrices as

$\displaystyle N_{i}=$
$\displaystyle\left[\begin{array}[]{cccc}a_{i}&b_{i}&c_{i}&d_{i}\\\
w_{i}&f_{i}&g_{i}&h_{i}\\\ p_{i}&q_{i}&r_{i}&s_{i}\\\
l_{i}&m_{i}&n_{i}&t_{i}\end{array}\right]\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\forall
i=1,2,3).$ (35)

Keeping in mind the symmetric conditions imposed on $N_{i}\,^{\prime}s(\forall
i=1,2,3)$ matrices, i.e., $N_{i}=N_{i}^{T}$, we get $b_{i}=w_{i}$,
$c_{i}=p_{i},$$q_{i}=g_{i}$, $l_{i}=d_{i}$, $h_{i}=m_{i}$, and $s_{i}=n_{i}$.
Hence, the $N_{1},N_{2},N_{3}$ matrices are defined as:

$\displaystyle N_{1}=\left[\begin{array}[]{cccc}a_{1}&b_{1}&c_{1}&d_{1}\\\
b_{1}&f_{1}&g_{1}&h_{1}\\\ c_{1}&g_{1}&r_{1}&s_{1}\\\
d_{1}&h_{1}&s_{1}&t_{1}\end{array}\right],\>\>N_{2}=\left[\begin{array}[]{cccc}a_{2}&b_{2}&c_{2}&d_{2}\\\
b_{2}&f_{2}&g_{2}&h_{2}\\\ c_{2}&g_{2}&r_{2}&s_{2}\\\
d_{2}&h_{2}&s_{2}&t_{2}\end{array}\right]$
$\displaystyle,\>\>N_{3}=\left[\begin{array}[]{cccc}a_{3}&b_{3}&c_{3}&d_{3}\\\
b_{3}&f_{3}&g_{3}&h_{3}\\\ c_{3}&g_{3}&r_{3}&s_{3}\\\
d_{3}&h_{3}&s_{3}&t_{3}\end{array}\right].$ (48)

From eq. (30), we have the following relations for $l=m$ $(\forall l=1,2,3)$:

$\displaystyle e_{1}N_{1}-N_{1}e_{1}=$ $\displaystyle 0$ $\displaystyle
e_{2}N_{2}-N_{2}e_{2}=$ $\displaystyle 0$ $\displaystyle
e_{3}N_{3}-N_{3}e_{3}=$ $\displaystyle 0.$ (49)

By putting the values of $N_{1}$ from eq. (48) and $e_{1}$ from eq.(26) into
the first relation of the eq.(30) above, we get the following conditions for
the matrix elements of $N_{1}$: $2b_{1}=0=2s_{1}$, $f_{1}-a_{1}=0$,
$g_{1}-d_{1}=0$, $h_{1}+c_{1}=0$, and $r_{1}-t_{1}=0$, hence we have the
following form of $N_{1}$:

$\displaystyle N_{1}=$
$\displaystyle\left[\begin{array}[]{cccc}a_{1}&0&c_{1}&d_{1}\\\
0&a_{1}&d_{1}&-c_{1}\\\ c_{1}&d_{1}&r_{1}&0\\\
d_{1}&-c_{1}&0&r_{1}\end{array}\right].$ (54)

Further, by evaluating the trace of the eq. (30), we have:

$\displaystyle Tr(e_{l}N_{m})-Tr(N_{m}e_{l})=$ $\displaystyle
2\in_{lmn}Tr(N{}_{n})\,\>\>\>\>\>\>\>\>(\forall l,m,n=1,2,3)$ $\displaystyle
0=$ $\displaystyle Tr(N_{n})\,\>\>\>\>\>\>\>\>(\forall n=1,2,3).$ (55)

Hence, the trace of $N_{n}^{\prime}s\>(\forall n=1,2,3)$ matrices must be
equal to zero. Keeping this in mind, we put $a_{1}=r_{1}=0$, and then the
$N_{1}$ matrix has the following form:

$\displaystyle N_{1}=$
$\displaystyle\left[\begin{array}[]{cccc}0&0&c_{1}&d_{1}\\\
0&0&d_{1}&-c_{1}\\\ c_{1}&d_{1}&0&0\\\ d_{1}&-c_{1}&0&0\end{array}\right].$
(60)

From a similar procedure, using the second relation of eq. (49) and the value
of $N_{2}$ from eq. (48), we evaluate the following conditions for the $N_{2}$
matrix elements: $2c_{2}=0=2h_{2}$, $g_{2}+d_{2}=0,$ $r_{2}-a_{2}$=0,
$s_{2}-b_{2}=0$, and $t_{2}-f_{2}=0,$ hence we have the following
representation for $N_{2}$:

$\displaystyle N_{2}$
$\displaystyle=\left[\begin{array}[]{cccc}a_{2}&b_{2}&0&d_{2}\\\
b_{2}&f_{2}&-d_{2}&0\\\ 0&-d_{2}&a_{2}&b_{2}\\\
d_{2}&0&b_{2}&f_{2}\end{array}\right].$ (65)

Noting the traceless nature of the $N_{2}$ matrix, we further impose the
conditions as $a_{2}=f_{2}=0;$ hence, we have the following form of the
$K_{2}$ matrix:

$\displaystyle N_{2}$
$\displaystyle=\left[\begin{array}[]{cccc}0&b_{2}&0&d_{2}\\\
b_{2}&0&-d_{2}&0\\\ 0&-d_{2}&0&b_{2}\\\ d_{2}&0&b_{2}&0\end{array}\right].$
(70)

Now, from eq. (30) for $l=3$ and $m=1$, and using the values of $N_{1}$,
$N_{2}$ from eqs. (65) and (70), and $e_{3}$ from eq. (26), we may have,

$\displaystyle e_{3}N_{1}-N_{1}e_{3}=$ $\displaystyle 2N_{2}$
$\displaystyle\left[\begin{array}[]{cccc}d_{1}&-c_{1}&0&0\\\
-c_{1}&-d_{1}&0&0\\\ 0&0&d_{1}&-c_{1}\\\ 0&0&-c_{1}&-d_{1}\end{array}\right]=$
$\displaystyle\left[\begin{array}[]{cccc}0&b_{2}&0&d_{2}\\\
b_{2}&0&-d_{2}&0\\\ 0&-d_{2}&0&b_{2}\\\ d_{2}&0&b_{2}&0\end{array}\right].$
(79)

Hence, we get the conditions: $d_{1}=d_{2}=0$ and $c_{1}=-b_{2}$. Therefore,
we have the following form of $N_{1}$ and $N_{2}$:

$\displaystyle N_{1}=$ $\displaystyle\left[\begin{array}[]{cccc}0&0&c_{1}&0\\\
0&0&0&-c_{1}\\\ c_{1}&0&0&0\\\
0&-c_{1}&0&0\end{array}\right],\,\,\,\,\,\,\,\,\,N_{2}=\left[\begin{array}[]{cccc}0&-c_{1}&0&0\\\
-c_{1}&0&0&0\\\ 0&0&0&-c_{1}\\\ 0&0&-c_{1}&0\end{array}\right].$ (88)

By using a similar procedure for the $N_{3}$ matrix and using the relations
evaluated from eq.(30) as

$\displaystyle e_{3}N_{3}-N_{3}e_{3}=$ $\displaystyle 0$ $\displaystyle
e_{1}N_{2}-N_{2}e_{1}=$ $\displaystyle 2N_{3}$ $\displaystyle
e_{3}N_{1}-N_{1}e_{3}=$ $\displaystyle 2N_{2},$ (89)

we have the following conditions for $N_{3}$ matrix elements :
$d_{3}=g_{3}=0,$ $h_{3}=-c_{3}=d_{2}=0,$ $s_{3}=-b_{3}$$=f_{2}=0$,
$t_{3}=b_{2}=a_{3}$ $=-f_{3}=-r_{3}=-c_{1}$, $b_{3}=f_{2}=0$, and
$a_{2}=-b_{3}=0.$ Hence, we have the following form of the $N_{3}$ matrix:

$\displaystyle N_{3}=$
$\displaystyle\left[\begin{array}[]{cccc}-c_{1}&0&0&0\\\ 0&c_{1}&0&0\\\
0&0&c_{1}&0\\\ 0&0&0&-c_{1}\end{array}\right].$ (94)

By eqs. (88) and (94), we see that the $c_{1}$ is arbitrary. Now taking the
value of $c_{1}=1$, we have the following matrix representations for
$N_{1},N_{2}$, and $N_{3}$:

$\displaystyle N_{1}=$ $\displaystyle\left[\begin{array}[]{cccc}0&0&1&0\\\
0&0&0&-1\\\ 1&0&0&0\\\
0&-1&0&0\end{array}\right],\,\,N_{2}=-\left[\begin{array}[]{cccc}0&1&0&0\\\
1&0&0&0\\\ 0&0&0&1\\\
0&0&1&0\end{array}\right],\,\,N_{3}=\left[\begin{array}[]{cccc}-1&0&0&0\\\
0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&-1\end{array}\right].$ (107)

Together with the matrices in eq.(26), they constitute the graded Lie algebra
space of quaternions (eq. (28)). The other Jacobi’s identity:

$\displaystyle[Q_{p},\\{Q_{q},Q_{r}\\}]+[Q_{q},\\{Q_{r},Q_{p}\\}]+[Q_{r},\\{Q_{p},Q_{q}\\}]=$
$\displaystyle 0\>\>\>\>\>(\forall p,q,r=1\,to\,4),$ (108)

can also be shown to be satisfied as well. Using eq. (28), the above identity
changes to:

$\displaystyle(N_{j})_{qr}(e_{j})_{ps}+(N_{j})_{rp}(e_{j})_{qs}+(N_{j})_{pq}(e_{j})_{rs}=$
$\displaystyle 0\>\>\>\>\>(\forall p,q,r,s=1\,to\,4\>\>\>and\>\>\>\forall
j=1,2,3).$ (109)

The above equation is again satisfied by the matrix representations of
$e_{j}\,$ and $N_{j}\,$, as can be shown by directly putting the values of
$e_{j}$ and $N_{j}$ matrices from eqs. (26) and (107) into it. The $N_{r}\,$
matrices defined in eq. (107) are symmetric and unitary as well. Also, they
are involutory, since we have $N_{r}^{2}=I\,(\forall r=1,2,3)$. Now one can
see that the $N_{r}\,$ matrices form the graded Lie algebra representation in
eq. (28), hence they will be further abbreviated as graded partners of
quaternion basis units $(e_{i}$) throughout the whole paper.

It can be seen by simple calculations from eq. (107) and (26) that the graded
partner matrices $N_{1},N_{2},N_{3}$ satisfy the following multiplication
rules:

$\displaystyle N_{r}N_{s}=$
$\displaystyle\delta_{rs}-\in_{rst}e_{t}\,\,\,\>\>\>\>\>\>\>(\forall
r,s,t=1\,to\,3)$ $\displaystyle N_{r}N_{s}+N_{s}N_{r}=$ $\displaystyle
2\delta_{rs}\,\,\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>(\forall
r,s=1\,to\,3).$ (110)

Hence, the graded partners $N_{r}$ anti-commute with each other. Similarly,
one can evaluate the multiplication rules between the matrix representation of
quaternion basis units $e_{t}\>(\forall t=1\,to\,3)$ and the symmetric partner
matrices $N_{r}\>(\forall r=1\,to\,3)$ as

$\displaystyle N_{r}e_{s}=$
$\displaystyle\delta_{rs}C+\in_{rst}N_{t}\,\,\,\,\,\,\,\,\,\>\>(\forall
r,s,t=1\,to\,3)$ $\displaystyle N_{r}e_{s}+e_{s}N_{r}=$ $\displaystyle
2\delta_{rs}C\>\>\>\>\>\>\>\>\>\>\>\>\,\,\,\,\,\,\,\,\,\>\>\>\>(\forall
r,s,t=1\,to\,3).$ (111)

Where the matrix $C$ is evaluated as

$\displaystyle C$ $\displaystyle=\left[\begin{array}[]{cccc}0&0&0&-1\\\
0&0&-1&0\\\ 0&1&0&0\\\ 1&0&0&0\end{array}\right].$ (116)

The matrix $C$ is unitary and has the following properties:
$C=-C^{T}=-C^{\dagger}=-C^{-1}$, with $C^{2}=-I$. The eq. (111) also shows
that the quaternion basis units $e_{t}$ and the symmetric partner matrices
$N_{r}$ anticommutate with each other. The multiplication operation of $C$
with the basis elements $e_{j}\,$ and $K_{r}\,$ maps them into each other, as
can be seen from eqs. (26), (107), and (116) as

$\displaystyle Ce_{j}=$ $\displaystyle-
N_{j}\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\forall j=1\,to\,3)$ $\displaystyle CN_{r}=$
$\displaystyle e_{r}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\forall
r=1\,to\,3).$ (117)

Now one can evaluate the commutator bracket relations for the quaternion units
$e_{j}\,(\forall j=1\,to\,3)$ and their graded partners $N_{r}\,(\forall
r=1\,to\,3)$ with $C$ as

$\displaystyle[e_{i},e_{j}]=$ $\displaystyle
2\in_{ijk}e_{k}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\forall i,j,k=1\,to\,3)$
$\displaystyle{}[N_{r},N_{s}]=$
$\displaystyle-2\in_{rst}e_{t}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\forall
r,s,t=1\,to\,3)$ $\displaystyle{}[N_{l},e_{m}]=$ $\displaystyle
2\in_{lmn}N_{n}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\forall l,m,n=1\,to\,3)$
$\displaystyle{}[C,e_{j}]=$ $\displaystyle
0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\forall
j=1\,to\,3)$ $\displaystyle{}[C,N_{r}]=$ $\displaystyle
0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\forall
r=1\,to\,3).$ (118)

Also, from the multiplication rules of eqs. (110) and (111), one can see that
the quaternion units $e_{j}\,(\forall j=1\,to\,3)$ and their graded partners
$N_{r}\,(\forall r=1\,to\,3)$ satisfy the following Jacobi identities of Lie
algebra:

$\displaystyle[e_{l},[e_{m},N_{n}]]+[e_{m},[N_{n},e_{l}]]+[N_{n},[e_{l},e_{m}]]=$
$\displaystyle 0\>\>\>\>\>\>\>(\forall l,m,n=1\,to\,3)$
$\displaystyle{}[e_{l},[N{}_{m},N_{n}]]+[N{}_{m},[N_{n},e_{l}]]+[N_{n},[e_{l},N{}_{m}]]=$
$\displaystyle 0\>\>\>\>\>\>\>(\forall l,m,n=1\,to\,3)$
$\displaystyle{}[N_{l},[N{}_{m},N_{n}]]+[N{}_{m},[N_{n},N_{l}]]+[N_{n},[N_{l},N{}_{m}]]=$
$\displaystyle 0\>\>\>\>\>\>\>(\forall l,m,n=1\,to\,3).$ (119)

It is to be noted that from the commutation relations of eq. (118), the unit
$C$ commutes with both the quaternion $(\mathbb{H})$ basis units $(e_{i})$ and
their graded partners $(N_{r})$; hence, it corresponds to the Casimir element
for the Lie algebra $(T=\\{e_{1},e_{2},e_{3},N_{1},N_{2},N_{3}\\})$ made by
the quaternion $(\mathbb{H})$ basis units $(e_{i})$ and their graded partners
$(N_{r})$. Any state in the linear space of this Lie algebra
$(T=\\{e_{1},e_{2},e_{3},N_{1},N_{2},N_{3}\\})$ is mapped by $C$ on to the
linear space itself as

$\displaystyle C$
$\displaystyle|e_{j},N_{l}>=\lambda|-N_{j},e_{l}>\>\>\>\>\>\>\>\>(\forall
j,l=1\,to\,3).$ (120)

Also, the Lie algebra $(T)$, which is a non-abelian Lie algebra, doesn’t have
any non-zero proper ideals. The Cartan-Killing form for this Lie algebra is
calculated and comes out to be invertible as
$B(x,y)=$$Tr(Ad(x)Ad(y))$$=16I_{4}$ {$\forall x,\,y\in T\\}.$

##  4 Quaternions $(\mathbb{H})$ and Graded Partner Algebra
$(\mathbb{\mathcal{F}}):$

Now one may define the graded partner algebra $(\mathbb{\mathcal{F}}),$
corresponding to the quaternion algebra $(\mathbb{H}),$ which has basis units
$\\{C,N_{1},N_{2},N_{3}\\}$. Any graded partner vector $(F)$ over the field of
real numbers is defined in $\mathbb{\mathcal{F}}$ as

$\displaystyle F=$ $\displaystyle
f^{0}N_{0}+f^{1}N_{1}+f^{2}N_{2}+f^{3}N_{3}=f^{0}N_{0}+f^{r}N_{r}\;\;\;\;\;\;\;\;\;\;\;\;(\forall
F\subset\mathbb{\mathcal{F}}),$ (121)

where $N_{0}=C$ and $N_{r}\>(\forall r=1\,to\,3)$ are the graded partner basis
units, and $f^{0},$$f^{r}\>(\forall r=1\,to\,3)$ are the real numbers. Now one
can evaluate that the product of any two graded partner vectors
$(F_{1},F_{2})$ in $\mathbb{\mathcal{F}}$ results in a quaternion by using the
multiplication rules of eq. (110) as

$\displaystyle F_{1}F_{2}=$
$\displaystyle(f_{1}^{0}N_{0}+f_{1}^{1}N_{1}+f_{1}^{2}N_{2}+f_{1}^{3}N_{3})(f_{2}^{0}N_{0}+f_{2}^{1}N_{1}+f_{2}^{2}N_{2}+f_{3}^{3}N_{3})$
$\displaystyle=$
$\displaystyle(-f_{1}^{0}f_{2}^{0}+f_{1}^{r}f_{2}^{r})e_{0}+(f_{1}^{0}f_{2}^{r}+f_{1}^{r}f_{2}^{0}-\in_{pqr}f_{1}^{p}f_{2}^{q})e_{r}\>\>\>$
(122)

where $e_{0}$ and $e_{r}\>\>(\forall r=1,2,3)$ are the quaternion basis units.
So, it is clear from eq. (122) that the multiplication operation in the linear
space of the graded partner algebra $(\mathbb{\mathcal{F}})$ results in the
quaternion linear space $(\mathbb{H})$ as

$\displaystyle:\mathbb{\mathcal{F}}\times\mathbb{\mathcal{F}}\longrightarrow$
$\displaystyle\mathbb{H}.$ (123)

Also, using the multiplication rules of eq.(111), one can confirm that any
multiplication between a quaternion $(Q)$ and graded partner vector $(F)$
results in a graded partner vector, as

$\displaystyle QF=$
$\displaystyle(q^{0}e_{0}+q^{1}e_{1}+q^{2}e_{2}+q^{3}e_{3})(f^{0}N_{0}+f^{1}N_{1}+f^{2}N_{2}+f^{3}N_{3})$
$\displaystyle=$
$\displaystyle(q^{0}f^{0}+q^{r}f^{r})N_{0}+(q^{0}f^{r}-q^{r}f^{0}+\in_{str}q^{s}f^{t})N_{r}\>\>\>\>\>\>\>(Q\in\mathbb{H},F\in\mathcal{F}),$
(124)

where $N_{0}$ and $N_{r}\>\>(\forall r=1,2,3)$ are the graded partner basis
units. So, the multiplication operation between any graded partner vector
$(F)$ in $\mathbb{\mathcal{F}}$ and any quaternion element $(Q)$ results in
the space of graded partner algebra $(\mathbb{\mathcal{F}})$ as

$\displaystyle:\mathbb{\mathcal{F}}\times\mathbb{\mathbb{H}}\longrightarrow$
$\displaystyle\mathbb{\mathbb{\mathcal{F}}}.$ (125)

Keeping in view the positivity and definiteness of the norm of any vector, we
define the norm of a graded partner vector in the space of graded partner
algebra $(\mathcal{F})$ as

$\displaystyle N(F)=F\widetilde{F}=$
$\displaystyle(f^{0})^{2}+(f^{1})^{2}+(f^{2})^{2}+(f^{3})^{2}=\left|F\right|\geq
0\>\>\>\>\>\>\>\>\>\>\>\>\>\>(\forall F\subset\mathcal{F}).$ (126)

Where $\widetilde{F}$ is the graded partner conjugate of $F$ defined as

$\displaystyle\widetilde{F}=$
$\displaystyle-f^{0}N_{0}+f^{1}N_{1}+f^{2}N_{2}+f^{3}N_{3}.$ (127)

The quaternion conjugate of the product of two graded partner vectors is
numerically equal to the product of those graded partner vectors conjugated
separately as

$\displaystyle\overline{F_{1}F_{2}}=$
$\displaystyle\widetilde{F_{2}}\widetilde{F_{1}},\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>(\forall
F_{1},F_{2}\subset\mathcal{F}).$ (128)

Where $\overline{F_{1}F_{2}}$ corresponds to the quaternion conjugate of
$F_{1}F_{2}$ since, according to the eq. (122), multiplication of two graded
partner vectors $(F_{1},\,F_{2})$ results in a quaternion, while
$\widetilde{F_{1}}$ and $\widetilde{F_{2}}$ are the graded partner conjugate
vectors of $F_{1}$ and $F_{2}$ defined in eq. (127). The graded partner
vectors are not commutative, but for real parts, they are as

$\displaystyle Re(F_{1}F_{2})=$ $\displaystyle
Re(F_{2}F_{1})=-f_{1}^{0}f_{2}^{0}+f_{1}^{1}f_{2}^{1}+f_{1}^{2}f_{2}^{2}+f_{1}^{3}f_{2}^{3}\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>(\forall
F_{1},F_{2}\subset\mathcal{F}).$ (129)

The inverse of the multiplication of two graded partners satisfies the
following relation:

$\displaystyle(F_{1}F_{2})^{-1}=$ $\displaystyle
F_{2}^{-1}F_{1}^{-1}\>\>\>\>\>\>\>\>\>\>\>\>\>\>(\forall
F_{1},F_{2}\subset\mathcal{F}).$ (130)

Where the inverse of a graded vector $(F^{-1})$ is defined by eq. (126) as

$\displaystyle F^{-1}=$ $\displaystyle\frac{\widetilde{F}}{\left|F\right|}.$
(131)

Now any graded partner valued $2\times 2$ matrix is defined as

$\displaystyle M=$ $\displaystyle\left[\begin{array}[]{cc}m_{1}&m_{2}\\\
m_{3}&m_{4}\end{array}\right],\>\>\>\>\>\>\>\>(\forall
m_{1},m_{2},m_{3},m_{4}\in\mathcal{F}).$ (134)

One can evaluate from eq. (122) that the multiplication of two graded partner-
valued matrices results in a quaternionic matrix. It can be seen that due to
the anticommutative nature of the multiplication between graded partner units
in eq. (110), $(XY)^{T}\neq Y^{T}X^{T}$, where $X$ and $Y$ are the graded
partner valued matrices. Let $X$ and $Y$ be two general graded valued matrices
compatible with the multiplication, then we have:

$\displaystyle(XY)_{ij}^{T}=$ $\displaystyle(XY)_{ji}$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{n}X_{jk}Y_{ki}$ $\displaystyle\neq$
$\displaystyle\sum_{k=1}^{n}Y_{ki}X_{jk}=\sum_{k=1}^{n}(Y^{T})_{ik}(X^{T})_{kj}=(Y^{T}X^{T})_{ij}$
$\displaystyle\therefore\>\>(XY)^{T}\neq$ $\displaystyle Y^{T}X^{T}.$ (135)

In a similar way to the composition rule of eq. (124), we evaluate that the
multiplication of a quaternionic matrix with a graded partner-valued matrix
results in a graded partner-valued matrix. Again, due to the anticommutative
nature of the multiplication between quaternion basis elements and graded
partner units in eq. (111), $(XH)^{T}\neq H^{T}X^{T},$ where $X$ is the graded
partner valued matrix and $H$ is the quaternion valued matrix.

The Hermitian conjugate for any $2\times 2$ graded partner-valued matrix $(X)$
is defined as

$\displaystyle M^{\dagger*}=\widetilde{M}^{T}=$
$\displaystyle\left[\begin{array}[]{cc}\widetilde{m}_{1}&\widetilde{m}_{3}\\\
\widetilde{m}_{2}&\widetilde{m}_{4}\end{array}\right],\>\>\>\>\>\>\>\>(\forall
m_{1},m_{2},m_{3},m_{4}\in\mathcal{F}).$ (138)

Where $\widetilde{m}_{j}$ is the graded partner conjugate of $m_{j}$ defined
in eq.(127).

Also, the quaternionic Hermitian conjugate of the multiplication of two graded
partner matrices is numerically equal to the product of the Hermitian
conjugates of those two graded partner valued matrices separately:

$\displaystyle\left(XY\right)^{\dagger}=$ $\displaystyle
Y^{\dagger\ast}X^{\dagger\ast}.$ (139)

Where $X$ and $Y$ are two graded partner matrices compatible with the
multiplication and ${}^{\prime}\dagger^{\prime}$ for the quaternionic
Hermitian conjugation operation (because the multiplication $XY$ results in a
quaternionic matrix ) defined in eq. (12). One can see that this matrix
multiplication property can be proved easily. Let the left-hand side of the
eq.(139) be calculated as

$\displaystyle(XY)_{ij}^{\dagger}=$
$\displaystyle(\overline{XY})_{ij}^{T}=\sum_{k=1}^{n}(\overline{X_{jk}Y_{ki}})$
$\displaystyle\quad\quad\quad\quad=\sum_{k=1}^{n}(\widetilde{Y_{ki}}\widetilde{X_{jk}})$
$\displaystyle\quad\quad\quad\quad=\sum_{k=1}^{n}(\widetilde{Y}^{T})_{ik}(\widetilde{X}^{T})_{kj}=(\widetilde{Y}^{T}\widetilde{X}^{T})_{ij}=(Y^{\dagger\ast}X^{\dagger\ast})_{ij}.$
(140)

The second line of the above equation is derived from the multiplication rule
defined in eq.(128). Since the multiplication of two graded partner vectors
$X_{jk}$ and $Y_{ki}$ results in a quaternion element,
$\overline{X_{jk}Y_{ki}}$ corresponds to the quaternion conjugation operation.

##  5 Superspace of Quaternions $(S)$ and Quaternionic Supergroups:

In the previous two sections we have extended the quaternion Lie algebra into
the $\mathbb{Z}_{2}-graded$ Lie algebra by introducing the graded-partner
matrices. In order to construct a menifest supersymmetric model, the
quaternionic linear vector space has to extended to a superspace. Basically
superspace is the coordinate space of a field theory which exihibits
supersymmetry. Superspace contains all the materials to define supersymmetric
dynamics of the system, same as the usual Minkowski $(x^{\mu})$ system has do
for a non-supersymmetric quantum field theory. Supersymmetry is the largest
extension of the Poincaré symmetry that contain not only the bosonic degree of
freedom but also the fermionic degree $(\chi_{a},\overline{\chi}_{a})$ of
freedom as well. These fermionic degrees $(\chi_{a},\overline{\chi}_{a})$ are
Grassmann numbers which follow anticommutator relations rather than the
commutator relations. We here use the algebraic properties of the derived
graded partner algebra $(\mathcal{F})$ to construct the superspace of
Quaternions $(\mathbb{H})$.

Now, in order to extend the quaternionic linear vector space $\mathbb{H}^{l}$
over the quaternionic field $\mathbb{H}$ (bosonic sector) [20, 30] to the
quaternionic superspace $(S^{l,m})$, we consider the corresponding fermionic
partners $(\chi_{a})$ in the graded partner vector space
$\mathbb{\mathcal{F}}^{m}$ over the field of graded partner algebra
$\mathbb{\mathcal{F}}$ as,

$\displaystyle\chi_{a}=$
$\displaystyle\chi_{a}^{0}N_{0}+\chi_{a}^{r}N_{r}\>\>\>\>\>(\forall
r=1\,to\,3),$ $\displaystyle\\{\chi_{a}^{s},\chi_{b}^{t}\\}=$ $\displaystyle
0\>\>\>\>\>\>\>\>(\forall s,t=0,1,2,3).$ (141)

Where $N_{0}=C$ and $N_{r}\>(\forall r=1\,to\,3)$ are the graded partner basis
units of algebra $\mathbb{\mathcal{F}}$. The $l+m$ dimensional quaternionic
superspace $S^{l,m}$ is considered to be described by the coordinates
$(q_{1},q_{2},...q_{l},$ $\chi_{1},\chi_{2},...\chi_{m})$ $=s_{A}\in S^{l,m}$
$(A=1,2,...l+m).$ Where $q_{1},q_{2},q_{3},...q_{l}$ are the corresponding
quaternionic coordinates in the space of $\mathbb{H}^{l}$ and
$\chi_{1},\chi_{2},\chi_{3},...,\chi_{m}$ in the graded partner vector space
$\mathbb{\mathcal{F}}^{m}.$ Therefore, the extended superspace is created by
the two sub-spaces, quaternionic space $\mathbb{H}^{l}$ and symmetric partner
space $\mathbb{\mathcal{F}}^{m}$,

$\displaystyle S^{l,m}=$
$\displaystyle\mathbb{H}^{l}\oplus\mathbb{\mathcal{F}}^{m}.$ (142)

The $S^{l,m}$ is a $\mathbb{Z}_{2}-graded$ superspace, where quaternionic
coordinates ($q_{i})$ construct the even sector $(S^{(0)})$of the superspace,
while the graded partner vectors $(\chi_{j})$ construct the odd sector
$(S^{(1)})$ of the superspace. The multiplication rules evaluated in eq. (122)
and (124) thus suggest that

$\displaystyle S^{(i)}\times S^{(j)}=$ $\displaystyle
S^{(i+j)mod\,2},\,\,\,\,\,\,i,j=0,1.$ (143)

Which is the charecteristics of $\mathbb{Z}_{2}-graded$ algebra. The possible
graded matrix representation in the graded linear space has the following
form:

$\displaystyle\varPi(\xi)=$
$\displaystyle\left(\begin{array}[]{cc}C_{1}(\mathbb{H})&D_{1}(\mathcal{F})\\\
D_{2}(\mathbb{\mathcal{F}})&C_{2}(\mathbb{H})\end{array}\right),\>\>\>\>\begin{array}[]{cc}C_{1}(\mathbb{H})\oplus
C_{2}(\mathbb{H})&\subset Bosonic\;sector\,(\mathbb{H})\\\
D_{1}(\mathbb{\mathcal{F}})\oplus D_{2}(\mathbb{\mathcal{F}})&\subset
fermionic\;sector\,(\mathcal{F})\end{array}.$ (148)

The matrix elements of $\varPi(\xi)$ belong to the $\mathbb{Z}_{2}-graded$
linear superspace $S^{l,m}$. Multiplication between the two graded matrix
matrices in the grded linear space of $S^{l,m}$ retain its form as the virtue
of the algebraic properties of graded partner algebra $(\mathcal{F})$ and
quaternions $(\mathbb{H})$ as

$\displaystyle\varPi_{1}\varPi_{2}=$
$\displaystyle\left(\begin{array}[]{cc}C_{1}^{1}&D_{1}^{1}\\\
D_{2}^{1}&C_{2}^{1}\end{array}\right)\left(\begin{array}[]{cc}C_{1}^{2}&D_{1}^{2}\\\
D_{2}^{2}&C_{2}^{2}\end{array}\right)=\left(\begin{array}[]{ccc}C_{1}^{1}C_{1}^{2}+D_{1}^{1}D_{2}^{2}&&C_{1}^{1}D_{1}^{2}+D_{1}^{1}C_{2}^{2}\\\
\\\
D_{2}^{1}C_{1}^{2}+C_{2}^{1}D_{2}^{2}&&D_{2}^{1}D_{1}^{2}+C_{2}^{1}C_{2}^{2}\end{array}\right)=\left(\begin{array}[]{cc}C_{1}^{3}&D_{1}^{3}\\\
D_{2}^{3}&C_{2}^{3}\end{array}\right).$ (158)

where by the multiplication properties defined in eq. (122) & (124);
$C_{1}^{3}=C_{1}^{1}C_{1}^{2}+D_{1}^{1}D_{2}^{2}$ &
$C_{2}^{3}=D_{2}^{1}D_{1}^{2}+C_{2}^{1}C_{2}^{2}$ $\in\mathbb{H}$, and
$D_{1}^{3}=C_{1}^{1}D_{1}^{2}+D_{1}^{1}C_{2}^{2}$ &
$D_{2}^{3}=D_{2}^{1}C_{1}^{2}+C_{2}^{1}D_{2}^{2}$ $\in\mathbb{\mathcal{F}}$.

Now, we consider the dynamics of the superspace $S^{l,m}$ under the
quaternionic supergroups. The supergroups are the groups of isometries of the
superspace. It has been seen that only two quaternionic supergroups are
possible [20]. One is $SL(l,m;\mathbb{H})=SU^{*}(2l,2m)$, which is not
corresponding to any metric-preserving group; the other is
$UU_{a}(l;m;\mathbb{H}),$ which preserves the quaternionic anti-unitary
product. In this case, we have defined the bosonic sector in quaternionic
space and the fermionic sector in graded partner space. The canonical metric
form in the $UU_{a}$ quaternionic supergroup is defined as
$g_{xy}=(I_{l},e_{2}I_{m})$. Now, keeping this in mind, one can extend the
quaternionic anti-unitary product in $\mathbb{H}$ supersymmetrically to
$\mathbb{S}^{l,m}$ space by using eq.(141) as

$\displaystyle(S,S^{\prime})_{UU_{a}}=\bar{S}_{x}g_{xy}S^{\prime}{}_{y}=$
$\displaystyle(q,q^{\prime})_{U}+(\chi,\chi^{\prime})_{U_{a}}$
$\displaystyle=($ $\displaystyle
q_{k}^{0}q^{\prime}{}_{k}^{0}+q_{k}^{r}q^{\prime}{}_{k}^{r}-\chi_{\alpha}^{0}\chi^{\prime}{}_{\alpha}^{2}+\chi_{\alpha}^{2}\chi^{\prime}{}_{\alpha}^{0}-\chi_{\alpha}^{3}\chi^{\prime}{}_{\alpha}^{1}+\chi_{\alpha}^{1}\chi^{\prime}{}_{\alpha}^{3}$
, $\displaystyle
q_{k}^{0}q^{\prime}{}_{k}^{1}-q_{k}^{1}q^{\prime}{}_{k}^{0}+q_{k}^{3}q^{\prime}{}_{k}^{2}-q_{k}^{2}q^{\prime}{}_{k}^{3}+\chi_{\alpha}^{0}\chi^{\prime}{}_{\alpha}^{3}+\chi_{\alpha}^{3}\chi^{\prime}{}_{\alpha}^{0}+\chi_{\alpha}^{1}\chi^{\prime}{}_{\alpha}^{2}+\chi_{\alpha}^{2}\chi^{\prime}{}_{\alpha}^{1}$
, $\displaystyle
q_{k}^{0}q^{\prime}{}_{k}^{2}-q_{k}^{2}q^{\prime}{}_{k}^{0}+q_{k}^{1}q^{\prime}{}_{k}^{3}-q_{k}^{3}q^{\prime}{}_{k}^{1}-\chi_{\alpha}^{0}\chi^{\prime}{}_{\alpha}^{0}+\chi_{\alpha}^{2}\chi^{\prime}{}_{\alpha}^{2}-\chi_{\alpha}^{1}\chi^{\prime}{}_{\alpha}^{1}-\chi_{\alpha}^{3}\chi^{\prime}{}_{\alpha}^{3}$
, $\displaystyle
q_{k}^{0}q^{\prime}{}_{k}^{3}-q_{k}^{3}q^{\prime}{}_{k}^{0}+q_{k}^{2}q^{\prime}{}_{k}^{1}-q_{k}^{1}q^{\prime}{}_{k}^{2}-\chi_{\alpha}^{0}\chi^{\prime}{}_{\alpha}^{1}-\chi_{\alpha}^{1}\chi^{\prime}{}_{\alpha}^{0}+\chi_{\alpha}^{3}\chi^{\prime}{}_{\alpha}^{2}+\chi_{\alpha}^{2}\chi^{\prime}{}_{\alpha}^{3}),$
(159)

where $S$ and $S^{\prime}$ are the elements of superspace $S^{l,m}.$ Here, the
unitary and anti-unitary quaternionic products are defined as:

$\displaystyle(q,q^{\prime})_{U}$
$\displaystyle=\bar{q}_{k}q^{\prime}{}_{k}=(q_{k}^{0}-q_{k}^{1}e_{1}-q_{k}^{2}e_{2}-q_{k}^{3}e_{3})(q^{\prime}{}_{k}^{0}+e_{1}q^{\prime}{}_{k}^{1}+e_{2}q^{\prime}{}_{k}^{2}+e_{3}q^{\prime}{}_{k}^{3})$
$\displaystyle(\chi,\chi^{\prime})_{U_{a}}$
$\displaystyle=\chi{}_{\alpha}e_{2}\chi^{\prime}{}_{\alpha}=(\chi_{\alpha}^{0}C+\chi_{\alpha}^{1}N_{1}+\chi_{\alpha}^{2}N_{2}+\chi_{\alpha}^{3}N_{3})e_{2}(\chi^{\prime}{}_{\alpha}^{0}C+\chi^{\prime}{}_{\alpha}^{1}N_{1}+\chi^{\prime}{}_{\alpha}^{2}N_{2}+\chi^{\prime}{}_{\alpha}^{3}N_{3}),$
(160)

where $\bar{q}_{k}$ corresponds to the quaternionic conjugation operation.
Comparing this result to Lukierski et al. [20], one can see that this
supersymmetric product is describing the isometries corresponding to the
transformations in the $UU_{a}(l;m;\mathbb{H})$ group, described by the
intersections of orthosymplectic supersymmetric groups as

$\displaystyle UU_{a}(l;m;\mathbb{H})=$ $\displaystyle OSp(4l;4m;R)\cap
OSp^{(1)}(2l,2l;4m;R)$ $\displaystyle\cap$ $\displaystyle
OSp^{(2)}(2l,2l;4m;R)\cap OSp^{(3)}(2l,2l;4m;R).$ (161)

##  6 Quaternionic Super conformal algebra in $D=4$ space:

A conformal transformation of the coordinates is a mapping that leaves
invariant the metric $\zeta_{\mu\nu}$ up to a scale [33],

$\displaystyle\zeta^{\prime}_{\mu\nu}(x^{\prime})=$
$\displaystyle\Lambda(x)\zeta_{\mu\nu}.$ (162)

The set of all conformal transformations form a group with the Poincaré group
as a subgroup corresponds to $\Lambda(x)=1$. A conformal group in $d$ with
$p+q=d$ in a metric $\\{-1.....,+1....\\}$ with signature $(p,q)$ is
isomorphic to the $SO(p+1,q+1)$ group. The conformal group of $D=4$
dimensional space $\mathbb{R}^{4,0}$ is thus isomorphic to the $SO(5,1)$ group
for Euclidean metric signature. So we have the $15$ generators for conformal
algebra in $D=4$ dimensions, containing the $6$ generators of rotation
$SO(4)$, $4$ generators of translations $P_{\mu}$ , $4$ generators of
conformal accelerations $K_{\mu}$, and one dilation $D$.

To construct the $6$ quaternionic generators of $SO(4)$, we first construct
the $2\times 2$ dimensional quaternionic valued $\Gamma$-matrices in Weyl
representation by extending the imaginary unit $i$ in the $\sigma_{2}-$Pauli
matrix to triplet quaternionic units $e_{i}$$(\forall i=1,2,\,and\,3)$ as

$\displaystyle\Gamma_{1}=$ $\displaystyle\left(\begin{array}[]{cc}0&-e_{1}\\\
e_{1}&0\end{array}\right),\;\Gamma_{2}=\left(\begin{array}[]{cc}0&-e_{2}\\\
e_{2}&0\end{array}\right),\;\Gamma_{3}=\left(\begin{array}[]{cc}0&-e_{3}\\\
e_{3}&0\end{array}\right),\Gamma_{0}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right).$ (171)

These $\Gamma-$matrices satisfy the following relation of Clifford algebra:

$\displaystyle\Gamma_{\mu}\Gamma_{\nu}+\Gamma_{\nu}\Gamma_{\mu}=$
$\displaystyle 2\eta_{\mu\nu}I_{2}.$ (172)

Where we have defined $\Gamma_{\mu}=(\Gamma_{k},\Gamma_{0}),$ $(\forall
k=1,2,3)$, and the metric $\eta_{\mu\nu}=\\{1,1,1,1\\}$. The $\Gamma_{\mu}$
are in Weyl representation, they satisfy the following properties:

(i) $\Gamma_{0}^{\dagger}=\Gamma_{0}$

(ii) $\Gamma_{k}^{\dagger}=\Gamma_{k}$$\,(\forall k=1,2,3)$

(iii) $\Gamma_{0}\Gamma_{\mu}\Gamma_{0}=\overline{\Gamma}_{\mu}$.

Any arbitrary space-time four vector $(x^{\mu})$ in space may be associated
with these quaternionic Hermitian $\Gamma-$ matrices as

$\displaystyle x^{\mu}\rightarrow X=$ $\displaystyle
x_{\mu}\Gamma_{\mu}=\left(\begin{array}[]{cc}0&x_{0}-x_{1}e_{1}-x_{2}e_{2}-x_{3}e_{3}\\\
x_{0}+x_{1}e_{1}+x_{2}e_{2}+x_{3}e_{3}&0\end{array}\right).$ (175)

the determinant of which is given [14, 30] as

$\displaystyle det(X)=$
$\displaystyle[(x_{0})^{2}+(x_{1})^{2}+(x_{2})^{2}+(x_{3})^{2}]^{2}=(\eta_{\mu\nu}x_{\mu}x_{\nu})^{2}$
(176)

where the metric is $\eta_{\mu\nu}=\\{1,1,1,1\\}$. The general transformation
from the Hermitian matrix $X$ to another Hermitian matrix $X^{\prime}$ without
changing the determinant is thus given by

$\displaystyle X\rightarrow X^{\prime}=$ $\displaystyle
TXT^{\dagger},\,\,\,\,\,\,\,\,\,\,\,(T\in SL(2,\mathbb{H}))$ (177)

where the quaternionic transformation matrix $T$ must have $det(T)=1$, and
$X^{\prime}=x^{\prime}_{\mu}\Gamma_{\mu}$. However it is to be noted that it
can’t be a general element of $SL(2,\mathbb{H})$. Since the general line
element that remain unchanged of the transformation under [30]
$SL(2,\mathbb{H})$ group is a six-dimensional space-time which involve the
matrices $\sigma_{0}=I_{2}$ and $\sigma_{3}$ with the $\Gamma-$matrices
defined in eq. (171) as well. $SL(2,\mathbb{H})$ group is the universal
covering group [19, 30] of $SO(5,1)$ i.e.
$SL(2,\mathbb{H})\cong\overline{SO(5,1)}$. Rather under certain conditions
matrices like $T$ form a subgroup of $SL(2,\mathbb{H}),$ we may call it
$Res\,SL(2,\mathbb{H})$ group. So we may have the following conditions for $T$
matrices transformations:

$\displaystyle T\sigma_{0}T^{\dagger}=\sigma_{0},\quad\quad\quad\quad\quad$
$\displaystyle T\sigma_{3}T^{\dagger}=\sigma_{3}.\,\,\,\,\,\,\,\,\,\,(\forall
T\in Res\,SL(2,\mathbb{H}))$ (178)

Now we define the $T$ matrix as

$\displaystyle T=$ $\displaystyle\left(\begin{array}[]{cc}P&Q\\\
R&S\end{array}\right),\,\,\,\,\,(P,Q,R,S\in\mathbb{H}),$ (181) $\displaystyle
and$
$\displaystyle\,\,\,\,det(T)=\left|P\right|^{2}\left|S\right|^{2}+\left|Q\right|^{2}\left|R\right|^{2}-2Re(P\overline{R}S\overline{Q})=1$
(182)

The eq.(178) shows that the $T$ matrices must be unitary hence

$\displaystyle TT^{\dagger}=\left(\begin{array}[]{cc}P&Q\\\
R&S\end{array}\right)\left(\begin{array}[]{cc}\overline{P}&\overline{R}\\\
\overline{Q}&\overline{S}\end{array}\right)=$
$\displaystyle\left(\begin{array}[]{cc}1&0\\\ 0&1\end{array}\right).$ (189)

Compairing both sides of the above eq. we get
$\left|P\right|^{2}+\left|Q\right|^{2}=\left|R\right|^{2}+\left|S\right|^{2}=1$
and $P\overline{R}+Q\overline{S}=R\overline{P}+S\overline{Q}=0$. Also by
$T^{\dagger}T=I$, we have the conditions
$\left|P\right|^{2}+\left|R\right|^{2}=\left|Q\right|^{2}+\left|S\right|^{2}=1$
and $\overline{P}Q+\overline{R}S=\overline{Q}P+\overline{S}R=0$. By a similar
calculations from the second relation of eq. (178) we have:

$\displaystyle T\sigma_{3}T^{\dagger}$ $\displaystyle=\sigma_{3}$
$\displaystyle\left(\begin{array}[]{cc}P&Q\\\
R&S\end{array}\right)\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)\left(\begin{array}[]{cc}\overline{P}&\overline{R}\\\
\overline{Q}&\overline{S}\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right),$ (198)

so, we must have
$\left|P\right|^{2}+\left|Q\right|^{2}=1,\left|R\right|^{2}-\left|S\right|^{2}=-1$
and $P\overline{R}-Q\overline{S}=R\overline{P}-S\overline{Q}=0$. Considering
all these conditions together it is concluded that
$\left|P\right|^{2}=\left|S\right|^{2}=1$ and $Q=R=0$. Hence the
transformation matrices like $T$ which constitute the group
$Res\,SL(2,\mathbb{H})$ must have the following properties:

$\displaystyle T=$ $\displaystyle\left(\begin{array}[]{cc}P&0\\\
0&S\end{array}\right),\,\,\,\,\,(P,S\in\mathbb{H}),\,\,\,\,\,\,where\,\,\,\left|P\right|^{2}=\left|S\right|^{2}=1\,\,\,\,\,$
(201) $\displaystyle and$
$\displaystyle\,\,\,\,\,\,TT^{\dagger}=T^{\dagger}T=I\>\>\>\>\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\forall
T\in Res\,SL(2,\mathbb{H})).$ (202)

It can be shown that the $Res\,SL(2,\mathbb{H})$ is a closed subgroup of
$SL(2,\mathbb{H})$. Because for any two matrices $T_{1}$, $T_{2}$ $\in
Res\,SL(2,\mathbb{H})$, we have,

$\displaystyle T_{1}T_{2}=$ $\displaystyle\left(\begin{array}[]{cc}P_{1}&0\\\
0&S_{1}\end{array}\right)\left(\begin{array}[]{cc}P_{2}&0\\\
0&S_{2}\end{array}\right)=\left(\begin{array}[]{cc}P_{1}P_{2}&0\\\
0&S_{1}S_{2}\end{array}\right)=\left(\begin{array}[]{cc}P_{3}&0\\\
0&S_{3}\end{array}\right)\in Res\,SL(2,\mathbb{H}).$ (211)

Where by eq.(189) it is easy to show that
$(T_{1}T_{2})^{\dagger}(T_{1}T_{2})=T_{2}^{\dagger}T_{1}^{\dagger}T_{1}T_{2}=I$
and $\left|P_{3}\right|^{2}=(\overline{P_{1}P_{2}})(P_{1}P_{2})$
$=\left|P_{1}\right|^{2}\left|P_{2}\right|^{2}=1$ and
$\left|S_{3}\right|^{2}=(\overline{S_{1}S_{2}})(S_{1}S_{2})$
$=\left|S_{1}\right|^{2}\left|S_{2}\right|^{2}=1$. Hence, if $T_{1}$ and
$T_{2}$ are the elements of $Res\,SL(2,\mathbb{H})$ than $T_{1}T_{2}$ is also
an element of this restricted $Res\,SL(2,\mathbb{H})$ group, which leaves the
interval defined in eq. (176) invariant. Also from eq. (202) we have
$\overline{T}=T^{-1}.$

Since according to the eq. (202), the group elements of
$Res\,SL(2,\mathbb{H})$ must be unitary, hence $Res\,SL(2,\mathbb{H})$
$\subset$ $U(2,\mathbb{H})\cong Sp(2)$. Now, it can be shown that the
$Res\,SL(2,\mathbb{H})$ group is homomorphic to the $SO(4)$ group. We may
write the $SO(4)$ transformations of a four vector $x_{\mu}$ in $D=4$
dimensions, which leaves the space interval defined in eq. (176) invariant as

$\displaystyle x^{\prime}_{\mu}=$
$\displaystyle\Lambda_{\mu\nu}x_{\nu}\,\,\,\,\,\,\,\,(\forall\mu,\nu=0,1,2,3)\,\,\,\,\,\,\,\,\,\,(\forall\Lambda\in
SO(4))$ (212)

Where $\Lambda_{\mu\nu}$ are the matrix components of $SO(4)$ transformation
matrix $(\Lambda)$. Putting the value of $x^{\prime}_{\mu}$from eq. (212) into
eq. (177) we have;

$\displaystyle\Lambda_{\mu\nu}\Gamma_{\mu}=$ $\displaystyle
T\Gamma_{\nu}T^{\dagger}\,\,\,\,\,\,\,\,\,\,\,(T\in Res\,SL(2,\mathbb{H}))$
(213)

Now using the eq. (172), into the eq. above, we get the following relation
between the $SO(4)$ transformation matrix $\Lambda$ and the matrix $T$ $(\in
Res\,SL(2,\mathbb{H})$):

$\displaystyle\Lambda_{\mu\nu}=$
$\displaystyle\frac{1}{2}Tr[T\Gamma_{\nu}T^{\dagger}\Gamma_{\mu}].$ (214)

Where the $Tr$ corresponding to the quaternionic trace, i.e. $Tr(Q)=ReTr(Q).$
Hence we may associate a $T$ matrix in $Res\,SL(2,\mathbb{H})$ group for each
and every $SO(4)$ rotation $\Lambda_{\mu\nu}$ in $D=4$ dimensional space. The
homomorphism between $SO(4)$ and $Res\,SL(2,\mathbb{H})$ then be described by
the relation between the components of $\Lambda_{\mu\nu}$ and $T(\in
Res\,SL(2,\mathbb{H}))$. By the simple cyclic properties of quaternionic trace
it is easy to evaluate
$\Lambda_{\mu\nu}(T_{1}T_{2})=\Lambda_{\mu\nu}(T_{1})\Lambda_{\mu\nu}(T_{2})$
$\forall\,T_{1},\,T_{2}\in Res\,SL(2,\mathbb{H}).$ Also from eq. (202), we
have $\Lambda_{\mu\nu}(T^{-1})=(\Lambda_{\mu\nu}(T))^{-1}$. The $det(\Lambda)$
is a continuous function of $T(\in Res\,SL(2,\mathbb{H}))$, since
$Res\,SL(2,\mathbb{H})$ is also a continuous group, because the domain of
these variables is simply connected, a discontinuous jump from
$det(\Lambda)=1$ to $det(\Lambda)=-1$ is excluded. By the consequence for the
restriction (177) on the values of $T^{\prime}s$, we have $det(\Lambda)=1$ for
all the elements of the $SO(4)$ defined in eq. (214). Hence the homomorphism
between the group $SO(4)$ and the group $Res\,SL(2,\mathbb{H})$ has been
established consistently. The vector $X$ defined in eq. (175) transform as a
vector under endomorphic transformation in $Res\,SL(2,\mathbb{H})$.

Similarly, the two-component quaternion spinors acting on
$Res\,SL(2,\mathbb{H})$ are defined as

$\displaystyle\varPsi_{\alpha}=$ $\displaystyle\left(\begin{array}[]{c}\phi\\\
\xi\end{array}\right)\>\>(\forall\phi,\,\xi\in\mathbb{H}),\>\>\>\>\>\>\>\>\>\varPsi_{\alpha}^{\dagger}=(\overline{\phi},\,\,\overline{\xi})$
(217)

where $\dagger$ is the quaternionic hermitian conjugate, and $\overline{\phi}$
& $\overline{\xi}$ are the quaternionic conjugate of $\phi$ & $\xi$
respectively. The transformation properties under $Res\,SL(2,\mathbb{H})$ of
undotted quaternionic spinor and its conjugate are such as

$\displaystyle\varPsi^{\prime}_{\alpha}=$ $\displaystyle
T_{\alpha}^{\>\,\beta}\varPsi_{\beta}\>\>\>\>\>\>\>\>\>\varPsi{}_{\alpha}^{{}^{\prime}\dagger}=\varPsi_{\beta}^{\dagger}T_{\>\>\alpha}^{\beta}\quad\quad\quad\quad\quad
T\in Res\,SL(2,\mathbb{H}).$ (218)

While the dotted spinors and its conjugate transform as

$\displaystyle\eta^{\overset{.}{\alpha}}=$
$\displaystyle\left(\begin{array}[]{c}\lambda\\\
\varsigma\end{array}\right)\>\>(\forall\lambda,\,\varsigma\in\mathbb{H}),\>\>\>\>\>\>\>\>\>\eta^{\overset{.}{\alpha\dagger}}=(\overline{\lambda},\,\,\overline{\varsigma})$
(221)

where transformation properties under $Res\,SL(2,\mathbb{H})$ are defined as

$\displaystyle\eta^{\overset{.}{\alpha^{\prime}}}=$
$\displaystyle(\overline{T})_{\>\>\overset{.}{\beta}}^{\overset{.}{\alpha}}\eta^{\overset{.}{\beta}},\>\>\>\>\>\>\eta^{\overset{.}{\alpha^{\prime}}\dagger}=\eta^{\overset{.}{\alpha}\dagger}(\overline{T})_{\overset{.}{\beta}}^{\>\>\overset{.}{\alpha}}.$
(222)

Where we have used the property $T^{-1}=\overline{T}$. The differential
operator is defined as

$\displaystyle\partial=\Gamma_{\mu}\partial_{\mu}=$
$\displaystyle\left(\begin{array}[]{cc}0&\partial_{0}-e_{1}\partial_{1}-e_{2}\partial_{2}-e_{3}\partial_{3}\\\
\partial_{0}+e_{1}\partial_{1}+e_{2}\partial_{2}+e_{3}\partial_{3}&0\end{array}\right).$
(225)

Now one can form the quaternionic $SO(4)$ generators $(\Sigma_{\mu\nu})$ using
$\Gamma-$matrices defined in eq. (171) as

$\displaystyle\Sigma_{\mu\nu}=\frac{1}{4}$
$\displaystyle[\Gamma_{\mu}\Gamma_{\nu}-\Gamma_{\nu}\Gamma_{\mu}]\>\>\>\>\>\>\>\>\>\>(\forall\mu,\nu=0\,to\,3).$
(226)

Which satisfy the following $SO(4)$ Lie algebra as

$\displaystyle[\Sigma_{\mu\nu},\Sigma_{\rho\sigma}]=$
$\displaystyle-(\eta_{\mu\rho}\Sigma_{\nu\sigma}-\eta_{\mu\sigma}\Sigma_{\nu\rho}-\eta_{\nu\rho}\Sigma_{\mu\sigma}+\eta_{\nu\sigma}\Sigma_{\mu\rho})\;\;\;\;\;\;(\forall\mu,\nu,\rho,\sigma=0\,to\,3).$
(227)

The 4 generators of translation or linear momentum in $D=4$ may then be
written as

$\displaystyle P_{\mu}=$
$\displaystyle\frac{1}{2}\left(\begin{array}[]{cc}0&0\\\
e_{\mu}&0\end{array}\right)\quad\quad\quad(\forall\mu=0,1,2,3).$ (230)

The 4 generators of conformal accelerations may also be written as

$\displaystyle K_{\mu}=$
$\displaystyle\frac{1}{2}\left(\begin{array}[]{cc}0&e_{\mu}\\\
0&0\end{array}\right)\quad\quad\quad(\forall\mu=0,1,2,3).$ (233)

The generator of dilation is defined as

$\displaystyle D=$ $\displaystyle\frac{1}{2}\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)$ (236)

One can evaluate that the Lie algebra of the $SO(5,1)$ conformal group in
$D=4$ as

$\displaystyle[p_{\mu},p_{\nu}]=$ $\displaystyle
0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(\forall\mu,\nu=0\,to\,3)$
$\displaystyle{}[K_{\mu},K_{\nu}]=$ $\displaystyle 0$
$\displaystyle{}[\Sigma_{\mu\nu},p_{\rho}]=$
$\displaystyle-(\eta_{\mu\rho}p_{\nu}-\eta_{\nu\rho}p_{\mu})\>\>\>\>\>\>(\forall\mu,\nu,\rho=0\,to\,3)$
$\displaystyle{}[\Sigma_{\mu\nu},\Sigma_{\rho\sigma}]=$
$\displaystyle-(\eta_{\mu\rho}\Sigma_{\nu\sigma}-\eta_{\mu\sigma}\Sigma_{\nu\rho}-\eta_{\nu\rho}\Sigma_{\mu\sigma}+\eta_{\nu\sigma}\Sigma_{\mu\rho}).\;\;\;\;\;\;(\forall\mu,\nu,\rho,\sigma=0\,to\,3).$
$\displaystyle{}[\Sigma_{\mu\nu},K_{\rho}]=$
$\displaystyle(\eta_{\mu\rho}K_{\nu}-\eta_{\nu\rho}K_{\mu})\>\>\>\>\>\>(\forall\mu,\nu,\rho=0\,to\,3)$
$\displaystyle{}[D,P_{\mu}]=$ $\displaystyle-P_{\mu}$
$\displaystyle{}[D,K_{\mu}]=$ $\displaystyle K_{\mu}$
$\displaystyle{}[p_{\mu},K_{\nu}]=$ $\displaystyle
D-\frac{1}{2}\Sigma_{\mu\nu}.$ (237)

Now we introduce the graded matrix representation to form the quaternionic
realization of super Poincaré algebra in $D=4$-dimensional space. Since
Section $5$, we have defined the graded matrix representation of super algebra
in super space $(S)$. We have asserted that the quaternionic $(\mathbb{H})$
bosonic sector of the super algebra is contained along the diagonal quadrant,
while the fermionic sector is spanned in the regime of the graded partner
algebra $(\mathbb{\mathcal{F}})$ in the off-diagonal quadrant. Hence, keeping
in view this form of graded matrix representation, we introduce the bosonic
sector of the superalgebra as follows:

$\displaystyle M_{\mu\nu}=$
$\displaystyle\left(\begin{array}[]{ccc}\Sigma_{\mu\nu}&\vdots&0\\\
\cdots&\cdots&\cdots\\\
0&\vdots&0\end{array}\right),P_{\mu}=\left(\begin{array}[]{ccc}p_{\mu}&\vdots&0\\\
\cdots&\cdots&\cdots\\\
0&\vdots&0\end{array}\right),\;\;\;\;\;(\forall\mu,\nu=0\,to\,3).$ (244)
$\displaystyle K_{\mu}$
$\displaystyle=\left(\begin{array}[]{ccc}K_{\mu}&\vdots&0\\\
\cdots&\cdots&\cdots\\\
0&\vdots&0\end{array}\right),D=\left(\begin{array}[]{ccc}D&\vdots&0\\\
\cdots&\cdots&\cdots\\\ 0&\vdots&0\end{array}\right)$ (251)

Now to construct the fermionic sector of the $N=1$ super conformal algebra in
$D=4$ dimensions, we introduce the fermionic $4$ super partners in $D=4$
dimensional space in terms of our graded partner algebra
$(\mathbb{\mathcal{F}})$ as

$\displaystyle Q_{1}(N_{\mu})=$
$\displaystyle\left(\begin{array}[]{cccc}0&0&\vdots&N_{\mu}\\\ 0&0&\vdots&0\\\
\cdots&\cdots&\cdots&\cdots\\\
0&0&\vdots&0\end{array}\right),\>\>\>\>\>\>Q_{2}(N_{\mu})=\left(\begin{array}[]{cccc}0&0&\vdots&0\\\
0&0&\vdots&N_{\mu}\\\ \cdots&\cdots&\cdots&\cdots\\\
0&0&\vdots&0\end{array}\right)$ (260) $\displaystyle S_{1}(N_{\nu})=$
$\displaystyle\left(\begin{array}[]{cccc}0&0&\vdots&0\\\ 0&0&\vdots&0\\\
\cdots&\cdots&\cdots&\cdots\\\
\widetilde{N}_{\nu}&0&\vdots&0\end{array}\right),\>\>\>\>\>\>S_{2}(N_{\nu})=\left(\begin{array}[]{cccc}0&0&\vdots&0\\\
0&0&\vdots&0\\\ \cdots&\cdots&\cdots&\cdots\\\
0&\widetilde{N}_{\nu}&\vdots&0\end{array}\right).$ (269)

Where $N_{\mu}=\\{N_{0}=C,N_{1},N_{2},N_{3}\\}$ are the basis units of the
graded partner algebra $(\mathbb{\mathcal{F}})$ that follow the multiplication
rules of eqs. (110), (111), and (117). Hence, keeping in view the graded form
of fermionic super-partners defined in eq.(269) and bosonic partners in eq.
(251), one can write the complete superalgebra in $D=4$ as

$\displaystyle\\{Q_{1}(N_{\mu}),S_{2}(N_{\nu})\\}=$
$\displaystyle\delta_{\mu\nu}N_{0}+\delta_{\mu 0}N_{\nu}+\delta_{\nu
0}N_{\mu}-\in_{\mu\nu\sigma}N_{\sigma}\;\;(\forall\mu,\nu,\rho,\sigma=0\,to\,3)$
$\displaystyle\\{Q_{2}(N_{\mu}),S_{1}(N_{\nu})\\}=$
$\displaystyle\delta_{\mu\nu}P_{0}+\delta_{\mu 0}P_{\nu}+\delta_{\nu
0}P_{\mu}-\in_{\mu\nu\sigma}P_{\sigma}\;\;(\forall\mu,\nu,\rho,\sigma=0\,to\,3)$
$\displaystyle\\{Q_{1}(N_{\mu}),Q_{2}(N_{\nu})\\}=$ $\displaystyle
0.\>\>\>\>\>\>\>\>\>\>\>\>\>(\forall\mu,\nu=0\,to\,3).$
$\displaystyle\\{S_{1}(N_{\mu}),S_{2}(N_{\nu})\\}=$ $\displaystyle
0\>\>\>\>\>\>\>\>\>\>\>\>\>\>(\forall\mu,\nu=0\,to\,3)$
$\displaystyle\\{Q_{1}(N_{\mu}),S_{1}(N_{\nu})\\}=$
$\displaystyle\frac{1}{2}[M_{\mu\nu}+A(\widetilde{N}_{\nu}N_{\mu})+2D(N_{\mu}\widetilde{N}_{\nu})]$
$\displaystyle\\{Q_{2}(N_{\mu}),S_{2}(N_{\nu})\\}=$
$\displaystyle\frac{1}{2}[M_{\mu\nu}+A(\widetilde{N}_{\nu}N_{\mu})-2D(N_{\mu}\widetilde{N}_{\nu})]$
$\displaystyle{}[Q_{1}(N_{\mu}),D]=$ $\displaystyle-\frac{1}{2}Q_{1}(N_{\mu})$
$\displaystyle{}[Q_{2}(N_{\mu}),D]=$ $\displaystyle\frac{1}{2}Q_{2}(N_{\mu})$
$\displaystyle{}[S_{1}(N_{\nu}),D]=$ $\displaystyle\frac{1}{2}S_{1}(N_{\nu})$
$\displaystyle{}[S_{2}(N_{\nu}),D]=$ $\displaystyle-\frac{1}{2}S_{2}(N_{\nu})$
$\displaystyle{}[X_{ij},Q_{k}]=$
$\displaystyle\delta_{jk}Q_{i}(e_{\mu}N_{\nu})$
$\displaystyle{}[X_{ij},S_{k}]=$
$\displaystyle-\delta_{jk}S_{i}(e_{\mu}\widetilde{N}_{\nu})$ (270)

where $N_{\mu}\widetilde{N}_{\nu}=\delta_{\mu\nu}e_{0}+\delta_{\mu
0}e_{\nu}+\delta_{\nu 0}e_{\mu}-\in_{\mu\nu\sigma}e_{\sigma}$ and
$e_{\mu}\widetilde{N}_{\nu}=-\delta_{\mu\nu}\widetilde{N}_{0}+\delta_{\mu
0}\widetilde{N}_{\nu}-\delta_{\nu
0}\widetilde{N}_{\mu}+\in_{\mu\nu\sigma}\widetilde{N}_{\sigma}$. The operator
$A=I_{3}$ is the $U(1)$ $R-$symmetry generator, while $X_{ij}$ are the
generators [20] of quaternionic $SL(2,\mathbb{H})\cong SU^{*}(4)$ $R-$
symmentry defined as

$\displaystyle(X_{ij})_{rs}=$ $\displaystyle e_{\mu}\delta_{is}\delta_{jr},$
(271)

that satisfy the algebra

$\displaystyle[X_{ij}(e_{\mu}),X_{ab}(e_{\nu})]=$
$\displaystyle\delta_{ib}X_{aj}(e_{\mu}e_{\nu})-\delta_{ja}X_{il}(e_{\mu}e_{\nu}).$
(272)

##  7 Discussion and Conclusions:

The $\mathbb{Z}_{2}-$ graded algebra of quaternions has been studied in the
matrix representations of real numbers. We have developed the graded partners
$K_{i}$$\,(i=1,2,3)$ by introducing the Lie algebra of quaternions as the
bosonic partner $(L_{0})$ of the graded algebra. Then, using the Jacobi
identities of $\mathbb{Z}_{2}-$ graded algebra, we have evaluated the matrix
representations of graded partners $K_{i}$. It has been shown that these three
matrices, along with the matrix representations of quaternion units,
constitute the $\mathbb{Z}_{2}-$ graded Lie algebra space of quaternions. The
multiplication rules between the graded partner units $K_{i}$ and quaternion
basis units $e_{i}$ have been studied. It has been shown that matrix $C$
commutes with each and every basis unit $(e_{i}$) of quaternions
$(\mathbb{H})$ and graded partner units $(K_{i}$), so it corresponds to the
Casimir element of the Lie algebra
$(T=\\{e_{1},e_{2},e_{3},K_{1},K_{2},K_{3}\\})$, which is formed by the basis
units $e_{i}$ and $K_{i}$. Then we have defined the graded partner algebra
$\mathbb{\mathcal{F}}$ as comprising the basis units $C$ and $K_{i}$ $(\forall
i=1,2,3).$

The quaternion space $(\mathbb{H}^{l})$ has been further extended to the
graded superspace $(S^{l,m})$ by considering the bosonic part of the
superspace as quaternionic, while the fermionic part is represented by the
graded partner algebra $(\mathcal{F})$ of quaternions $(\mathbb{H})$. Then we
have defined the graded matrix representation for this superspace, which has
principal diagonal elements as the bosonic part (hence quaternionic
$(\mathbb{H})$) and off-diagonal elements as the fermionic part (made from
$(\mathbb{\mathcal{F}})$). The quaternionic supergroups have been studied in
context with this structure of superalgebra and superspace by extending the
anti-unitary quaternionic product supersymmetrically.

After constructing the superspace as the combined space of quaternion algebra
$(\mathbb{H})$ and graded partner algebra $(\mathcal{F})$, we have studied the
$D=4$ super conformal algebra. Where the bosonic part of the superalgebra
contains the generators of the quaternionic conformal group, however, the
fermionic part of the superalgebra has been constructed by introducing fermion
partners $Q_{a}$ in the field of graded partner algebra
$(\mathbb{\mathcal{F}}).$ Finally, the super conformal algebra in $D=4$
dimensions has been established.

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# Persistence of Topological Order and Formation of Quantum Well States in
Topological Insulators Bi2(Se,Te)3 under Ambient Conditions

Chaoyu Chen1, Shaolong He1, Hongming Weng1, Wentao Zhang1, Lin Zhao1, Haiyun
Liu1, Xiaowen Jia1, Daixiang Mou1, Shanyu Liu1, Junfeng He1, Yingying Peng1,
Ya Feng1, Zhuojin Xie1, Guodong Liu1, Xiaoli Dong1, Jun Zhang1, Xiaoyang
Wang2, Qinjun Peng2, Zhimin Wang2, Shenjin Zhang2, Feng Yang2, Chuangtian
Chen2, Zuyan Xu2, Xi Dai1, Zhong Fang1, X. J. Zhou1,∗
1Beijing National Laboratory for Condensed Matter Physics, Institute of
Physics, Chinese Academy of Sciences, Beijing 100190, China
2Technical Institute of Physics and Chemistry, Chinese Academy of Sciences,
Beijing 100190, China

(July 28, 2011)

The topological insulators represent a unique state of matter where the bulk
is insulating with an energy gap while the surface is metallic with a Dirac
cone protected by the time reversal symmetryTin3D ; ZhangQiPT ; HasanKaneRev ;
ZhangQiRMP ; JMoorePer . These characteristics provide a venue to explore
novel quantum phenomena in fundamental physicsXLQiNP ; RDLiAxions ; XLQiMomo ;
LFuMajorana ; RYuSience  and show potential applications in spintronics and
quantum computingRevnextgeneration ; ZhangQiPT ; JMoorePer . One critical
issue directly related with the applications as well as the fundamental
studies is how the topological surface state will behave under ambient
conditions (1 atmosphere air and room temperature). In this paper, we report
high resolution angle-resolved photoemission measurements on the surface state
of the prototypical topological insulators, Bi2Se3, Bi2Te3 and
Bi2(Se0.4Te2.6), upon exposing to ambient conditions. We find that the
topological order persists even when the surface is exposed to air at room
temperature. However, the surface state is strongly modified after such an
exposure. Particularly, we have observed the formation of two-dimensional
quantum well states near the surface of the topological insulators after the
exposure which depends sensitively on the original composition, x, in
Bi2(Se3-xTex). These rich information are crucial in utilizing the surface
state and in probing its physical properties under ambient conditions.

The angle-resolved photoemission spectroscopy (ARPES) is a powerful
experimental tool to directly identify and characterize topological
insulatorsHasanARPES . A number of three-dimensional topological insulators
have been theoretically predicted and experimentally identified by
ARPESBiSbARPES ; BiSeTheo ; Bi2Se3ARPES ; Bi2Te3ARPES ; TlBiSe2Theo ;
TlBiSe2ARPES ; some of their peculiar properties have been revealed by
scanning tunneling microscopy (STM)TZhangSTM ; YazdaniSTM ; KaptulnikSTM ;
PCheng ; Hanaguri . The application of the topological surface states depends
on the surface engineering that can be manipulated by incorporation of non-
magneticLAWraypn ; mnmi ; ccvb ; crs  or magneticYLChenFe ; ZHasanPertur ;
LAWraypn ; mnmi  impurities or gas adsorptionsZHasanPertur ; LPlucinski ;
LAWraypn ; HMBenia . While the ARPES and STM measurements usually involve the
fresh surface obtained by cleaving samples in situ under ultra-high vacuum,
for the transport measurements which are widely used to investigate the
intrinsic quantum behaviors of the topological surface stateDXQu ; JGAnalytis
; ZRen ; JChen , and particularly the ultimate utilizations of the topological
insulators, the surface may be exposed to ambient conditions (1 atmosphere air
and room temperature) or some gas protection environment. It is therefore
crucial to investigate whether the topological order can survive under the
ambient conditions, and furthermore, whether and how the surface state may be
modified after such exposures.

We start by first looking at the electronic structure of the prototypical
topological insulators Bi2(Se,Te)3 under ultra-high vacuum. The Fermi surface
and the band structure of the Bi2(Se3-xTex) topological insulators depend
sensitively on the composition, x, as shown in Fig. 1. The single crystal
samples here were all cleaved in situ and measured at 30 K in an ultra-high
vacuum (UHV) chamber with a base pressure better than 5 $\times$ 10-11 Torr.
For Bi2Se3, a clear Dirac cone appears near -0.36 eV (Figs. 1d and 1e); the
corresponding Fermi surface (Fig. 1a) is nearly circular but with a clear
hexagon-shape in the measured dataHexBi2Se3 . It is apparently of n-type
because the Fermi level intersects with the bulk conduction band. On the other
hand, the Dirac cone of the Bi2Te3 sample lies near -0.08 eV (Figs. 1h and
1i), much closer to the Fermi level than that reported before (-0.34 eV in
Bi2Te3ARPES ). The corresponding Fermi surface (Fig. 1c) becomes rather small,
accompanied by the appearance of six petal-like bulk Fermi surface sheets.
These results indicate that our Bi2Te3 sample is of p-type because the Fermi
level intersects the bulk valence band along the
$\overline{\Gamma}$-$\overline{M}$ direction. This is also consistent with the
positive Hall coefficient measured on the same Bi2Te3 sampleCZhang . This
difference of the Fermi surface topology and the location of the Dirac cone
from othersBi2Te3ARPES  may be attributed to the different carrier
concentration in Bi2Te3 due to different sample preparation conditions. In our
Bi2(Se3-xTex) samples, we have seen a crossover from n-type Bi2Se3 to p-type
Bi2Te3. In order to eliminate the interference of the bulk bands on the
surface state near the Fermi level, we fine tuned the composition x in
Bi2(Se3-xTex) and found that, for x=2.6, nearly no spectral weight can be
discerned from the bulk conduction band, as seen from both the Fermi surface
(Fig. 1b) and the band structure (Figs. 1f and 1g). A slight substitution of
Te by Se in Bi2(Se0.4Te2.6) causes a dramatic drop of the Dirac point to -0.31
eV (Figs. 1f and 1g) and an obvious hexagon-shaped Fermi surface (Fig. 1b). It
is interesting to note that the hexagon-shape of Bi2(Se0.4Te2.6) (Fig. 1b) is
rather pronounced, although its Fermi surface size is smaller than that of
Bi2Se3 (Fig. 1a). The hexagonally-shaped Fermi surface observed in the
topological surface states reflects the hybridization of surface electronic
states with the bulk states and can be theoretically explained by considering
the higher order terms in the k $\cdot$ p HamiltonianHexWarp .

In order to directly examine how the topological surface state behaves under
ambient conditions in the topological insulators, we carried out our ARPES
measurements in different ways. (1). We first cleaved the sample _in situ_ and
performed ARPES measurement in the ultra-high vacuum (UHV) chamber. The sample
was then pulled out to another chamber filled with 1 atmosphere N2 gas,
exposed for about 5 minutes, before transferring back to the UHV chamber to do
ARPES measurements; (2). We cleaved and measured the sample in the UHV
chamber, and then pulled the sample out to air for 5 minutes before
transferring back to the UHV chamber for the ARPES measurements; (3). We
cleaved the sample in air and then transferred it to the UHV chamber to do the
ARPES measurement. Our measurements show that the above procedures of exposure
to air or N2 produce similar and reproducible results for a given sample.

The surface exposure of the topological insulators to air or N2 gives rise to
a dramatic alteration of the surface state, as shown in Figs. 2, 3 and 4, for
Bi2Se3, Bi2(Se0.4Te2.6), and Bi2Te3, respectively, when compared with those
for the fresh surface (Fig. 1). The first obvious change is the shifting of
the Dirac cone position relative to the Fermi level. For Bi2Se3,
Bi2(Se0.4Te2.6) and Bi2Te3, it shifts from the original -0.36 eV (Figs. 1d and
1e), -0.31 eV (Figs. 1f and 1g), -0.08 eV (Figs. 1h and 1i) for the fresh
surface to -0.48 eV (Fig. 2b), -0.40 eV (Figs. 3a and 3b), and -0.28 eV [Figs.
4(c-f)] at 30 K for the exposed surface, respectively. In all these cases, the
shift of the Dirac cone to a larger binding energy indicates an additional
doping of electrons into the surface state. The exposure also gives rise to a
dramatic change of the surface Fermi surface. For Bi2Se3, in addition to a
slight Fermi surface size increase, an obvious change occurs in the Fermi
surface shape that the hexagon-shape becomes much more pronounced in the
exposed surface (Fig. 2d) than that in the fresh sample (Fig. 1a). For
Bi2(Se0.4Te2.6), one clearly observes the much-enhanced warping effect in the
exposed surface (Fig. 3c) when compared with the nearly standard hexagon in
the fresh surface (Fig. 1b). The most dramatic change occurs for Bi2Te3 where
not only the Fermi surface size increases significantly, but also the warping
effect in the exposed surface (Fig. 4i) becomes much stronger. Overall, the
exposure causes the lowering of the Dirac cone position, an increase of the
surface Fermi surface size, and an obvious enhancement of the Fermi surface
warping effect in the Bi2(Se3-xTex) system.

The topological order in the Bi2(Se3-xTex) topological insulators is robust
even when the surface is exposed to ambient conditions, in spite of all the
alterations mentioned above. One clearly observes the persistence of the Dirac
cone in the exposed surface as in Bi2Se3 (Figs. 2b and 2c), in Bi2(Se0.4Te2.6)
(Figs. 3a and 3b), and Bi2Te3 (Figs. 4(c-h)). Particularly, this is the case
for the surface exposed to air and measured at room temperature (Fig. 2c for
Bi2Se3, and Figs. 4g and 4h for Bi2Te3). On the other hand, after the
exposure, although the signal of the surface state gets weaker for Bi2Se3
(Fig. 2), it remains rather strong for Bi2(Se0.4Te2.6) (Fig. 3) and Bi2Te3
(Fig. 4). This is in a stark contrast to the conventional trivial surface
state where minor surface contamination will cause the extinction of the
surface stateHuefner . The robustness of the topological order to Coulomb,
magnetic and disorder perturbations has been reported beforeZHasanPertur ;
LPlucinski . Our present observations directly demonstrate the robustness of
the topological order against absorption and thermal process under ambient
conditions due to the protection of the time-reversal symmetryHasanKaneRev ;
ZhangQiRMP .

The surface exposure to air or N2 in the Bi2(Se3-xTex) topological insulators
produces two-dimensional electronic states near the surface. In Bi2Se3, the
exposure gives rise to additional parabolic bands, as schematically marked by
the dashed line in Figs. 2b and 2c. Correspondingly, this leads to additional
Fermi surface sheet(s) inside the regular topological surface state (Figs. 2d
and 2e). In Bi2(Se0.4Te2.6), this effect gets more pronounced and the newly
emerged bulk conduction band splits into several discrete bands, as marked by
the dashed lines in Fig. 3b. While the band quantization effect occurs in the
bulk conduction band in Bi2(Se0.4Te2.6), it shows up in the valence band in
the exposed Bi2Te3 surface, as shown in Figs. 4(c-h), where one can see a
couple of discrete “M”-shaped bands. The quantized bands are obvious at low
temperature and get slightly smeared out when the temperature rises to room
temperature (Figs. 4g and 4h).

The formation of the split bands in the exposed surface of the topological
insulators is reminiscent of the quantum well states observed in the quantum
confined systemsSiAg  and in some topological insulatorstdeg ; mnmi ; crs ;
ccvb ; HMBenia ; KingReshba . There are a couple of possibilities that the
quantum well states may be formed. One usual way is due to band bending
effect. As mentioned before, the surface exposure to air or N2 causes an
electron transfer to the surface of the topological insulators. The
accumulation of these additional electrons near the surface would lead to a
downward bending of the bulk bands near the surface region, as schematically
shown in Fig. 3d, resulting in a “V”-shaped potential well where the bulk
conduction band of electrons can be confined. This picture, as proposed
beforetdeg ; ccvb , seems to be able to explain the two-dimensional quantum
well states in the conduction bands in Bi2(Se0.4Te2.6) (Figs. 3a and 3b).
However, it becomes questionable to explain the quantum well states observed
in the bulk valence band of Bi2Te3 (Figs. 4c-j). In this case, the downward
band bending no longer acts as a quantum well potential for the valence band
top because the charge carriers are hole-like. The band-bending is therefore
not a general picture that can explain the formation of the two-dimensional
quantum well states in Bi2(Se0.4Te2.6) (Figs. 3a and 3b) and Bi2Te3 (Fig. 4)
topological insulators on the same footing. An alternative scenario is the
expansion of van der Waals spacings in between the quintuple layers (QLs)
caused by the intercalation of gasesSVEremeev . The observation of multiple
split bands with different spacings would ask for multiple van der Waals gaps
with different expansions. Whether and how these can be realized in the
exposed surface remains to be investigated. We note that our observations of
multiple split bands are similar to those seen in the ultra-thin films of
Bi2Se3YZhang  and Bi2Te3YYLi . From our first principle band structure
calculations on Bi2Te3 with different number of quintuple layers, we also find
that a detached slab with a thickness of 7 quintuple layers can give a rather
consistent description (Fig. 4l) of our observed results in terms of the
quantitative spacings between the 3 resolved bands (VB0, VB1, and VB2 bands as
marked in Figs. 4c and 4l). In addition, the distance between the conduction
band bottom (CB0 band in Figs. 4i and 4l) and the first valence sub-band
bottom (VB0 band in Figs. 4i and 4l) is rather consistent between the measured
and calculated results. These seem to suggest that a “confined surface slab”
with nearly 7 quintuple layers may be formed after the exposure that acts more
or less independently from the bulk. More work needs to be done to further
investigate whether such a confined surface slab can be thermodynamically
stable. Overall, the formation of the two-dimensional quantum well states is a
general phenomenon for the exposed surface of the Bi2(Se3-xTex) topological
insulators; the effect depends sensitively on the composition x of the samples
which may facilitate manipulation of these quantum well states.

The present work has significant implications on the fundamental study and
ultimate applications of the topological insulators. Many experimental
measurements, such as some transport measurements, involve samples exposed to
ambient conditions. The practical applications may involve sample surface
either exposed to ambient condition, or be in contact with other magnetic or
superconducting materials. On the one hand, the robustness of the topological
order under ambient conditions sends a good signal for these experimental
characterization and practical utilizations. The formation of the quantum well
states may give rise to new phenomena to be studied and utilized. The
sensitivity of the surface state to the Bi2(Se3-xTex) composition provides a
handle to manipulate these quantum states. On the other hand, the strong
modification of the electronic structure and the formation of additional
quantum well states in the exposed surface have to be considered seriously in
interpreting experimental data and in surface engineering. It is critical to
realize before hand that the surface under study or to be utilized may exhibit
totally different behaviors as those from the fresh surface cleaved in ultra-
high vacuum. In addition to the alteration of electronic states upon exposure,
the transport properties of the topological surface state may be further
complicated by the formation of quantum well states.

We thank Prof. X. H. Chen for providing us samples at the initial stage of the
project, and Prof. Liling Sun and Prof. Zhong-xian Zhao for their help in the
characterization of the samples. This work is supported by the NSFC (91021006)
and the MOST of China (973 program No: 2011CB921703).

METHODS

Crystal growth methods  Single crystals of $Bi_{2}(Se_{3-x}Te_{x})$ (x=0, 2.6
and 3) were grown by the self-flux method. Bismuth, selenium and tellurium
powders were weighed according to the stoichiometric $Bi_{2}(Se_{3-x}Te_{x})$
(x=0, 2.6 and 3) composition. After mixing thoroughly, the powder was placed
in alumina crucibles and sealed in a quartz tube under vacuum. The materials
were heated to 1000 ∘C, held for 12 hours to obtain a high degree of mixing,
and then slowly cooled down to 500 ∘C over 100 hours before cooling to room
temperature. Single crystals of several millimeters in size were obtained. The
crystal structure of the resulting crystals was examined by use of a rotating
anode x-ray diffractometer with Cu _K α_ radiation ($\lambda$ = 1.5418 Å). The
chemical composition of the crystals was analyzed by the energy dispersive
X-ray spectroscopy (EDAX) and the induction-coupled plasma atomic emission
spectroscopy (ICP-AES). The resistivity of the crystals was measured by the
standard four-probe method.

Laser-ARPES methods.  The angle-resolved photoemission measurements were
carried out on our vacuum ultra-violet (VUV) laser-based angle-resolved
photoemission systemLiuARPES . The photon energy of the laser is 6.994 eV with
a bandwidth of 0.26 meV. The energy resolution of the electron energy analyzer
(Scienta R4000) is set at 1 meV, giving rise to an overall energy resolution
of $\sim$1 meV which is significantly improved from 10$\sim$15 meV from
regular synchrotron radiation systemsBi2Se3ARPES ; Bi2Te3ARPES . The angular
resolution is $\sim$0.3∘, corresponding to a momentum resolution $\sim$0.004
$\AA$-1 at the photon energy of 6.994 eV, more than twice improved from 0.009
$\AA$-1 at a regular photon energy of 21.2 eV for the same angular resolution.
Our superior instrumental resolution of laser ARPES has made the measured
features of topological insulators in this work much sharper. The Fermi level
is referenced by measuring on a clean polycrystalline gold that is
electrically connected to the sample. The samples were all measured in vacuum
with a base pressure better than 5$\times$10-11 Torr.

∗Corresponding author<EMAIL_ADDRESS>
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Figure 1: Fermi surface and band structure of Bi2(Se3-xTex) (x=0, 2.6, 3)
topological insulators cleaved in situ and measured at 30 K in ultra-high
vacuum. (a-c) show Fermi surface of Bi2Se3, Bi2(Se0.4Te2.6) and Bi2Te3,
respectively. The Fermi surface here, and in other figures below, are original
data without involving artificial symmetrization. The band structures along
two high symmetry lines $\bar{\Gamma}-\bar{K}$ and $\bar{\Gamma}-\bar{M}$ are
shown in (d,e) for Bi2Se3, in (f,g) for Bi2(Se0.4Te2.6) and (h,i) for Bi2Te3.
Figure 2: Fermi surface and band structure of Bi2Se3 cleaved in air and
measured in the ultra-high vacuum (UHV) chamber. (a). Band structure of the
fresh Bi2Se3 cleaved and measured in the UHV chamber at 30 K along
$\bar{\Gamma}-\bar{M}$ direction. (b). Band structure of Bi2Se3 cleaved in air
and measured in UHV at 30 K along $\bar{\Gamma}-\bar{M}$ direction. (c). Band
structure of Bi2Se3 cleaved in air and measured in UHV at 300 K along
$\bar{\Gamma}-\bar{M}$ direction. (d,e). Fermi surface of Bi2Se3 cleaved in
air and measured in UHV at 30K and 300 K, respectively. Black dashed lines in
(b) and (c) mark the parabolic bands above the Dirac point from the two-
dimensional electron gas. Figure 3: Emergence of quantum well states in
Bi2Te2.6Se0.4 after exposing to N2. (a,b). Band structure measured at 30 K
along $\bar{\Gamma}-\bar{K}$ and $\bar{\Gamma}-\bar{M}$, respectively. Black
dashed lines in (b) mark the quantum well states formed in the bulk conduction
band (BCB) above the Dirac point. (c). The corresponding Fermi surface. It
shows three-fold symmetry where three corners of M points are strong while the
other three are weak. This is also in agreement with the asymmetric band
structure in Fig. 3b. (d). Schematic band structure showing the possible
formation of the quantum well states near the sample surface in the bulk
conduction band. The blue dotted lines between the bulk valence band (BVB) and
bulk conduction band (BCB) represent the topological surface states while the
blue solid lines represent quantum well states.  Figure 4: Persistence of
topological surface state and formation of quantum well states in Bi2Te3 after
exposure to N2 or air. The sample was first cleaved and measured in UHV at 30
K. (a,b) show the corresponding band structure along the
$\bar{\Gamma}-\bar{K}$ and $\bar{\Gamma}-\bar{M}$ directions. The sample was
then pulled out from the UHV chamber and exposed to N2 at 1 atmosphere for 5
minutes before transferring back into UHV chamber for the ARPES measurement.
(c,d) show the band structure of the N2-exposed sample along the
$\bar{\Gamma}-\bar{K}$ and $\bar{\Gamma}-\bar{M}$ directions. The black dashed
lines in (c) illustrate the quantum well states formed in the bulk valence
band below the Dirac point. The sample was then pulled out again and exposed
to air for 5 minutes before putting back in vacuum for ARPES measurement.
(e,f) show the band structure of the air-exposed sample at 30 K along the
$\bar{\Gamma}-\bar{K}$ and $\bar{\Gamma}-\bar{M}$ directions. (g,h) show the
measurements at 300 K and (i,j) show their corresponding second-derivative
images in order to highlight the bands. (k) Fermi surface of N2-exposed
sample. (l). First principle calculation of the band structure of Bi2Te3 slab
with seven quintuple layers.