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\def\norm #1{{\left\Vert\,#1\,\right\Vert}}
\def\g {{\frak g}}
\def\R {{\Bbb R}}
\def\C {{\Bbb C}}
\def\h{{\frak h}}
\def\m{{\frak m}}
\def\N{{\Bbb N}}
\def\e{{\epsilon}}
\def\l{{\frak l}}
\def\L{{\Lambda_\infty}}
\def\K{{\Bbb K}}
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\topmatter
\title
On enlargability of infinite-dimensional
Lie superalgebras
\endtitle
\author Vladimir G. Pestov
\endauthor
\affil
Department of Mathematics\\
Victoria University of Wellington \\
P.O. Box 600 \\
Wellington, New Zealand
\endaffil
\abstract{We show that in infinite dimensions
a Lie superalgebra coming from
a Rogers supergroup may not come from a DeWitt one.
Thus, we produce an evidence that
a whole class of Lie superalgebras
can be enlarged only by means of the ``na\"\i ve''
approach to supermanifolds, which therefore cannot be merely thrown away.}
\endabstract
\subjclass{17B70, 58A50, 58C50, 22E65}
\endsubjclass
\endtopmatter
\document
\smallpagebreak
\heading Introduction
\endheading
\smallpagebreak
The term ``enlargability'' in Lie theory means a possibility to
associate a Lie group to a Lie algebra;
those Lie algebras coming from Lie groups are
called ``enlargable Lie algebras'' \cite{E, vEK}.
Similarly,
one of important goals of supermanifold theory is
to provide means for enlarging a Lie superalgebra,
by attributing to it a supergroup - that is, a group object in a
category of supermanifolds.
This is done with clear purpose
``to
recapture explicitely the geometry implicit in the algebraic
structure of Lie superalgebras'' \cite{Ba}.
>From this viewpoint, the wider a class of ``enlargable'' Lie superalgebras
is, the better.
The diversity of the approaches to
supermanifold theory existing at
present
has not been given a unified treatment yet.
Although the Berezin-Le\u\i tes-Kostant version
\cite{BerL, Ko, Ber2, L, BnL, Ma, BBHR, BSV1}
is probably the nearest one to the comprehensive viewpoint,
we still believe that any other approach
is also a contribution towards a thorough
insight into the ``true'' notion of
a supermanifold (and, thereby, a supergroup) rather than a ``half-baked
{\it ad hoc} definition'' \cite{BMFS}.
A wide-spread point of view (shared by us)
is that the ``na\"\i ve'' theory with a non-trivial
``ground algebra'' of coefficients
\cite{DeW, Ro1,2,3,4, Ba, BoG, HQRMdU,JP2, ChB1,2, RaC, ChBDM,
Rn, BBHR, BBHRP, VV, Kh, Br, KoN, MK}
can be most probably rewritten in the language of relative categories of
graded manifolds --- namely, if $M$ is a supermanifold over a
graded-commutative algebra, $\Lambda$, of
``supernumbers,'' then $M$ can be identified with a
superbundle $X\to Spec~\Lambda$ where
$X$ is a (possibly, infinite dimensional \cite{Mo, Schm})
Berezin-Le\u\i tes-Kostant
supermanifold. This idea was repeatedly uttered
by Le\u\i tes, mostly in private communications.
Unfortunately, the fact is that almost no written evidence has been
produced in support of the idea
(apart from some initial work done by Penkov \cite{P}),
so its status remains that of a plausible conjecture
(although backed by the prominence similar constructions have won in
category theory and theory of topoi \cite{J})
and by no means that of a mathematical result.
The well-known Batchelor's theorem \cite{Ba}, which is indeed one of the
highlights of ``supermathematics,''
states that every
smooth finite-dimensional
DeWitt supermanifold
\cite{DeW, ChB1,2, RaC, ChBDM, BBHR}
over
a finite-dimensional Grassmann algebra of coefficients is an image
under the change of base functor of
a smooth finite-dimensional graded manifold.
However, it is very important to stress that the change of base functor
is not an equivalence of categories ---
there are more morphisms between DeWitt supermanifolds than between
the corresponding graded manifolds --- one of the corollaries
being the fact that the category of
DeWitt supergroups is richer than that of Berezin-Le\u\i tes-Kostant
supergroups. While one can still hope to describe DeWitt
supergroups as deformations
of Berezin-Le\u\i tes-Kostant
supergroups over the base $Spec~\Lambda$ (in the spirit of, say, \cite{FF}),
the supergroups resulting from a more general class of
supermanifolds - the Rogers supermanifolds
\cite{Ro1,2,3,4, Ba, BoG, HQRMdU, JP2, RaC, Rn, BBHR, BBHRP, VV, Kh, KoN, MK}
- seemingly are not amenable to such a treatment.
The ultimate question arising in this connection is whether the Rogers
supergroups are worthy of studying. (Compare the points of
view in, say, \cite{Ro2,3} and \cite{Pi}.)
Indeed, it is known that any
finite dimensional Lie superalgebra comes from a supergroup
which belongs at widest to the DeWitt category (and thus falls within the
bed of deformation theory approach)
\cite{BerL, Ko, Ro2, Pe1,3,4}.
It is shown in the present work that in infinite dimensions
a Lie superalgebra $\frak g$ coming from
a Rogers supergroup may not come from a DeWitt one.
Thus, we produce an evidence that
a whole class of Lie superalgebras
can be enlarged only by means of the most
general ``na\"\i ve''
approach to supermanifolds, which therefore cannot be merely thrown away.
Our conclusion is that the problem of rewording the ``na\"\i ve'' approach
in the language
of Berezin-Le\u\i tes-Kostant
supermanifolds (possibly infinite-dimensional)
and their deformations becomes substantial.
In this paper we follow the ``na\"\i ve'' approach to
supermanifold theory, developed along the usual lines:
ground algebra $\Lambda$ - graded modules over $\Lambda$ -
linear superspaces as even sectors of graded modules -
calculus over linear superspaces - supermanifolds -
supergroups and Lie superalgebras. In order to be able
to construct our examples (Sec. 6), we need a great deal
of background constructions and results belonging to
``global superanalysis;''
most of them
can be proved by a direct analogy with similar
finite-dimensional results
\cite{DeW, Ro1,3,4, BoG, HQRMdU, JP2, ChB1,2, ChBDM, Rn, BBHR, BBHRP, VV, Kh, KoN, MK}
and/or infinite-dimensional results in purely even case
\cite{Bou, BS, E, ChBDM}, and this is why in most cases we
merely sketch the proofs.
The term ``na\"\i ve'' does not mean any lack of mathematical rigour.
In particular, the DeWitt approach \cite{DeW}, which we extend to an
infinite-dimensional setting in this paper, in its finite-dimensional
form satisfies the Rothstein axiomatics
for supermanifolds \cite{Rn} as it is presented in \cite{BBHRPe},
\footnote{Although the papers \cite{Rn} and \cite{BBHRPe} deal with Banach ground
algebras only, the results can be almost {\it verbatim}
carried over the case of the
graded local Arens-Michael algebras, studied in \cite{Pe3,4};
this case already includes
the DeWitt ground algebra $\Lambda_\infty$.}
and it is ``na\"\i ve'' only in the sense that morphisms between supermanifolds
are completely determined by underlying set-theoretic mappings.
There is one subtle point about all this
generalization worth mentioning.
It is known that
in finite dimensions, in order to produce substantial results,
one needs to consider only {\it free} graded modules over
$\Lambda$, that is, the modules of the form
$\Lambda\otimes E$, where $E$ is a (finite dimensional) graded
linear space. Of course, the same is true in infinite dimensions
as well (what is unnoticed in \cite{Kh}), and one faces the problem of
right choice of the notion of a free graded module over $\Lambda$,
because the tensor product should be appropriately
topologized and completed
thereafter;
as it was shown by the author \cite{Pe3},
in case where $\Lambda$ is Banach, neither the projective nor
the injective tensor product result in a satisfactory concept
of a free graded $\Lambda$-module.
In fact, one is bound to consider ground algebras $\Lambda$
which are {\it nuclear} as locally convex spaces: in this
case all the reasonable notions of topological tensor products
coincide, and the resulting notion of a graded $\Lambda$-module
behaves immaculately. One can show that there is a whole
natural class of nuclear graded-commutative algebras,
resulting from the concept of the free graded local Arens-Michael
(GLAM) algebra over a locally convex space \cite{Pe3,4}.
Here
we choose as the ground algebra the
DeWitt algebra, $\L$, which is a nuclear Fr\'echet
graded commutative algebra with a number of
fascinating properties.
Our examples occur in dimension $(\infty, 0)$ and
we do not enter into a systematic investigation of
supermanifolds and supergroups infinite-dimensional in odd sector;
already the $(0,\infty)$-dimensional
supermanifolds are still enigmatic.
\smallpagebreak
\heading
1. The DeWitt algebra
\endheading
\smallpagebreak
The basic field $\Bbb K$ in this paper is $\R$.
(However, all results from Sec. 1-5 remain true for more general
valuation fields $\Bbb K$, including $\C$ and ${\Bbb Q}_p$, and
the results of Section 6 make sense for $\C$ as well.)
The term {\it ``graded''} in this paper means {\it ``${\Bbb Z}_2$-graded''}.
A {\it graded} locally convex space (LCS), $E$, is an LCS
over the basic field $\Bbb K$ together with a
fixed decomposition $E \cong E^0 \oplus E^1$ into a direct
sum of closed linear subspaces,
where $E^0$ is called the {\it even} and $E^1$ the {\it odd
part (sector)} of $E$. The {\it parity}
$\tilde x$ of an element $x \in E^0 \cup E^1$
is defined by letting $x \in E^{\tilde x},
\tilde x \in \lbrace 0,1 \rbrace = \Bbb Z_2$.
A {\it (locally convex) graded-commutative algebra} is
an associative algebra carrying a structure of a
graded locally convex space,
$\Lambda \cong \Lambda^0 \oplus \Lambda^1$,
in such a way that the operations are continuous and
$$\tilde{xy} = \tilde x + \tilde y,~~ x,y \in \Lambda^0 \cup
\Lambda^1 \tag 1$$
$$ xy = (-1)^{\tilde x\tilde y}yx,~~ x,y \in \Lambda^0 \cup
\Lambda^1 \tag 2$$
\noindent Usually a graded commutative algebra is supposed to be unital.
A {\it local} algebra has a unique maximal ideal
(which coincides with its radical).
The quotient of a local algebra $\Lambda$ by
its radical is isomorphic to the basic field $\Bbb K$.
The corresponding homomorphism of augmentation
$\beta_{\Lambda}: \Lambda \to \Bbb K$ is called
the {\it number part map} \cite{Ber2} or the {\it body map}
\cite{DeW}.
If $\Lambda$ is a Banach local
algebra then $\beta_{\Lambda}$ is continuous
\cite{He}.)
The Grassmann algebra, $\G q$, with odd generators
$\xi_1, \xi_2, \dots ,\xi_q$ is the most well-known
example of a graded commutative algebra.
We call a graded-commutative algebra $\Lambda$ {\it effective}
if the common left annihilator of the odd sector of $\Lambda$
is zero:
$$^\perp(\Lambda^1)=_{def}
\{x\in\Lambda : \forall y\in \Lambda^1, xy=0\} = (0) \tag 3$$
In other terms, $\Lambda$ is effective iff the left regular
representation of $\Lambda$ in $\Lambda^{(\Lambda^1)}$ is faithful.
(Cf. \cite{ChB1,2, JP2, Kh, VV, Pe3}.)
The DeWitt algebra, $\L$,
is the projective limit of an inverse sequence of finite dimensional
Grassmann algebras
$$\G0\leftarrow \G1 \leftarrow \G2 \leftarrow \dots \leftarrow
\G q \leftarrow \G{q+1} \leftarrow \dots \tag 4$$
\noindent where the homomorphism $\G{q+1}\to\G q$ sends the generators
$\xi_1, \xi_2, \dots ,\xi_q$ to themselves and $\xi_{q+1}$ to $0$.
In other terms, it is formed by all formal power series
$$\sum_{\mu}\xi^\mu \tag 5$$
\noindent in countably infinitely many
free anticommuting generators $\xi_1, \xi_2, \dots ,\xi_q, \dots $.
Here $\mu$ denotes a multiindex of the form $(\mu_1,\mu_2, \dots, \mu_k), ~k\in\Bbb N$,
running over the collection of all finite subsets of the set of natural
numbers $\Bbb N$ arranged in an increasing order;
by $\xi^\mu$ one denotes the monomial $\xi_1^{\mu_1}\dots \xi_k^{\mu_k}$
\cite{BBHR}.
For every $q$ there is a canonical unital graded algebra homomorphism
$\pi_q : \L\to\G q$ defined by the conditions
$\pi_q(\xi_i)=\xi_i$ if $i\leq q$ and $\pi_q(\xi_i)=0$ if $i> q$.
The topology is introduced by letting a sequence
$(x_n)$ converge to an element $x$ if and only if
for every $q$ the sequence $(\pi_qx_n)$ converges to $\pi_n$ in
a finite-dimensional algebra $\G q$.
The algebra $\L$ is complete metrizable
locally convex (= Fr\'echet)
locally multiplicatively convex \cite{He}
graded commutative algebra. It is effective
(contrary to $\G q$).
We denote by $I_q$ the kernel of $\pi_q$; it is a closed graded
ideal of $\L$.
The algebra $\L$ is {\it nuclear} as a locally convex space, being a
projective limit of a sequence of finite-dimensional normed spaces
$\G q$ (see \cite{Scha}).
It means that for any locally convex space $E$,
all the ``reasonable'' locally convex topologizations of the
tensor product $\L\otimes E$ coincide (including the projective
topological tensor product, weak topological tensor product, etc.)
By $\L\hat\otimes E$, as usual, we will denote
the completed projective topological tensor product.
For more on $\L$, see \cite{Ber1, DeW, ChB1,2, ChBDM, MK, Pe3}.
\smallpagebreak
\heading
2. Locally convex superspaces
\endheading
\smallpagebreak
We call by a {\it free graded $\L$-module} a graded topological
module $M$
over $\L$ of the form $M\cong \L\hat\otimes E$ where
$E$ is a graded locally convex space.
For every $q\in\N$, the free
graded (topological) $\G q$-module
$red_qM\cong \G q\otimes E$ is called the $q${\it -th reduction}
of $M$. The $0$-th reduction of $M$ is just $E$ and it is called
the {\it body} of $M$; we denote $M_B = red_0M$.
The canonical {\it reduction mappings}
$\pi_q^M =_{def} \pi_q\hat\otimes id_M : M \to red_qM$ are continuous and linear,
and the topology of a free graded $\L$-module $M$ is
{\it projective} \cite{Scha}
with respect to the family of reduction mappings in the
sense that it is the coarsest topology making the reduction mappings
continuous.
Denote the kernel of $\pi_q^M$ by $I^M_q$.
These kernels --- and by the same token,
the reduction mappings --- can be defined
independently of a particular choice of a tensor product decomposition
of $M$. Namely, $I^M_q$ is the closed submodule of $M$
generated by $I_q\cdot M$.
If the body space $M_B$ is purely even (that is, $M_B^1=(0)$)
then $M$ is called a {\it purely even module}.
If $E$ is Banach (Fr\'echet, etc.) then we call $M$
a free graded {\it Banach (Fr\'echet, etc.)} $\L$-module, although, say,
$M$ is never normable as a locally convex space.
For every two free graded $\L$-modules, $M$ and $N$,
the graded $\L$-module of all $\L$-linear mappings from $M$ to $N$
is
denoted by $L_\L(M,N)$.
If $M$ and $N$ are Banach then $L_\L(M,N)$ is endowed with
the (locally convex) {\it topology of uniform
convergence on bounded subsets,}
or the $\beta${\it -topology}. Its base is formed by the sets
$$[B; V] =_{def} \{f\in L_\L(M,N): f(B)\subset V\},$$
\noindent where $B$ runs over the family of all bounded subsets of $M$,
and $V$- over the family of all open subsets of $N$ \cite{Scha}.
\proclaim{Theorem 1}
Let $M$ and $N$ be free graded Banach $\L$-modules.
Then $L_\L(M,N)$ endowed with the
$\beta$-topology is a free graded Banach $\L$-module, the graded Banach
space $L(M_B,N_B)$ with the $\beta$-topology
being its body.
\endproclaim
\demo{Proof}
We will show that the $\beta$-topology on $L_\L(M,N)$ is projective
with respect to the family of canonical mappings
$\pi_q^L\equiv
\pi_q^{L_\L(M,N)} : L_\L(M,N)\to L_{\G q}(red_qM, red_qN)$, and since
the latter
modules, $L_{\G q}(red_qM, red_qN)$, with the $\beta$-topology
are verified directly to be
isomorphic to $\G q\otimes L(M_B,N_B)$, the statement follows.
Indeed, let $B$ be bounded and $V$ be open in $\L$. One can assume
that for some $q$, $V=V+I^M_q= (\pi_q^{L})^{-1}(\pi_q(V))$. This implies
$[B; V] = [B+I^M_q; V]$. The latter set coincides with
$(\pi_q^{L})^{-1}([\pi_q^{L}(B); \pi_q^{L}(V)])$.
Since the set $\pi_q^{L}(B)$
is bounded in $\G q$, then $[\pi_q^{L}(B); \pi_q^{L}(V)]$ is open in
$L_{\G q}(red_qM, red_qN)$.
\qed\enddemo
A subset $U\subset M$ is called {\it DeWitt open}
if $U=(\pi_0^M)^{-1}\pi_0^M(U)$ and $\pi_0^M(U)$ is open in $E\cong M_B$.
The arising {\it DeWitt topology} on $M$ is obviously non-Hausdorff.
Denote for a subset $U\subset M$ by $U^\sim$ the set
$(\pi_0^M)^{-1}U$; then a subset $U\subset M$ is DeWitt open iff $U$ is open
and $U=U^\sim$. In most cases we will consider the restriction of the DeWitt
topology to the even sector, $M^0$, of $M$.
In contrast to the DeWitt (or {\it coarse})
topology on $M$, the ordinary topology of tensor product is
usually called the {\it Rogers}, or {\it fine}, topology.
The even sector, $M^0$, of a free graded $\L$-module is referred to
as a {\it (locally convex) topological linear superspace}.
(In a terminology consistent with the existence of
the change of base functor,
a locally convex topological superspace is rather the {\it space of
$\L$-points of a superspace} than the superspace itself.)
In particular, any locally convex topological linear superspace, $M^0$,
carries a natural structure of a locally convex topological
module over the topological algebra $\L^0$.
Morphisms between locally convex topological superspaces
are the so-called superlinear mappings:
a mapping $f : M^0\to N^0$ between two locally convex superspaces
is called {\it superlinear} if it is the restriction of a
continuous even $\L$-module morphism $\hat f : M\to N$
\cite{DeW, Ro1,4, BoG, HQRMdU, JP2, ChB1,2, ChBDM, Rn, BBHR, BBHRP, VV, Kh, KoN, MK}.
Let $M$ be a free graded topological $\L$-module.
Denote by $M^\dag$ a graded $\L$-module $L_{\L^0}(M^0; \L)$
endowed with the $\beta$-topology, and by $M^{\dag\dag}$ a
graded $\L$-module $L_{\L}(M^\dag ; \L)$ (cf. \cite{JP2}).
The graded topological module $M^\dag$ is canonically isomorphic to the
free graded module $L_{\L}(M; \L)\cong \L\hat\otimes M_B'$, and
the body of a free graded $\L$-module $M^{\dag\dag}$ is the second Banach
dual space $(M_B)''$; here the prime $'$ denotes, as usual,
the strong dual of a Banach space \cite{Scha}.
The Banach superspace $M^0$ embeds into $M^{\dag\dag}$ by means of the
canonical evaluation mapping $\kappa$ given by $\kappa (f)(x) =_{def} f(x)$
for $f\in M^\dag$ and $x\in M^0$.
We identify $M^0$ with its image under $\kappa$ in $M^{\dag\dag}$.
Set
$$ (M^0)^\kappa =_{def} \{ x\in M^{\dag\dag} : \forall \lambda \in
(\L)^{\tilde x},~ \lambda x\in M^0\} $$
The set $ (M^0)^\kappa$ forms a graded topological $\L$-submodule of
$M^{\dag\dag}$, and
a canonical graded topological
$\L$-module isomorphism $\hat\kappa : M \cong (M^0)^\kappa$
can be established such that $\hat\kappa\vert M^0 = \kappa$.
The correspondence $M^0\mapsto (M^0)^\kappa$ is a covariant
functor from the category
of locally convex superspaces and $\L^0$-linear mappings
into the category of graded topological modules and even continuous
$\L$-linear mappings.
Since every superlinear mapping between two locally convex superspaces is in
particular $\L^0$-linear, and the above functor is inverse
to the natural defining
functor of restriction $M\mapsto M^0, ~ f \mapsto f\vert M^0$, one
comes to the following result.
\proclaim{Theorem 2}
The categories of locally convex superspaces and superlinear mappings
and free graded topological modules and even continuous $\L$-module
homomorphisms are equivalent.
\endproclaim\qed
\proclaim{Corollary 1}
A mapping $f$ between two free topological graded $\L$-modules
is $\L$-linear if and only if it is $\L^0$-linear.
\endproclaim\qed
\proclaim{Corollary 2}
A mapping $f$ between two locally convex superspaces $M^0$ and $N_0$ is
superlinear if and only if it is $\L^0$-linear.
\endproclaim\qed
In fact, the above construction $M^0\mapsto M^\dag \mapsto M^{\dag\dag}$
makes sense for an arbitrary graded topological $\L$-module $M$;
however, in this case the evaluation mapping $\kappa$, still
continuous, is not necessarily a topological embedding
(it may even fail to be one-to-one).
This extended functor enables one to prove the following
\proclaim{Lemma 1}
If the even part of a graded topological $\L$-module $N$ is isomorphic as
a topological $\L^0$-module to a Banach superspace $M^0$,
and for every $x\in N_1$ there is a $\lambda\in\L^1$ with
$\lambda x\neq 0$ then
$N$ is isomorphic to $M$.
\endproclaim\qed
A mapping $f$ from an open subset $U$ in a topological linear superspace
$M^0$ to a topological linear superspace $N_0$ is called
{\it superdifferentiable at a point} $x\in U$ if
$f$ is G\^ateaux differentiable at $x$ and the
G\^ateaux differential, $D_xf$, is a superlinear mapping from
$M^0$ to $N^0$ and thus can be represented (uniquely) by an element
of $L_\L(M,N)^0$.
A mapping $f$ which is superdifferentiable
at every point $x\in U$ in such a way that
the superdifferential mapping $x\mapsto D_xf$
from $M^0$ to $L_\L(M,N)^0$ is continuous,
is also called a $G^1${\it -mapping} on $U$ and is said to
belong to the graded $\L$-algebra $G^1(U)$.
It is clear how to define recursively
$G^k$-mappings for all $k\in\N$, as well as $G^\infty$-mappings.
If $M^0$ and $N^0$ are Banach superspaces then a superdifferentiable
mapping is actually Fr\'echet differentiable, and a
$G^\infty$ mapping is Fr\'echet smooth
(but not {\it vice versa}).
A mapping $f$ from an open subset $U$ in a Banach superspace
$M^0$ to a Banach superspace $N_0$ is called
{\it superanalytic at a point} $x\in U$ if it is analytic
as a mapping between Fr\'echet spaces and in addition
the Fr\'echet differential, $D_xf$, is a superlinear mapping.
Every superanalytic mapping is $G^\infty$.
All the usual properties of supersmoothness and superanalyticity
\cite{DeW, Ro1,4, BoG, HQRMdU, JP2, ChB1,2, ChBDM, Rn, BBHR, BBHRP, VV, Kh, KoN, MK,
BSV1}
are carried over to the Banach case.
\smallpagebreak
\heading
3. Supermanifolds
\endheading
\smallpagebreak
Let $M$ be a free graded Banach $\L$-module.
A {\it supersmooth/superanalytic
Banach supermanifold modeled over} $M$ is
a Hausdorff topological space $X$ together with a fixed
supersmooth/superanalytic
{\it atlas} on it, ${\Cal A}$, that is, a family of homeomorphisms
on their images,
$f_\alpha : U_\alpha \to X$, where $U_\alpha\subset M^0$ are open
convex neighbourhoods of the origin, such that the transition functions
$f_\alpha^{-1}\circ f_\beta$ are supersmooth/superanalytic in their
natural domain of definition.
If there exists an atlas such that
every set $U_\alpha\equiv dom~f_\alpha$ is DeWitt open
in $M^0$ then the supermanifold $X$ is called
a {\it DeWitt supermanifolds}
(cf. \cite{BBHR, RaC}).
Simplest examples of supermanifolds are the
{\it (DeWitt) superdomains} which are just (DeWitt)
open convex subsets of $M^0$ with their natural supersmooth/superanalytic structure.
It is clear how to define morphisms between supermanifolds.
Any supersmooth/superanalytic
Banach supermanifold $X$ carries an underlying structure of an
infinite dimensional
smooth/analytic Fr\'echet manifold. We will denote the
underlying manifold by $X^0$. Any morphism between supermanifolds
determines a morphism between underlying Fr\'echet manifolds, thereby
a forgetful functor from the category of supermanifolds to the
category of Fr\'echet manifolds comes into being.
Graded derivations of the sheaf ${\Cal S}_X$
of germs of supersmooth/superanalytic
mappings $X\to\L$ are called {\it graded vector fields}
on $X$. Graded derivations of the stalk ${\Cal S}_{X,x}$
are called {\it tangent vectors to $X$ at a point} $x\in X$.
\proclaim{Theorem 3} The totality of tangent vectors to a Banach supermanifold
$X$ at a point $x\in X$ forms a free graded Banach $\L$-module $T_xX$
canonically isomorphic to $M$.
\endproclaim
\demo{Proof}
One can assume that $X=U\subset M^0$ is a superdomain and $x=0\in U$.
The $\L^0$-module $M^0$ is isomorphic to the even part
of $T_xX$ as a topological $\L^0$-module; this isomorphism is given
by the map
$$M^0\ni x \mapsto [f \mapsto
{\partial (f\vert_{\{t\cdot x : t\in \K\}})\over \partial t} \in \L] $$
\noindent where the derivative is the usual derivative (in any sense) of a
mapping $f\vert_{\{t\cdot x : t\in \K\}}: \R\to\L$ at $0$.
Now one can apply Lemma 1.
\enddemo\qed
We call $T_xX$ the {\it tangent module to $X$ at $x$.}
\proclaim{Theorem 4}
The even part $T_xX^0$ of the tangent module $T_xX$
is canonically isomorphic as a locally convex space to
the tangent space to the Fr\'echet manifold $X^0$ at the point $x$.
\endproclaim
\demo{Proof}
There exists a canonical isomorphism, say $j$,
between the tangent space to the
underlying Fr\'echet manifold $X^0$ at a point
$x$ and the model locally convex space,
which in this case is the underlying Fr\'echet space, $M^0_+$, of $M^0$.
(See, e.g., \cite{Mi}.)
Now take as the desired isomorphism the composition of $j$
with the isomorphism from Theorem 3.
\enddemo\qed
\smallpagebreak
\heading
4. Lie superalgebras
\endheading
\smallpagebreak
A {\it Lie superalgebra} over $\L$ is a
free graded $\L$-module, $\g$, endowed with a
graded Lie bracket, which is bi-$\L$-linear, anticommutative
and satisfies the graded Jacobi identity.
For every $q\in\N$ the submodule $I_q^\g$ is a graded Lie ideal in
the Lie superalgebra $\g$ viewed as a Lie superalgebra over the ground
field $\Bbb K$. The quotient Lie superalgebra $red_q\g=_{def}
\g/I_q^\g$ carries a natural structure of a free graded
$\G q\equiv \L/I_q$-module; in addition, the graded Lie bracket on
$red_q\g$ is bi-$\G q$-linear. This means that the $q${\it -th
reduction} $red_q\g$ of $\g$ is a Lie superalgebra over the
finite-dimensional
Grassmann algebra $\G q$. As a Lie $\Bbb K$-superalgebra,
$\g$ can be represented as the projective limit
$\g\cong lim_{\leftarrow}red_q\g$.
If the body Lie superalgebra $\g_B$ is Banach, then we refer to $\g$
as a {\it Banach-Lie superalgebra}. In this case,
the underlying graded locally convex space of $\g$ is Fr\'echet
(non-Banach unless $\g=(0)$),
and all the $q$-reductions $red_q\g$ are Banach-Lie $\Bbb K$-superalgebras.
In particular, for every $q$ the even sector $red_q\g^0$ is an ordinary
Banach-Lie algebra.
DeWitt \cite{DeW} calls the Lie algebras $red_q\g^0$ $q${\it -skeletons}
of $\g$ and denotes by $S_q(\g)$.
The even sector $\g^0$ is a Fr\'echet-Lie algebra of a
particular kind, the so-called {\it Arens-Michael}
Fr\'echet-Lie algebra \cite{He}, that is, it is embeddable as
a closed subalgebra into the direct product of a family of
Banach-Lie algebras (indeed,
$\g^0\cong lim_{\leftarrow}S_q(\g)$).
The change of base functor from the category of
Banach-Lie $\K$-superalgebras to the category
of Banach-Lie $\L$-superalgebras
takes a particularily simple form: $\h\mapsto \L\hat\otimes_{\K}\h$.
DeWitt calls the Banach-Lie $\L$-superalgebras images of the base
change functor {\it conventional} Lie superalgebras.
Unconventional
Lie superalgebras exist already in the dimension $(0,1)$
\cite{DeW}.
>From the point of view of deformation theory
\cite{FF}, any Lie superalgebra
$\g$ over $\L$ is a {\it deformation} of the body Lie
$\Bbb K$-superalgebra $\g_B$ over the base $Spec~\L$.
(The locally ringed superspace in the sense of \cite{Ma}, $Spec~\L$,
is a pair
consisting of a one-point topological space, $\star$, and a sheaf
over it with the algebra of global sections isomorphic to $\L$.)
Usually the deformations of Lie superalgebras are considered over
Grassmann algebras $\G q$, and it is clear that
such deformations of Banach-Lie
$\Bbb K$-superalgebras
are exhausted by all
$q$-reductions of Banach-Lie superalgebras $\g$ over $\L$.
\proclaim{Lemma 2}
Let $\g$ and $\h$ be Banach-Lie superalgebras over $\L$.
If the even sectors, $\g^0$ and $\h^0$, are isomorphic as
topological Lie algebras over $\L^0$, then
$\g$ and $\h$ are isomoprhic as Banach-Lie superalgebras over $\L$.
\endproclaim
\demo{Proof}
The canonical isomorphism $\hat\kappa$ between $\g$ and $\h$ viewed as
free graded $\L$-modules is proved
to preserve a super Lie bracket on the odd sector.
The following fact, together with
the effectiveness of $\L$, is used: for any $x,y\in\g$ and any
$\lambda\in\L^{\tilde x},~\mu\in\L^{\tilde y}$,
one has $\lambda\mu\hat\kappa ([x,y]) = \hat\kappa ([\lambda x,\mu y]) =
\lambda \mu [\hat\kappa (x), \hat\kappa (y)]$.
\qed\enddemo
We find it more fair to use the term {\it Schur-Baker-Campbell-Hausdorff-Dynkin
series}, or just {\it SBCHD series}, for what is usually referred to as
the Hausdorff series.
\proclaim{Theorem 5}
Let $\g$ be a Banach-Lie superalgebra over $\L$.
The SBCHD series converges in a DeWitt open neighbourhood of zero in
$\g^0$ and makes it into a local analytic Lie group.
\endproclaim
\demo{Proof}
It is sufficient to prove that for every $q$ the series $x.y \equiv H(x,y)$
converges in the skeleton Banach-Lie algebra $S_q(\g)^0$ as $x,y\in U^\sim$,
where $U$ is an open ball in $\g^0$ of radius ${3\over 2}log~2$.
One can show that it is sufficient
to prove the convergence in the following two special cases:
a) $H(x,\alpha)$, b) $H(-\alpha, H(\alpha,x))$,
where $\alpha\in I_0^0$ and $x\in U^\sim$.
Due to the nilpotency of $\alpha$
(indeed, $(ad_\alpha)^{2^q}= 0$), there exists a very simple majorization
of the SBCHD series:
$$ \norm{H_{r,s}(x,\alpha)} \leq C\cdot M^{r+s+1}\norm x^s $$
\noindent where $C=\eta\cdot
max\{\norm{\alpha}, \norm{\alpha}^2,\dots , \norm{\alpha}^{2^q} \}$
if $r\leq 2^q$, and $C=0$ for $r> 2^q$,
and $M,~\eta$ are positive
constants taken directly from \cite{Bou}, ch. II, $\S 7,~n^o2$.
This inequality assures the convergence of the series $H(x,\alpha)$.
A similar majorization is true for the series $H(-\alpha, H(\alpha,x))$.
To show that the SBCHD series makes $U^{\sim}$ into a local
analytic Lie group, it is sufficient to prove the analyticity of the
emerging local group operation $(x,y)\mapsto x.y$.
This is done once again by reducing the consideration
to the skeletons
$S_q(\g)$. They are Banach-Lie algebras, and therefore
the SBCHD series determines an analytic local group law as soon as it
converges at all points an open neighbourhood of zero.
In particular, the local Lie group law in each skeleton $S_q(\g)$
is continuous.
Analyticity of a mapping
in a Fr\'echet space means local representability as a sum of
polynomial series (which has been just demonstrated)
plus continuity \cite{BS}. The continuity follows from that for every
$x,y\in U^{\sim}$ one has $\pi_q(x.y) = (\pi_qx).(\pi_qy)$, that is,
the local group law in $U^{\sim}$ is represented as the inverse limit of
a sequence of continuous mappings.
\qed\enddemo
Notice that for every $q\in\N$, the Lie ideal
$I_q^0\cong (I_q^\g)^0\subset\g^0$ is a subset
of every DeWitt neighbourhood, $V$,
of sero in
$\g^0$; this means that every $I_q^0$ becomes a Fr\'echet-Lie
subgroup of the local Lie group $V$.
We will denote the Lie ideal $I_q^0$ viewed as a Fr\'echet-Lie
group by $\dot I_q^0$. The Lie algebra of $\dot I_q^0$
is canonically isomorphic to the underlying Fr\'echet-Lie algebra
of $I_q^0$, and the corresponding exponential mapping is identity,
${\Bbb I}d: I_q^0\to\dot I_q^0$.
The local Lie group quotient of $V$ by $\dot I_q^0$ is isomorphic to
a local Lie group attached to the Lie algebra $S_q\g$.
Let $M$ be a free graded $\L$-module.
Then the free graded $\L$-module $L_\L(M,M)$
with its topology becomes a Banach-Lie
superalgebra over $\L$ if being endowed with the supercommutator:
$[f,g]=_{def}fg-(-1)^{\tilde f \tilde g}gf$.
This Banach-Lie superalgebra is denoted by ${\frak gl}(M)$
and called the {\it general linear superalgebra (of $M$)}.
It is conventional because
${\frak gl}(M)\cong \L\hat\otimes {\frak gl}(M_B)$.
In a categorial approach \cite{L, BnL} ${\frak gl}(M)$
is referred to as the {\it superalgebra of $\L$-points} of a general linear
superalgebra.
\smallpagebreak
\heading
5. Lie supergroups
\endheading
\smallpagebreak
A {\it Banach-Lie supergroup} modeled over a free graded $\L$-module $M$
is a group object in the category of Banach-Lie supermanifolds
over $\L$. In this case, it is just a supermanifold endowed with
superanalytic (or supersmooth) group operations.
(The corresponding structure morphisms are fully restored from the
set-theoretic mappings due to the property of
$\L$ being effective.)
\smallskip
{\it In this paper we will consider
superanalytic Banach-Lie supergroups only.}
\smallskip
If the underlying supermanifold of a Banach-Lie superalgebra
$G$ is a DeWitt supermanifold then $G$ is called a
{\it DeWitt} Banach-Lie supergroup.
Heuristically, DeWitt supergroups may be considered as deformations
over $Spec~\L$ of Berezin-Le\u\i tes-Kostant supergroups
(= graded Lie groups).
\footnote{Once again, to the best of our knowledge,
this idea has not been shaped as a precise mathematical result yet
- and there are definitely certain
topologo-algebraic
subtleties to be surmounted.}
To every Banach-Lie supergroup, $G$, there is associated the underlying
Fr\'echet-Lie group, which we will denote by $G^0$.
The totality of all left-invariant graded vector fields on a
Banach-Lie supergroup $G$,
endowed with the graded Lie bracket of graded
vector fields, forms a Banach-Lie superalgebra which is
denoted by $sLie~G$.
(As a graded $\L$-module, it is naturally isomorphic, by means of
left translations, to the tangent module $T_eG$ to $G$ at $e_G$, and thus it
is free.)
\proclaim{Theorem 6}
The even sector, $(sLie~G)^0$, of the Banach-Lie superalgebra $sLie~G$ is
canonically isomorphic as a Fr\'echet-Lie algebra to $~Lie~(G^0)$.
\endproclaim
\demo{Proof}
The canonical isomorphism between the corresponding tangent spaces at $e$ from
Theorem 4 preserves the Lie bracket and thus is a Lie algebra isomorphism.
\qed\enddemo
For this and other basic results on supergroups,
cf. \cite{Ber2, BerL, BnL, BSV2, ChB1,2, DeW, Ko, L, P, Pe1,3,4, Ro2,3, Schw, SV1,2}.
Milnor \cite{Mi} calls a Lie group $G$ modeled over a locally convex space
a {\it Baker-Campbell-Hausdorff Lie group} if the (Schur-)Baker-Campbell-
Hausdorff(-Dynkin) series converges on a neighbourhood of zero, $V$, in
the locally convex Lie algebra $Lie~G$ of $G$ making it into
a local analytical Lie group, and in addition $G$
possesses an exponential
mapping $exp_G: Lie~G\to G$ which is a local Lie group isomorphism.
We will refer to such groups as {\it Schur-Baker-Campbell-Hausdorff-Dynkin}
(or just {\it SBCHD}) Lie groups.
An SBCHD Lie group is always analytical. Examples of such groups are all
Banach-Lie groups and (most of) groups of currents, in particular, loop
groups \cite{PrS}. However, not all Fr\'echet-Lie groups have the
SBCHD property: the group $Diff~S^1$ has not.
Since any connected simply connected
SBCHD Lie group $G$ is {\it regular} \cite{Mi}, then any
locally convex Lie algebra
morphism from $Lie~G$ to the Lie algebra $Lie~H$ of any Lie group, $H$,
modeled over a complete locally convex space,
gives rise to a Lie group morphism $G\to H$.
\proclaim{Theorem 7}
Let $G$ be a superanalytic Banach-Lie supergroup.
The underlying
Fr\'echet-Lie group $G^0$ is an SBCHD Lie group.
\endproclaim
\demo{Proof} Let a chart $\phi$ be defined in an open ball $B$ in a free graded $\L$-module $M$
and its values cover
an open neighbourhood $U$ of $e$ in $G$. Then $M$, first, can be
identified with the tangent module to $G$ at $e$ and second,
becomes a Banach-Lie superalgebra (isomorphic to the Lie superalgebra of $G$)
in the most natural way. The uniqueness of an analytic mapping with a given
set of derivatives
at a point implies that for every $x,y\in B$
one has
$$H(x,-y)\phi^{-1}[\phi(x)\phi(y)^{-1}]=e$$
\noindent and therefore
$\phi$ establishes a local Lie group isomorphism
between $G$ and an SBCHD local Lie group associated to $\g^0$ (Theorem 5).
This means that actually $\phi$ is an exponential mapping from $M^0\cong (sLie ~G)^0$
to the underlying Fr\'echet-Lie group of $G$, and the conditions of the Milnor's
definition are fulfilled.
\qed\enddemo
\proclaim{Theorem 8}
Let $\g$ be a Banach-Lie superalgebra over $\L$ such that the even sector
$\g^0$ is enlargable as a Fr\'echet-Lie algebra to a
SBCHD Lie group, $G$. Then $G$ carries a structure of a
Banach-Lie supergroup with $sLie~G\cong\g$.
Furthermore, if the neighbourhood $V$ in the definition of a
SBCHD Lie group can be chosen DeWitt open, then $G$ can be made into
a DeWitt Banach-Lie supergroup with the property $sLie~G\cong\g$.
\endproclaim
\demo{Proof}
The canonical atlas of the form $\{\phi_g\}_{g\in G},~
\phi_q(x)=_{def}exp(gx), x\in B_{{3\over 2}log~2}(0)$
is used to make $G$ into a
supermanifold, where $U$ is a neighbourhood of $e$ such that the restriction
of $exp_G$ to $U$ is a diffeomorphism. Both the superanalyticity of
the group operations and
the isomorphism $sLie~G\cong\g$ are verified by direct computation based on the
local isomorphism between $G$ an a local Lie group associated to $\g^0$
(Theorem 5) with Lemma 2 involved.
\qed\enddemo
We will say that a Banach-Lie $\L$-superalgebra $\g$
is {\it (Rogers) enlargable} if it comes from a Banach-Lie supergroup, $G$,
over $\L$ (that is, $sLie~G\cong\g$),
and that $\g$ is {\it DeWitt enlargable} if it comes from
a DeWitt Banach-Lie supergroup.
\proclaim{Theorem 9} Every finite-dimensional Lie superalgebra $\g$
over $\L$ is DeWitt enlargable.
\endproclaim
\demo{Proof}
It was proved in \cite{Pe3} that the even sector $\g^0$ of
a finite-dimensional Lie superalgebra $\g$
over $\L$ comes from an SBCHD Lie group and that
the neighbourhood $V\subset\g^0$ in the definition of a
SBCHD Lie group can be chosen DeWitt open.
\qed\enddemo
\proclaim{Theorem 10}
Let $\g$ be a Banach-Lie superalgebra.
Suppose all $q$-skeletons are enlargable Banach-Lie algebras
and let $exp_q : S_q(\g)\to G_q$ be the corresponding exponential
mappings to the Banach-Lie groups.
If there exists an open neighbourhood of zero, $U\subset\g$, such that
for every $q$ the respriction of $exp_q$ to $\pi_q(U)$ is
one-to-one, then $\g$ is an enlargable Lie superalgebra.
If there exists a DeWitt open $U$ with the above property then
$\g$ is DeWitt enlargable.
\endproclaim
\demo{Proof}
Set $G=_{def}lim_{\leftarrow}G_q$
and apply Theorem 8.
\qed\enddemo
\proclaim{Theorem 11}
A conventional purely even Banach-Lie superalgebra $\g$ is DeWitt enlargable
if and only if the body $\g_B$ is an enlargable Banach-Lie algebra.
\endproclaim
\demo{Proof}
The even sector $\g^0$ is isomoprhic to the semidirect product
$\g_B^0 \ltimes_\tau I_0^\g$.
The Lie algebra action $\tau : \g_B^0 \to Der~I_0^\g$
gives rise to a smooth action
$\dot\tau : G_0\to Aut~\dot I_0^\g$ where $G_0$ is a connected simply
connected Banach-Lie group attached to $\g_B^0$.
(This is proved at the level of $q$-redictions of the Lie ideal $I_0^\g$,
by means of Banach-Lie theory \cite{Bou}.)
Now the desired SBCDH Fr\'echet-Lie group assigned to $\g^0$ is
the semidirect product $G^0 \cong G_0 \ltimes_{\dot\tau} \dot I_0^\g$,
the exponential map being defined by
$exp_{G^0}(x,y)=_{def} (exp_{G_0}(x), y)$.
\qed\enddemo
In particular, the following is true.
\proclaim{Theorem 12}
The general linear superalgebra ${\frak gl}(M)$ is DeWitt enlargable
for an arbitrary free graded $\L$-module $M$, and the corresponding
Banach-Lie group is the general linear group $GL(M)$.
\endproclaim
\qed
\proclaim{Theorem 13}
A Banach-Lie superalgebra $\g$, admitting a continuous graded
Lie monomorphism into an enlargable Banach-Lie superalgebra $\h$,
is enlargable. Moreover, if $\h$ is DeWitt enlargable,
then $\g$ is so.
\endproclaim
\demo{Proof}
Based on the Theorem on extension of analytic structure
\cite{\`S} and Theorem 8.
\qed\enddemo
A Lie superalgebra $\g$ is {\it centerless}
if its {\it supercenter},
${\frak sz(\g)} = \{x\in\g :\forall y\in\g, [x,y]=0\}$, is zero.
Although it is known that every centerless Banach-Lie algebra
is enlargable \cite{vEK}, the even sector $\g^0$ of
a centerless Banach-Lie superalgebra $\g$ is not
necessarily a centerless Lie algebra. Therefore, the following result
is of interest.
\proclaim{Theorem 14}
Every centerless Banach-Lie superalgebra $\g$ is DeWitt enlargable.
\endproclaim
\demo{Proof}
The proof is similar to the proof of the corresponding
purely even result and it is
based on the fact that $\g$ admits a faithful linear representation ---
which is, of course, the adjoint representation $x\mapsto ad_x$ ---
and thus a continuous monomorphism
from $\g$ to the general linear superalgebra ${\frak gl}(\g_+)$
comes into being where
$\g_+$ stands for the underlying free graded $\L$-module of $\g$.
Now we use Theorems 8 and 13.
\qed\enddemo
\smallpagebreak
\heading
6. Examples
\endheading
\smallpagebreak
Fix an enlargable Banach-Lie algebra $\k$ and a
topological 2-cocycle $\eta\in H^2(\k; \Bbb R)$
such that the period group \cite{vEK}
of this cocycle, $Per~(\eta)$, is an infinite cyclic
subgroup of $\R$; we can assume that $Per~(\eta)\cong \Bbb Z$.
In other words, this means that
a one-dimensional central extension of $\k$ by means of the cocycle $\eta$,
a Banach-Lie algebra $\l\cong\k\times_\eta\R$,
is enlargable, and the connected simply connected Banach-Lie group $I$
corresponding to $\k\times_\eta\R$ is a central
extension of the connected
simply connected Banach-Lie group $K$ corresponding to
$\k$ by means of the 1-dimensional
toroidal group $U(1)\cong\R/\Bbb Z$:
$$e\to U(1) \to I \to K \to e $$
Different examples of this kind can be found in
\cite{vEK, LaT, Bou, E, PrS}.
Recall that
$\xi_1,\xi_2,\dots ,\xi_q, \dots$ denote a fixed
system of topologically free odd generators for the algebra $\L$.
\medskip
\noindent {\bf Example 1.}
{\it Enlargable Banach-Lie superalgebra which is non DeWitt enlargable.}
Put $\h =_{def} \L\hat\otimes\k$ and let
$\theta = \xi_1\xi_2\eta\in H_\L^2(\h;\L)$
be a bi-$\L$-linear even topological 2-cocycle on $\h$ with coefficients in
$\L$. Let $\g=_{def} \L\times_\theta\h$ be a
Banach-Lie superalgebra one-dimensional central
extension (over $\L$) of $\h$ by means of the 2-cocycle $\theta$.
For every $q\in\N$, the $q$-skeleton algebra $S_q\g$ is enlargable
and the restriction of the corresponding exponential mapping
$exp_q: S_q\g\to G_q$ to the open unit ball is one-to-one.
Indeed, the period group of the extension
$$0\to \L^0 \to S_q(\g)\to S_q(\h)\cong\G q\otimes\k \to 0$$
\noindent is isomorphic to $\xi_1\xi_2\Bbb Z$, and
therefore the corresponding
Banach-Lie group extension is well-defined:
$$e\to \L^0/\xi_1\xi_2\Bbb Z \to G_q\to S_q(I) \to e$$
\noindent where by $I_q$ we denote the $q$-skeleton of a connected
simply connected Fr\'echet-Lie group associated to the conventional
Lie algebra $\h^0$.
Now an application of the Theorem 10 implies that $\g$ is an enlargable
Banach-Lie superalgebra.
Suppose $\g$ is a DeWitt enlargable Banach-Lie superalgebra.
Then there exists an analytic DeWitt Lie supergroup $G$ attached to
$\g$. According to Theorems 6 and 7, the underlying Fr\'echet-Lie group
$G^0$ of $G$ is a SBCHD Fr\'echet-Lie group associated to the even sector
$\g^0$.
The mapping $i: (r,x)\mapsto (r\xi_1\xi_2, 1_\L\otimes x)$ determines
an embedding of the Banach-Lie algebra $\l$ into the Fr\'echet-Lie
algebra $\g$ as a (closed) locally convex Lie subalgebra.
Therefore, there exists a Lie group morphism $\hat i$
from $I$ to $G^0$ such that the corresponding exponential mappings
commute: $exp_{G^0}\circ i = \hat i \circ exp_I$.
However, for an element $y=_{def}(1, 0)\in\l$ one has
$exp_{G^0}\circ i(y) = exp_{G^0}(\xi_1\xi_2, e) \neq 0$,
because of injectivity of $exp_{G^0}$ along the ``soul direction,''
while $\hat i \circ exp_I(y)= \hat i(e_I) = e_{G^0}$.
This contradiction means that $\g$ is not DeWitt enlargable.
\medskip
\noindent {\bf Example 2.}
{\it Non-enlargable Banach-Lie superalgebra of which all $q$-skeletons
are enlargable Banach-Lie algebras.}
Define a Banach-Lie algebra $\m$ as an $l_1$-type sum
of countably many copies of the algebra $\k$, that is,
$\m$ is isomorhic to the completion of the Lie algebra
$\oplus_{i\in\Bbb N} \k_{(i)}$ endowed with the norm
$$\Vert (x_i)_{i=1}^\infty\Vert_{\oplus_{i=1}^\infty \k_{(i)}}
=_{def} \sum _{i=1}^\infty \Vert x_i \Vert_\k$$
\noindent
Define a topological
bi-$\L$-linear even
2-cocycle $\theta$ on $\m$ with coefficients in
$\L$ by
$$\theta[(x_i)_{i=1}^\infty, (y_j)_{j=1}^\infty]=_{def}
\sum_{n=1}^\infty\xi_n\xi_{n+1}\eta (x_n,y_n) $$
\noindent Set $\h =_{def} \L\hat\otimes\m$ and
let $\g=_{def} \L\times_\theta\h$ be a
Banach-Lie superalgebra one-dimensional central
extension (over $\L$) of $\m$ by means of the 2-cocycle $\theta$.
For every $q\in\N$, the $q$-skeleton Banach-Lie algebra $S_q(\g)$ is enlargable.
Indeed, the period group of the extension
$$0\to \L^0 \to S_q(\g)\to S_q(\m) \to 0$$
\noindent is isomorphic to the discrete subgroup
$(\sum_{n=1}^q\xi_n\xi_{n+1})\Bbb Z$,
and therefore the corresponding
Banach-Lie group extension is well-defined.
At the same time, the Banach-Lie superalgebra $\g$ is non enlargable.
Indeed, for every $n\in\Bbb N$, there exists an embedding
$$i_k: (r,x)\mapsto (r\xi_n\xi_{n+1}, 1_\L\otimes x)$$
\noindent of the Banach-Lie algebra $\l$ in $\g^0$ as a closed locally convex subalgebra,
and the arguments similar to those used in Example 1 show that
if $\g^0$ were enlargable then the exponential mapping
on $\g^0$ would send all elements of the form
$(\xi_n\xi_{n+1}, 0)$ to the identity of the corresponding
Fr\'echet-Lie group. Since elements of the form
$(\xi_n\xi_{n+1}, 0)$ are to be found in every neighbourhood of zero
in $\g^0$, then the exponential map would not be a local diffeomorphism,
which is impossible in view of Theorem 7.
\smallpagebreak
\heading
Acknowledgments
\endheading
\smallpagebreak
I am grateful
to the following Institutions for supporting the present
investigation in 1989--1992:
D.Sc. Training Program at
Tomsk State University; Institute of Mathematics, Novosibirsk
Science Centre (both --- Russian Federation,
then USSR); Department of Mathematics, University of Genoa;
Group of Mathematical Physics of the
Italian National Research Council (both Italy);
Department of Mathematics, University of Victoria (Canada);
and Department of Mathematics, Victoria University of Wellington
(New Zealand). A 1992 research grant V212/451/RGNT/594/153
from the Internal Grant Committee of the latter University
is acknowledged.
My special thanks also go to S.M. Berger, U. Bruzzo,
A.I. and N.S. Chistyakov, B.S. DeWitt, R.I. Goldblatt,
S.S. Kutateladze, A.E. Hurd, L.C. Johnston, and D. Leeming
for their encouragement and/or hospitality
at various stages of this work.
In addition, I am grateful to
Ugo Bruzzo for reading this manuscript prior to publication and
suggesting a large number of improvements.
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\endRefs
\bye