alasdairforsythe/text-english-code-fiction-nonfiction

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We give the
formulas for the calculi corresponding to coirreducible ones.

Another interesting aspect of viewing \ensuremath{U(\lalg{b_+})}{} as a quantum function
algebra is the connection to quantum deformed models of space-time and
its symmetries. In particular, the $\kappa$-deformed Minkowski space
coming from the $\kappa$-deformed Poincar\'e algebra
\cite{LuNoRu}\cite{MaRu} is just a simple generalisation of \ensuremath{U(\lalg{b_+})}.
We use this in section \ref{sec:kappa} to give
a natural $4$-dimensional differential calculus. Then we show (in a
formal context) that integration is given by
the usual Lesbegue integral on $\mathbb{R}^n$ after normal ordering.
This is obtained in an intrinsic context different from the standard
$\kappa$-Poincar\'e approach.

A further important motivation for the investigation of differential
calculi on
\ensuremath{U(\lalg{b_+})}{} and \ensuremath{C(B_+)}{} is the relation of those objects to the Planck-scale
Hopf algebra \cite{Majid_Planck}\cite{Majid_book}. This shall be
developed elsewhere.

In the remaining parts of this introduction we will specify our
conventions and provide preliminaries on the quantum group \ensuremath{U_q(\lalg{b_+})}, its
deformations, and differential calculi.


\subsection{Conventions}

Throughout, $\k$ denotes a field of characteristic 0 and
$\k(q)$ denotes the field of rational
functions in one parameter $q$ over $\k$.
$\k(q)$ is our ground field in
the $q$-deformed setting, while $\k$ is the
ground field in the ``classical'' settings.
Within section \ref{sec:q} one could equally well view $\k$ as the ground
field with $q\in\k^*$ not a root of unity. This point of view is
problematic, however, when obtaining ``classical limits'' as
in sections \ref{sec:class} and \ref{sec:dual}.

The positive integers are denoted by $\mathbb{N}$ while the non-negative
integers are denoted by $\mathbb{N}_0$.
We define $q$-integers, $q$-factorials and
$q$-binomials as follows:
\begin{gather*}
[n]_q=\sum_{i=0}^{n-1} q^i\qquad
[n]_q!=[1]_q [2]_q\cdots [n]_q\qquad
\binomq{n}{m}=\frac{[n]_q!}{[m]_q! [n-m]_q!}
\end{gather*}
For a function of several variables (among
them $x$) over $\k$ we define
\begin{gather*}
(T_{a,x} f)(x) = f(x+a)\\
(\fdiff_{a,x} f)(x) = \frac{f(x+a)-f(x)}{a}
\end{gather*}
with $a\in\k$ and similarly over $\k(q)$
\begin{gather*}
(Q_{m,x} f)(x) = f(q^m x)\\
(\partial_{q,x} f)(x) = \frac{f(x)-f(qx)}{x(1-q)}\\
\end{gather*}
with $m\in\mathbb{Z}$.

We frequently use the notion of a polynomial in an extended
sense. Namely, if we have an algebra with an element $g$ and its
inverse $g^{-1}$ (as
in \ensuremath{U_q(\lalg{b_+})}{}) we will mean by a polynomial in $g,g^{-1}$ a finite power
series in $g$ with exponents in $\mathbb{Z}$. The length of such a polynomial
is the difference between highest and lowest degree.

If $H$ is a Hopf algebra, then $H^{op}$ will denote the Hopf algebra
with the opposite product.

\subsection{\ensuremath{U_q(\lalg{b_+})}{} and its Classical Limits}
\label{sec:intro_limits}

We recall that,
in the framework of quantum groups, the duality between enveloping algebra
$U(\lalg{g})$ of the Lie algebra and algebra of functions $C(G)$ on the Lie
group carries over to $q$-deformations.
In the case of
$\lalg{b_+}$, the
$q$-deformed enveloping algebra \ensuremath{U_q(\lalg{b_+})}{} defined over $\k(q)$ as
\begin{gather*}
U_q(\lalg{b_+})=\k(q)\langle X,g,g^{-1}\rangle \qquad
\text{with relations} \\
g g^{-1}=1 \qquad Xg=qgX \\
\cop X=X\otimes 1 + g\otimes X \qquad
\cop g=g\otimes g \\
\cou (X)=0 \qquad \cou (g)=1 \qquad
\antip X=-g^{-1}X \qquad \antip g=g^{-1}
\end{gather*}
is self-dual. Consequently, it
may alternatively be viewed as the quantum algebra \ensuremath{C_q(B_+)}{} of
functions on the Lie group $B_+$ associated with $\lalg{b_+}$.
It has two classical limits, the enveloping algebra \ensuremath{U(\lalg{b_+})}{}
and the function algebra $C(B_+)$.
The transition to the classical enveloping algebra is achieved by
replacing $q$
by $e^{-t}$ and $g$ by $e^{tH}$ in a formal power series setting in
$t$, introducing a new generator $H$. Now, all expressions are written in
the form $\sum_j a_j t^j$ and only the lowest order in $t$ is kept.

We give the

formulas for the calculi corresponding to coirreducible ones.

Another interesting aspect of viewing \ensuremath{U(\lalg{b_+})}{} as a quantum function

algebra is the connection to quantum deformed models of space-time and

its symmetries. In particular, the $\kappa$-deformed Minkowski space

coming from the $\kappa$-deformed Poincar\'e algebra

\cite{LuNoRu}\cite{MaRu} is just a simple generalisation of \ensuremath{U(\lalg{b_+})}.

We use this in section \ref{sec:kappa} to give

a natural $4$-dimensional differential calculus. Then we show (in a

formal context) that integration is given by

the usual Lesbegue integral on $\mathbb{R}^n$ after normal ordering.

This is obtained in an intrinsic context different from the standard

$\kappa$-Poincar\'e approach.

A further important motivation for the investigation of differential

calculi on

\ensuremath{U(\lalg{b_+})}{} and \ensuremath{C(B_+)}{} is the relation of those objects to the Planck-scale

Hopf algebra \cite{Majid_Planck}\cite{Majid_book}. This shall be

developed elsewhere.

In the remaining parts of this introduction we will specify our

conventions and provide preliminaries on the quantum group \ensuremath{U_q(\lalg{b_+})}, its

deformations, and differential calculi.

\subsection{Conventions}

Throughout, $\k$ denotes a field of characteristic 0 and

$\k(q)$ denotes the field of rational

functions in one parameter $q$ over $\k$.

$\k(q)$ is our ground field in

the $q$-deformed setting, while $\k$ is the

ground field in the ``classical'' settings.

Within section \ref{sec:q} one could equally well view $\k$ as the ground

field with $q\in\k^*$ not a root of unity. This point of view is

problematic, however, when obtaining ``classical limits'' as

in sections \ref{sec:class} and \ref{sec:dual}.

The positive integers are denoted by $\mathbb{N}$ while the non-negative

integers are denoted by $\mathbb{N}_0$.

We define $q$-integers, $q$-factorials and

$q$-binomials as follows:

\begin{gather*}

[n]_q=\sum_{i=0}^{n-1} q^i\qquad

[n]_q!=[1]_q [2]_q\cdots [n]_q\qquad

\binomq{n}{m}=\frac{[n]_q!}{[m]_q! [n-m]_q!}

\end{gather*}

For a function of several variables (among

them $x$) over $\k$ we define

\begin{gather*}

(T_{a,x} f)(x) = f(x+a)\\

(\fdiff_{a,x} f)(x) = \frac{f(x+a)-f(x)}{a}

\end{gather*}

with $a\in\k$ and similarly over $\k(q)$

\begin{gather*}

(Q_{m,x} f)(x) = f(q^m x)\\

(\partial_{q,x} f)(x) = \frac{f(x)-f(qx)}{x(1-q)}\\

\end{gather*}

with $m\in\mathbb{Z}$.

We frequently use the notion of a polynomial in an extended

sense. Namely, if we have an algebra with an element $g$ and its

inverse $g^{-1}$ (as

in \ensuremath{U_q(\lalg{b_+})}{}) we will mean by a polynomial in $g,g^{-1}$ a finite power

series in $g$ with exponents in $\mathbb{Z}$. The length of such a polynomial

is the difference between highest and lowest degree.

If $H$ is a Hopf algebra, then $H^{op}$ will denote the Hopf algebra

with the opposite product.

\subsection{\ensuremath{U_q(\lalg{b_+})}{} and its Classical Limits}

\label{sec:intro_limits}

We recall that,

in the framework of quantum groups, the duality between enveloping algebra

$U(\lalg{g})$ of the Lie algebra and algebra of functions $C(G)$ on the Lie

group carries over to $q$-deformations.

In the case of

$\lalg{b_+}$, the

$q$-deformed enveloping algebra \ensuremath{U_q(\lalg{b_+})}{} defined over $\k(q)$ as

\begin{gather*}

U_q(\lalg{b_+})=\k(q)\langle X,g,g^{-1}\rangle \qquad

\text{with relations} \\

g g^{-1}=1 \qquad Xg=qgX \\

\cop X=X\otimes 1 + g\otimes X \qquad

\cop g=g\otimes g \\

\cou (X)=0 \qquad \cou (g)=1 \qquad

\antip X=-g^{-1}X \qquad \antip g=g^{-1}

\end{gather*}

is self-dual. Consequently, it

may alternatively be viewed as the quantum algebra \ensuremath{C_q(B_+)}{} of

functions on the Lie group $B_+$ associated with $\lalg{b_+}$.

It has two classical limits, the enveloping algebra \ensuremath{U(\lalg{b_+})}{}

and the function algebra $C(B_+)$.

The transition to the classical enveloping algebra is achieved by

replacing $q$

by $e^{-t}$ and $g$ by $e^{tH}$ in a formal power series setting in

$t$, introducing a new generator $H$. Now, all expressions are written in

the form $\sum_j a_j t^j$ and only the lowest order in $t$ is kept.