alasdairforsythe/text-english-code-fiction-nonfiction

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Let us try to push the analogy with the commutative case further and
take a look at the notion of integration. The natural way to encode
the condition of translation invariance from the classical context
in the quantum group context
is given by the condition
\[(\int\otimes\id)\circ\cop a=1 \int a\qquad\forall a\in A\]
which defines a right integral on a quantum group $A$
\cite{Sweedler}.
(Correspondingly, we have the notion of a left integral.)
Let us
formulate a slightly
weaker version of this equation
in the context of a Hopf algebra $H$ dually paired with
$A$. We write
\[\int (h-\cou(h))\triangleright a = 0\qquad \forall h\in H, a\in A\]
where the action of $H$ on $A$ is the coregular action
$h\triangleright a = a_{(1)}\langle a_{(2)}, h\rangle$
given by the pairing.

In the present context we set $A=\ensuremath{U(\lalg{b}_{n+})}$ and $H=\ensuremath{C(B_{n+})}$. We define the
latter as a generalisation of \ensuremath{C(B_+)}{} with commuting
generators $g,p_1,\dots,p_{n-1}$ and coproducts
\[\cop p_i=p_i\otimes 1+g\otimes p_i\qquad \cop g=g\otimes g\]
This can be identified (upon rescaling) as the momentum sector of the
full $\kappa$-Poincar\'e algebra (with $g=e^{p_0}$).
The pairing is the natural extension of (\ref{eq:pair_class}):
\[\langle x_{n-1}^{m_{n-1}}\cdots x_1^{m_1} x_0^{k},
p_{n-1}^{r_{n-1}}\cdots p_1^{r_1} g^s\rangle
= \delta_{m_{n-1},r_{n-1}}\cdots\delta_{m_1,r_1} m_{n-1}!\cdots m_1!
s^k\]
The resulting coregular
action is conveniently expressed as (see also \cite{MaRu})
\[p_i\triangleright\no{f}=\no{\frac{\partial}{\partial x_i} f}\qquad
g\triangleright\no{f}=\no{T_{1,x_0} f}\]
with $f\in\k[x_0,\dots,x_{n-1}]$.
Due to cocommutativity, the notions of left and right integral
coincide. The invariance conditions for integration become
\[\int \no{\frac{\partial}{\partial x_i} f}=0\quad
\forall i\in\{1,\dots,n-1\}
\qquad\text{and}\qquad \int \no{\fdiff_{1,x_0} f}=0\]
The condition on the left is familiar and states the invariance under
infinitesimal translations in the $x_i$. The condition on the right states the
invariance under integer translations in $x_0$. However, we should
remember that we use a certain algebraic model of \ensuremath{C(B_{n+})}{}. We might add,
for example, a generator $p_0$
to \ensuremath{C(B_{n+})}{}
that is dual to $x_0$ and behaves
as the ``logarithm'' of $g$, i.e.\ acts as an infinitesimal
translation in $x_0$. We then have the condition of infinitesimal
translation invariance
\[\int \no{\frac{\partial}{\partial x_{\mu}} f}=0\]
for all $\mu\in\{0,1,\dots,{n-1}\}$.

In the present purely algebraic context these conditions do not make
much sense. In fact they would force the integral to be zero on the
whole algebra. This is not surprising, since we are dealing only with
polynomial functions which would not be integrable in the classical
case either.
In contrast, if we had for example the algebra of smooth functions
in two real variables, the conditions just characterise the usual
Lesbegue integral (up to normalisation).
Let us assume $\k=\mathbb{R}$ and suppose that we have extended the normal
ordering vector
space isomorphism $\mathbb{R}[x_0,\dots,x_{n-1}]\cong \ensuremath{U(\lalg{b}_{n+})}$ to a vector space
isomorphism of some sufficiently large class of functions on $\mathbb{R}^n$ with a
suitable completion $\hat{U}(\lalg{b_{n+}})$ in a functional
analytic framework (embedding \ensuremath{U(\lalg{b}_{n+})}{} in some operator algebra on a
Hilbert space). It is then natural to define the integration on
$\hat{U}(\lalg{b_{n+}})$ by
\[\int \no{f}=\int_{\mathbb{R}^n} f\ dx_0\cdots dx_{n-1}\]
where the right hand side is just the usual Lesbegue integral in $n$
real variables $x_0,\dots,x_{n-1}$. This
integral is unique (up to normalisation) in
satisfying the covariance condition since, as we have seen,
these correspond
just to the usual translation invariance in the classical case via normal
ordering, for which the Lesbegue integral is the unique solution.
It is also the $q\to 1$ limit of the translation invariant integral on
\ensuremath{U_q(\lalg{b_+})}{} obtained in \cite{Majid_qreg}.

We see that the natural differential calculus in corollary
\ref{cor:nat_bnp} is
compatible with this integration in that the appearing braided
derivations are exactly the actions of the translation generators
$p_{\mu}$. However, we should stress that this calculus is not
covariant under the full $\kappa$-Poincar\'e algebra, since it was
shown in \cite{GoKoMa} that in $n=4$ there is no such
calculus of dimension $4$. Our results therefore indicate a new
intrinsic approach to $\kappa$-Minkowski space that allows a
bicovariant
differential calculus of dimension $4$ and a unique translation
invariant integral by normal ordering and Lesbegue integration.



\section*{Acknowledgements}
I would like to thank S.~Majid for proposing this project,
and for fruitful discussions during the preparation of this paper.

Let us try to push the analogy with the commutative case further and

take a look at the notion of integration. The natural way to encode

the condition of translation invariance from the classical context

in the quantum group context

is given by the condition

\[(\int\otimes\id)\circ\cop a=1 \int a\qquad\forall a\in A\]

which defines a right integral on a quantum group $A$

\cite{Sweedler}.

(Correspondingly, we have the notion of a left integral.)

Let us

formulate a slightly

weaker version of this equation

in the context of a Hopf algebra $H$ dually paired with

$A$. We write

\[\int (h-\cou(h))\triangleright a = 0\qquad \forall h\in H, a\in A\]

where the action of $H$ on $A$ is the coregular action

$h\triangleright a = a_{(1)}\langle a_{(2)}, h\rangle$

given by the pairing.

In the present context we set $A=\ensuremath{U(\lalg{b}_{n+})}$ and $H=\ensuremath{C(B_{n+})}$. We define the

latter as a generalisation of \ensuremath{C(B_+)}{} with commuting

generators $g,p_1,\dots,p_{n-1}$ and coproducts

\[\cop p_i=p_i\otimes 1+g\otimes p_i\qquad \cop g=g\otimes g\]

This can be identified (upon rescaling) as the momentum sector of the

full $\kappa$-Poincar\'e algebra (with $g=e^{p_0}$).

The pairing is the natural extension of (\ref{eq:pair_class}):

\[\langle x_{n-1}^{m_{n-1}}\cdots x_1^{m_1} x_0^{k},

p_{n-1}^{r_{n-1}}\cdots p_1^{r_1} g^s\rangle

= \delta_{m_{n-1},r_{n-1}}\cdots\delta_{m_1,r_1} m_{n-1}!\cdots m_1!

s^k\]

The resulting coregular

action is conveniently expressed as (see also \cite{MaRu})

\[p_i\triangleright\no{f}=\no{\frac{\partial}{\partial x_i} f}\qquad

g\triangleright\no{f}=\no{T_{1,x_0} f}\]

with $f\in\k[x_0,\dots,x_{n-1}]$.

Due to cocommutativity, the notions of left and right integral

coincide. The invariance conditions for integration become

\[\int \no{\frac{\partial}{\partial x_i} f}=0\quad

\forall i\in\{1,\dots,n-1\}

\qquad\text{and}\qquad \int \no{\fdiff_{1,x_0} f}=0\]

The condition on the left is familiar and states the invariance under

infinitesimal translations in the $x_i$. The condition on the right states the

invariance under integer translations in $x_0$. However, we should

remember that we use a certain algebraic model of \ensuremath{C(B_{n+})}{}. We might add,

for example, a generator $p_0$

to \ensuremath{C(B_{n+})}{}

that is dual to $x_0$ and behaves

as the ``logarithm'' of $g$, i.e.\ acts as an infinitesimal

translation in $x_0$. We then have the condition of infinitesimal

translation invariance

\[\int \no{\frac{\partial}{\partial x_{\mu}} f}=0\]

for all $\mu\in\{0,1,\dots,{n-1}\}$.

In the present purely algebraic context these conditions do not make

much sense. In fact they would force the integral to be zero on the

whole algebra. This is not surprising, since we are dealing only with

polynomial functions which would not be integrable in the classical

case either.

In contrast, if we had for example the algebra of smooth functions

in two real variables, the conditions just characterise the usual

Lesbegue integral (up to normalisation).

Let us assume $\k=\mathbb{R}$ and suppose that we have extended the normal

ordering vector

space isomorphism $\mathbb{R}[x_0,\dots,x_{n-1}]\cong \ensuremath{U(\lalg{b}_{n+})}$ to a vector space

isomorphism of some sufficiently large class of functions on $\mathbb{R}^n$ with a

suitable completion $\hat{U}(\lalg{b_{n+}})$ in a functional

analytic framework (embedding \ensuremath{U(\lalg{b}_{n+})}{} in some operator algebra on a

Hilbert space). It is then natural to define the integration on

$\hat{U}(\lalg{b_{n+}})$ by

\[\int \no{f}=\int_{\mathbb{R}^n} f\ dx_0\cdots dx_{n-1}\]

where the right hand side is just the usual Lesbegue integral in $n$

real variables $x_0,\dots,x_{n-1}$. This

integral is unique (up to normalisation) in

satisfying the covariance condition since, as we have seen,

these correspond

just to the usual translation invariance in the classical case via normal

ordering, for which the Lesbegue integral is the unique solution.

It is also the $q\to 1$ limit of the translation invariant integral on

\ensuremath{U_q(\lalg{b_+})}{} obtained in \cite{Majid_qreg}.

We see that the natural differential calculus in corollary

\ref{cor:nat_bnp} is

compatible with this integration in that the appearing braided

derivations are exactly the actions of the translation generators

$p_{\mu}$. However, we should stress that this calculus is not

covariant under the full $\kappa$-Poincar\'e algebra, since it was

shown in \cite{GoKoMa} that in $n=4$ there is no such

calculus of dimension $4$. Our results therefore indicate a new

intrinsic approach to $\kappa$-Minkowski space that allows a

bicovariant

differential calculus of dimension $4$ and a unique translation

invariant integral by normal ordering and Lesbegue integration.

\section*{Acknowledgements}

I would like to thank S.~Majid for proposing this project,

and for fruitful discussions during the preparation of this paper.