%Paper: hep-th/9308047
%From: Tatsuo Kobayashi <kobayasi@hep.s.kanazawa-u.ac.jp>
%Date: Tue, 10 Aug 93 16:43:39 +0900



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%%%%%%  "Differential Calculi on h-deformed Bosnic        %%%%%%%
%%%%%%   and Fermionic Quantum Planes                     %%%%%%%
%%%%%%    by T. Kobayashi                                 %%%%%%%
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\begin{flushright}

        {\normalsize
 KANAZAWA-93-07\\

   August,1993   }\\
\end{flushright}
\vfill
\begin{center}
{\large \bf Differential Calculi on h-deformed Bosonic and Fermionic Quantum
Planes}
\vfill
%\renewcommand{\thefootnote}{\fnsymbol{footnote}}
{\bf Tatsuo Kobayashi}

       Department of Physics, Kanazawa University, \\
       Kanazawa, 920-11, Japan \\
\vfill



\end{center}


\vfill
\nopagebreak
\begin{abstract}
We study differential calculus on h-deformed bosonic and fermionic quantum
space.
It is shown that the fermionic quantum space involves a parafermionic variable
as well as a classical fermionic one.
Further we construct the classical $su(2)$ algebra on the fermionic quantum
space and discuss a mapping between the classical $su(2)$ and the h-deformed
$su(2)$ algebras.

\end{abstract}

\vfill
\end{titlepage}
\pagestyle{plain}
\newpage
\voffset = -2.5 cm
\leftline{\large \bf 1. Introduction}
\vspace{0.8 cm}



Quantum groups and quantum algebras [1-4] have attracted much attention in
theoretical physics and mathematics, such as statistical models, integrable
models, conformal field theories, knot theory and so on [5-8].
Quantum space was introduced to represent quatum groups \cite {Manin}.
Differential calculus on the quantum space was also studied [10-15].
It is intriguing also from the viewpoint of non-commutative geometry.
The quantum differential calculus is closely related with q-oscillators,
which have been applied to various fields [16-18].
Further, we can obtain quantum groups $Sp_q(2n)$ and $SO_q(2n)$ using the
q-deformed phase space which is defined through the differential calculus
on the quantum space for $SL_q(n)$ \cite {Zumino}.
Furthermore, on the quantum space quantum deformed algebras have been
constructed, e.g., q-deformed Lorentz, Poincar{\'e}, conformal and
superconformal algebras [20-25].
These analyses imply that the quantum space is interesting as applications of
quantum groups and useful to show some aspects of quantum groups.


Recently other than the conventional q-deformation and its multi-parametric
extension, another deformation was discovered in [26-29] and is called
h-deformation.
Further, differential calculus on an h-deformed quantum bosonic space has been
 considered in \cite {Karimi,Agham}.
The purpose of this article is to investigate in detail bosonic and fermionic
differential calculus on h-deformed quantum planes, extending the above
analyses on the q-deformed differential calculus to the h-deformed case.
The differential calculus on the bosonic quantum space is studied and
a comment is obtained from the viewpoint of a constrainted system.
In addition, we discuss a differential calculus on an h-deformed fermionic
space.
Ref. \cite{Koba} shows that q-deformed fermionic coordinates and derivatives
represent $su_q(2)$ algebra, which is related with Drinfeld-Jimbo basis
by some mapping.
Following the approach, we investigate algebra which is constructed in terms
of the h-deformed fermionic elements.


This paper is organized as follows.
In section two we review on general non-commutative differential calculus
including the quantum space.
Then using a new solution of the Yang-Baxter equation, we derive h-deformed
bosonic differential calculus.
Further some comments on the h-deformed space are given.
In section three, similarly we derive h-deformed differential calculus on the
fermionic quantum space.
It is shown that we can represent the classical Lie algebra $su(2)$,
using their fermionic variables and derivatives.
Also we obtain a mapping between the classical algebra $su(2)$ and the
h-deformed algebra $su_h(2)$.
Further we discuss an h-deformed $SO(4)$ group.
The last section is devoted to conclusion and discussion.


\vspace{0.8 cm}
\leftline{\large \bf 2. Differential calculus on h-deformed bosonic space}
\vspace{0.8 cm}


A quantum space is a non-commutative space representing the corresponding
quantum group.
Ref. \cite{WZ} clarified general non-commutative differential calculus
including the quantum space and its analysis was extended to superspaces in
\cite{KU1}.
We set up commutation relations between coordinates $x^i$ and derivatives
$\partial_{x^i}$ as follows,
$$x^ix^j=B^{ij}_{\ \ k\ell}x^kx^\ell, \quad
\partial_{x^k} x^i=\delta^i_k+C^{ij}_{\ \ k\ell}x^\ell
\partial_{x^j}.
\eqno(2.1)$$
These matrices should satisfied the following relations:
$$B^{ij}_{\ \ pr}B^{rk}_{\ \ tn}B^{pt}_{\ \ \ell m}=
B^{jk}_{\ \ pr}B^{ip}_{\ \ \ell t}B^{tr}_{\ \ mn}, \quad
(\delta^i_k \delta^j_\ell-B^{ij}_{\ \ k\ell})(\delta^k_m\delta^\ell_n
+C^{k\ell}_{\ \ mn})=0.
\eqno(2.2)$$
The former equation is called Yang-Baxter equation.
We consider the case where $B^{ij}_{\ \ k\ell}$ is proportinal to
$C^{ij}_{\ \ k\ell}$.
In this case we can write commutation relations of derivatives by $B$-matrix
as follows,
$$\partial_{x^\ell}\partial_{x^k}=
B^{ij}_{\ \ k\ell}\partial_{x^j}\partial_{x^i}.
\eqno(2.3)$$


Recently a new solution of the Yang-Baxter equantion was discovered in
\cite{Ewen} as follows,
$$
 \widehat R =
\left(\begin{array}{cccc}
1 & -h' & h' & hh' \\
0 & 0 & 1 & h \\
0 & 1 &0 & -h  \\
0 & 0 & 0 & 1
\end{array}\right).
\eqno(2.4)$$
Also in Ref. \cite{Zak} and \cite{Dem}, the same type of the
$\widehat R$-matrix was discovered in the case where $h=h'$ and $h=h'=-1$,
respectively.
New type of a quantum group $SU_h(2)$ was studied in [26-28] and the
corresponding deformed algebra $su_h(2)$ has been constructed in \cite{Ohn}.
We identify $\widehat R$ with $B$ in order to obtain h-deformed differential
calculus.
The $\widehat R$-matrix has $\pm 1$ as eigenvalues.
This fact leads to a condition that we should identify $C$ with $\widehat R$,
too.
Namely we hereafter study the case where
$$ B^{ij}_{\ \ k\ell}=C^{ij}_{\ \ k\ell}=\widehat R^{ij}_{\ \ k\ell}.
\eqno(2.5)$$
We restrict ourselves to the case where $h=h'$.
It is easy to extend the following analysis to the case with two
independent parameters, $h$ and $h'$.
We substitute eq.(2.4) into (2.1) and (2.3), so that we obtain the following
commutation relations;
$$[x^1,x^2]=h(x^2)^2,\quad [\partial_{x^1},\partial_{x^2}]=h(\partial_{x^1})^2,
$$
$$[\partial_{x^2},x^1]=hx^1\partial_{x^1}+hx^2\partial_{x^2}
+h^2x^2\partial_{x^1},
\eqno(2.6)$$
$$[x^i\partial_{x^i},x^i]=1-hx^2\partial_{x^1}, \quad
[\partial_{x^1},x^2]=0.$$
The above algebra becomes classical in the limit, i.e.,
$h \rightarrow 0$.


We can introduce h-deformed quantum group $T^i_{\ j}$ such that the elements
transform the coordinates and the derivatives like $T^i_{\ j}x^j$.
Covariance of the commutation relations (2.6) under the transformation
requires the following commutation relation of $T^i_{\ j}$;
$$\widehat R^{ij}_{\ \ k\ell}T^k_{\ m}T^\ell_{\ n}=
T^i_{\ k}T^j_{\ \ell}\widehat R^{k\ell}_{\ \ mn}.
\eqno(2.7)$$


Next we consider conjugation consistent with (2.6).
Suppose that $h$ is pure imaginary, i.e., $\overline h=-h$.
Then we obtain the consistent conjugation as follows,
$$\overline {x^1}=x^1+hx^2, \quad \overline {x^2}=x^2,$$
$$ \overline {\partial_{x^1}}=-\partial_{x^1}, \quad
\overline {\partial_{x^2}}=-\partial_{x^2}-h\partial_{x^1}.
\eqno(2.8)$$
Through the conjugation we can define real coordinates and momenta as follows,
$$\widehat x^1=x^1+{h \over 2}x^2, \quad \widehat x^2 =x^2,$$
$$\widehat p_1=-i\partial_{x^1}, \quad
\widehat p_2=-i(\partial_{x^2}+{h \over 2}\partial_{x^1}).
\eqno(2.9)$$
They satisfy the following commutation relations;
$$[\widehat x^1,\widehat x^2]=h(\widehat x^2)^2, \quad
[\widehat p_1,\widehat p_2]=h(\widehat p_1)^2,$$
$$[\widehat p_j,\widehat x^j]=-i-h\widehat x^2\widehat p_1, \quad
[\widehat p_1,\widehat x^2]=0,
\eqno(2.10)$$
$$[\widehat p_2,\widehat x^1]=h(-i+\widehat x^1 \widehat p_1+
\widehat x^2 \widehat p_2-h\widehat x^2 \widehat p_1).$$
The similar phase space algebra as well as (2.6) has been obtained in
\cite{Karimi}.


These h-deformed coordinates and momenta can be represented in terms of
classical ones $\tilde x$ and $\tilde p$.
Suppose that $\widehat x^2=\tilde x^2$ and $\widehat p_1=\tilde p_1$, then we
have
$$\widehat x^1=\tilde x^1(1-ih\tilde p_1\tilde x^2)+ih(\tilde x^2)^2\tilde p_2,
 \quad
\widehat p_2=\tilde p_2(1-ih\tilde p_1\tilde x^2)+ih(\tilde p_1)^2\tilde x^2.
\eqno(2.11)$$


The deformed phase space algebra such as eq.(2.10) might remind us of Dirac
brackets of a constrainted system \cite{Dirac}.
 Actually we can find a \lq \lq constraint" which we can make identically
 equal to zero through the above commutation relations (2.10), as follows,
$$f=1-2ih\widehat p_1\widehat x^2.
\eqno(2.12)$$
The function $f$ commutes with $\widehat x^2$ and $\widehat p_1$ and satisfies
commutation relations with the other elements as follows,
$$[f,\widehat x^1]=i\widehat x^2 f,\quad [\widehat p_2,f]=i\widehat p_1f.
\eqno(2.13)$$
These relations seems to be somewhat different from the Dirac bracket, whose
right hand side vanishes completely.
But we can make $f$ equal to zero identically.
The above fact seems to show that $SU_h(2)$ is a `group' which transforms the
\lq \lq constraint" $f$ and its relations (2.13) covariantly.


\vspace{0.8 cm}
\leftline{\large \bf 3. Differential calculus on h-deformed fermionic space}
\vspace{0.8 cm}


In this section we discuss differential calculus on an h-deformed quantum
fermionic space.
Commutation relations of h-deformed coordinates $\theta^\alpha$ and
derivatives $\partial_\alpha$ are obtained by replacing $\widehat R$ in (2.1)
and (2.3) in terms of $-\widehat R$ as follows,
$$\theta^\alpha \theta^\beta=-\widehat R^{\alpha \beta}_{\ \ \mu \nu}
\theta^\mu \theta^\nu,
\quad \partial_\nu \partial_\mu =-\widehat R^{\alpha \beta}_{\ \ \mu \nu}
\partial_\beta \partial_\alpha,$$
$$\partial_\mu \theta^\alpha=\delta^\alpha_\mu
-\widehat R^{\alpha \beta}_{\ \ \mu \nu} \theta^\nu \partial_\beta.
\eqno(3.1)$$
These relations are written explicitly as follows,
$$(\theta^1)^2=h\theta^1\theta^2,\quad \{\theta^1,\theta^2\}=(\theta^2)^2=0,$$
$$(\partial_1)^2=\{\partial_1,\partial_2\}=0,\quad (\partial_2)^2=h\partial_1
\partial_2,$$
$$\{\partial_\alpha,\theta^\alpha \}=1+h\theta^2 \partial_1, \quad
\{\partial_1, \theta^2 \}=0,
\eqno(3.2)$$
$$\{\partial_2,\theta^1 \}=-h\theta^1 \partial_1-h\theta^2 \partial_2
-h^2\theta^2\partial_1.$$
Further eq.(3.2) leads to the following trilinear relations;
$$(\theta^1)^3=(\partial_2)^3=0.
\eqno(3.3)$$
The coordinate $\theta^1$ and the derivative $\partial_2$ are
parafermionic \cite{Para}, while its derivatives $\partial_1$ and its
coordinates $\theta^2$ are nilpotent.


As discussed in the previous section, we can introduce h-deformed quantum group
elements $T^\alpha_{\ \beta}$ which transform $\theta^\alpha$ into
$\theta'^\alpha=T^\alpha_{\ \beta}\theta^\beta$.
The elements satisfy the same commutation relation as (2.7).
This h-deformed quantum group is interesting also from the aspect that it
transforms fermionic and parafermionic variables, i.e., $\theta^1$ and
$\theta^2$.
Further, we can define a determinant of $T^\alpha_{\ \beta}$ using the
h-deformed fermionic space as follows,
$$\theta'^1 \theta'^2={\rm det} T \cdot \theta^1 \theta^2.
\eqno(3.4)$$
The definition leads to
$${\rm det}T=T^1_{\ 1}T^2_{\ 2}-T^1_{\ 2}T^2_{\ 1}+hT^1_{\ 1}T^2_{\ 1}.
\eqno(3.5)$$
Eq.(3.5) coinsides with the definition of the determinant in \cite{Dem}.


Ref.\cite{Koba} shows that q-deformed fermionic coordinates and derivatives
can represent q-deformed quantum algebra, e.g., $su_q(2)$.
Here we apply the similar analysis to the h-deformed case.
In the similar way to \cite{Koba}, we define the following generators;
$$L_+=\theta^2 \partial_1, \quad L_-=\theta^1 \partial_2.
\eqno(3.6)$$
Using their commutation relation, we introduce a Cartan generator as follows,
$$L_0=[L_+,L_-]=\theta^2\partial_2-\theta^1\partial_1.
\eqno(3.7)$$
In addition to (3.7), these generators satisfy the following relations;
$$[L_0,L_\pm]=\pm 2L_\pm.
\eqno(3.8)$$
Eqs. (3.7) and (3.8) are nothing but the classical $su(2)$ algebra up to the
normalization factor, although the coordinates and the derivatives are
deformed.
The generators act on the coordinates and the derivatives as follows,
$$[L_0,\theta^\alpha]=(-1)^\alpha \theta^\alpha, \quad
[L_0,\partial_\alpha]=-(-1)^\alpha \partial_\alpha,$$
$$[L_+,\theta^1]=\theta^2, \quad [L_+,\partial_2]=-\partial_1,$$
$$[L_+,\theta^2]=[L_+,\partial_1]=0,$$
$$[L_-,\theta^1]=-h\theta^1 L_0-h^1\theta^2 L_+,
\eqno(3.9)$$
$$[L_-,\theta^2]=\theta^1 +h\theta^2 L_0,$$
$$[L_-,\partial_1]=-\partial_2+h\partial_1 L_0+h\partial_1 L_0,$$
$$[L_-,\partial_2]=-h\partial_2 L_0-3h^2\partial_1+3h^2\partial_2 L_+.$$
The actions of $L_+$ and $L_0$ are never deformed, while those of $L_-$ are
deformed.


In Ref.\cite{Ohn} a h-deformed $su(2)$ algebra has been constructed as follows,
$$[H,X]={2 {\rm sinh} (hX) \over h},$$
$$[H,Y]=-\{Y,{\rm cosh} (hX) \},
\eqno(3.10)$$
$$[X,Y]=H.$$
The h-deformed algebra $su_h(2)$ can be related with the classical algebra
$su(2)$ through the following mapping;
$$X={{\rm log}\tilde L_+ \over h},$$
$$H=(\tilde L_+ -(\tilde L_+)^{-1})\tilde L_0, \eqno(3.11)$$
$$Y={h \over 2}(\tilde L_+ -(\tilde L_+)^{-1})(2\tilde L_+\tilde L_- -h
\tilde L_0),$$
where $\tilde L_0 =L_0/2$ and $\tilde L_\pm =L_\pm /\sqrt 2$.


In Ref.\cite{Zumino} Zumino derived quantum groups $Sp_q(2n)$ and $SO_q(2n)$
from
q-deformed bosonic and fermionic phase space algebra for $SL_q(n)$.
That was extended to the supersymmetric case in \cite{KU1}.
In the similar way, here we discuss h-deformation of $SO(4)$.
At first, we define h-deformed gamma matrices $\gamma^\alpha$ as
$$\gamma^\alpha = \partial_\alpha, \quad \gamma^{5-\alpha}=\theta^\alpha.
\eqno(3.12)$$
They satisfy the following h-deformed Clifford algebra;
$$ \gamma^\alpha \gamma^\beta +\tilde B^{\alpha \beta}_{\ \ \mu \nu}
\gamma^\mu \gamma^\nu=\eta^{\alpha \beta},
\eqno(3.13)$$
where $\eta^{\alpha \beta}$ is an SO(4) metric.
The matrix $\tilde B$ is composed of $\widehat R$-matrix (2.4) as follows,
$$\tilde B^{\alpha'\beta'}_{\ \ \mu' \nu'}=
-\widehat R^{\alpha \beta}_{\ \ \mu \nu}, \quad
\tilde B^{\alpha \beta}_{\ \ \mu \nu}=
-\widehat R^{\nu \mu}_{\ \ \beta \alpha},$$
$$\tilde B^{\alpha \beta'}_{\ \ \mu' \nu}=
-\widehat R^{\beta \nu}_{\ \ \alpha \mu }, \quad
\tilde B^{\alpha' \beta}_{\ \ \mu \nu'}=
-(\widehat R^{-1})^{\alpha \mu}_{\ \ \beta \nu},
\eqno(3.14)$$
where $\alpha,\beta,\mu,\nu=1,2$ and $\alpha'=5-\alpha$.
Ref.\cite{KU1} shows that $\tilde B$ constructed through the above procedure
satisfy the Yang-Baxter equation, if $\widehat R$ is the solution of the
Yang-Baxter equation.
We could introduce h-deformed $SO(4)$ group which transform the relation
(3.13) covariantly.
Instead of deriving explicitly $SO_h(4)$, we here introduce h-deformed
$SO(4)$ quantum space $X^i$ $(i=1\sim 4)$ whose commutation relations are
written as $X^iX^j=\tilde B^{ij}_{\ \ k \ell}X^kX^\ell$.
We have explicitly
$$[X^1,X^2]=h(X^1)^2, \quad [X^1,X^3]=0,$$
$$[X^1,X^4]=[X^2,X^3]=-hX^3X^1,
\eqno(3.15)$$
$$[X^2,X^4]=hX^4X^1+hX^3X^2+h^2X^3X^1, \quad
[X^3,X^4]=-h(X^3)^2.$$
The algebra has a center element $C\equiv X^4X^1+X^3X^2$.
Differential calculus on the above h-deformed space $X^i$ could be similarly
obtained.
Their commutation relations are covariant under the $SO_h(4)$ transformation,
as said the above.
Elements of $SO_h(4)$ satisfy the same relation as (2.7) except
$\widehat R$-matrix replacing $\tilde B$-matrix.


\vspace{0.8 cm}
\leftline{\large \bf 4. Conclusion}
\vspace{0.8 cm}

We have studied here the differential calculi on the h-deformed bosonic and
fermionic spaces.
%In the bosonic case, we have found a constraint.
%It might be possible to recognize that $SU_h(2)$ is a transformation for the
%constrainted system.
We have constructed the classical $su(2)$ algebra on the h-deformed fermionic
space.
The algebra is related with the h-deformed algebra $su_h(2)$.
It is shown that $SU_h(2)$ is a 'group' which transforms fermionic and
parafermionic variables into each other.
This fact is very interesting in applications to the parastatistics.
Also $SO_h(4)$ was discussed.
Supersymmetric extension is also intriguing.
It is easy to introduce h-deformed oscillators in the similar way to the above
 h-deformed differential calculus.
For example, we can define a sort of deformed oscillators by identifying
the h-deformed coordinates and derivatives with deformed anihilation and
creation operators, respectively.
Their applications to various fields are very interesting.


\vspace{0.8 cm}
\leftline{\large \bf Acknowledgement}
\vspace{0.8 cm}


The author would like to thank T.~Uematsu for numerous valuable discussions
and reading the manuscript.
He also thanks to P.~P.~Kulish, R.~Sasaki and H.~Terao for helpful discussions.




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