alasdairforsythe/text-english-code-fiction-nonfiction

text stringlengths 0 634k

We did not systematically remove known bot accounts~\cite{lebeuf2018swbots} from the author dataset, but we did check for the presence of software bots among the top committers of each year. We only found limited traces of continuous integration (CI) bots, used primarily to automate merge commits. After removing CI bots from the dataset the observed global trends were unchanged, therefore this paper presents unfiltered data.


\medskip
\emph{Future work.}
To some extent the above limitations are the price to pay to study such a large dataset: there exists a trade-off between large-scale analysis and accuracy.
We plan nonetheless to further investigate and mitigate them in future work.
Multi-method approaches, merging data mining with social science methods, could be applied to address some of the questions raised in this exploratory study.
While they do not scale to the whole dataset, multi-methods can be adopted to dig deeper into specific aspects, specifically those related to social phenomena.
Software is a social artifact, it is no wonder that aspects related to sociocultural evolution emerge when analyzing its evolution at this scale.





\clearpage



\section{Introduction}

One of the fundamental ingredients in the theory of non-commutative or
quantum geometry is the notion of a differential calculus.
In the framework of quantum groups the natural notion
is that of a
bicovariant differential calculus as introduced by Woronowicz
\cite{Wor_calculi}. Due to the allowance of non-commutativity
the uniqueness of a canonical calculus is lost.
It is therefore desirable to classify the possible choices.
The most important piece is the space of one-forms or ``first
order differential calculus'' to which we will restrict our attention
in the following. (From this point on we will use the term
``differential calculus'' to denote a
bicovariant first order differential calculus).

Much attention has been devoted to the investigation of differential
calculi on quantum groups $C_q(G)$ of function algebra type for
$G$ a simple Lie group.
Natural differential calculi on matrix quantum groups were obtained by
Jurco \cite{Jur} and
Carow-Watamura et al.\
\cite{CaScWaWe}. A partial classification of calculi of the same
dimension as the natural ones
was obtained by
Schm\"udgen and Sch\"uler \cite{ScSc2}.
More recently, a classification theorem for factorisable
cosemisimple quantum groups was obtained by Majid \cite{Majid_calculi},
covering the general $C_q(G)$ case. A similar result was
obtained later by Baumann and Schmitt \cite{BaSc}.
Also, Heckenberger and Schm\"udgen \cite{HeSc} gave a
complete classification on $C_q(SL(N))$ and $C_q(Sp(N))$.


In contrast, for $G$ not simple or semisimple the differential calculi
on $C_q(G)$
are largely unknown. A particularly basic case is the Lie group $B_+$
associated with the Lie algebra $\lalg{b_+}$ generated by two elements
$X,H$ with the relation $[H,X]=X$. The quantum enveloping algebra
\ensuremath{U_q(\lalg{b_+})}{}
is self-dual, i.e.\ is non-degenerately paired with itself \cite{Drinfeld}.
This has an interesting consequence: \ensuremath{U_q(\lalg{b_+})}{} may be identified with (a
certain algebraic model of) \ensuremath{C_q(B_+)}. The differential calculi on this
quantum group and on its ``classical limits'' \ensuremath{C(B_+)}{} and \ensuremath{U(\lalg{b_+})}{}
will be the main concern of this paper. We pay hereby equal attention
to the dual notion of ``quantum tangent space''.

In section \ref{sec:q} we obtain the complete classification of differential
calculi on \ensuremath{C_q(B_+)}{}. It turns out that (finite
dimensional) differential
calculi are characterised by finite subsets $I\subset\mathbb{N}$.
These
sets determine the decomposition into coirreducible (i.e.\ not
admitting quotients) differential calculi
characterised by single integers. For the coirreducible calculi the
explicit formulas for the commutation relations and braided
derivations are given.

In section \ref{sec:class} we give the complete classification for the
classical function algebra \ensuremath{C(B_+)}{}. It is essentially the same as in the
$q$-deformed setting and we stress this by giving an almost
one-to-one correspondence of differential calculi to those obtained in
the previous section. In contrast, however, the decomposition and
coirreducibility properties do not hold at all. (One may even say that
they are maximally violated). We give the explicit formulas for those
calculi corresponding to coirreducible ones.

More interesting perhaps is the ``dual'' classical limit. I.e.\ we
view \ensuremath{U(\lalg{b_+})}{} as a quantum function algebra with quantum enveloping
algebra \ensuremath{C(B_+)}{}. This is investigated in section \ref{sec:dual}. It
turns out that in this setting we have considerably more freedom in
choosing a
differential calculus since the bicovariance condition becomes much
weaker. This shows that this dual classical limit is in a sense
``unnatural'' as compared to the ordinary classical limit of section
\ref{sec:class}.
However, we can still establish a correspondence of certain
differential calculi to those of section \ref{sec:q}. The
decomposition properties are conserved while the coirreducibility
properties are not.

We did not systematically remove known bot accounts~\cite{lebeuf2018swbots} from the author dataset, but we did check for the presence of software bots among the top committers of each year. We only found limited traces of continuous integration (CI) bots, used primarily to automate merge commits. After removing CI bots from the dataset the observed global trends were unchanged, therefore this paper presents unfiltered data.

\medskip

\emph{Future work.}

To some extent the above limitations are the price to pay to study such a large dataset: there exists a trade-off between large-scale analysis and accuracy.

We plan nonetheless to further investigate and mitigate them in future work.

Multi-method approaches, merging data mining with social science methods, could be applied to address some of the questions raised in this exploratory study.

While they do not scale to the whole dataset, multi-methods can be adopted to dig deeper into specific aspects, specifically those related to social phenomena.

Software is a social artifact, it is no wonder that aspects related to sociocultural evolution emerge when analyzing its evolution at this scale.

\clearpage

\section{Introduction}

One of the fundamental ingredients in the theory of non-commutative or

quantum geometry is the notion of a differential calculus.

In the framework of quantum groups the natural notion

is that of a

bicovariant differential calculus as introduced by Woronowicz

\cite{Wor_calculi}. Due to the allowance of non-commutativity

the uniqueness of a canonical calculus is lost.

It is therefore desirable to classify the possible choices.

The most important piece is the space of one-forms or ``first

order differential calculus'' to which we will restrict our attention

in the following. (From this point on we will use the term

``differential calculus'' to denote a

bicovariant first order differential calculus).

Much attention has been devoted to the investigation of differential

calculi on quantum groups $C_q(G)$ of function algebra type for

$G$ a simple Lie group.

Natural differential calculi on matrix quantum groups were obtained by

Jurco \cite{Jur} and

Carow-Watamura et al.\

\cite{CaScWaWe}. A partial classification of calculi of the same

dimension as the natural ones

was obtained by

Schm\"udgen and Sch\"uler \cite{ScSc2}.

More recently, a classification theorem for factorisable

cosemisimple quantum groups was obtained by Majid \cite{Majid_calculi},

covering the general $C_q(G)$ case. A similar result was

obtained later by Baumann and Schmitt \cite{BaSc}.

Also, Heckenberger and Schm\"udgen \cite{HeSc} gave a

complete classification on $C_q(SL(N))$ and $C_q(Sp(N))$.

In contrast, for $G$ not simple or semisimple the differential calculi

on $C_q(G)$

are largely unknown. A particularly basic case is the Lie group $B_+$

associated with the Lie algebra $\lalg{b_+}$ generated by two elements

$X,H$ with the relation $[H,X]=X$. The quantum enveloping algebra

\ensuremath{U_q(\lalg{b_+})}{}

is self-dual, i.e.\ is non-degenerately paired with itself \cite{Drinfeld}.

This has an interesting consequence: \ensuremath{U_q(\lalg{b_+})}{} may be identified with (a

certain algebraic model of) \ensuremath{C_q(B_+)}. The differential calculi on this

quantum group and on its ``classical limits'' \ensuremath{C(B_+)}{} and \ensuremath{U(\lalg{b_+})}{}

will be the main concern of this paper. We pay hereby equal attention

to the dual notion of ``quantum tangent space''.

In section \ref{sec:q} we obtain the complete classification of differential

calculi on \ensuremath{C_q(B_+)}{}. It turns out that (finite

dimensional) differential

calculi are characterised by finite subsets $I\subset\mathbb{N}$.

These

sets determine the decomposition into coirreducible (i.e.\ not

admitting quotients) differential calculi

characterised by single integers. For the coirreducible calculi the

explicit formulas for the commutation relations and braided

derivations are given.

In section \ref{sec:class} we give the complete classification for the

classical function algebra \ensuremath{C(B_+)}{}. It is essentially the same as in the

$q$-deformed setting and we stress this by giving an almost

one-to-one correspondence of differential calculi to those obtained in

the previous section. In contrast, however, the decomposition and

coirreducibility properties do not hold at all. (One may even say that

they are maximally violated). We give the explicit formulas for those

calculi corresponding to coirreducible ones.

More interesting perhaps is the ``dual'' classical limit. I.e.\ we

view \ensuremath{U(\lalg{b_+})}{} as a quantum function algebra with quantum enveloping

algebra \ensuremath{C(B_+)}{}. This is investigated in section \ref{sec:dual}. It

turns out that in this setting we have considerably more freedom in

choosing a

differential calculus since the bicovariance condition becomes much

weaker. This shows that this dual classical limit is in a sense

``unnatural'' as compared to the ordinary classical limit of section

\ref{sec:class}.

However, we can still establish a correspondence of certain

differential calculi to those of section \ref{sec:q}. The

decomposition properties are conserved while the coirreducibility

properties are not.