Tài liệu Quaternionic Structures in Mathematics and Physics docx

Proceedings of the Second Meeting
Quaternionic Structures
in Mathematics
and Physics
r Editors
Stefano Marchiafava
Paolo Piccinni
Massimiliano Pontecorvo
i
2
=j
2
=k
2
=-l, ij=-ji=kjk=-kj=i, ki=-ik=j
Proceedings of die Second Meeting
Quaternionic Structures in
Mathematics and Physics
Proceedings
of
the Second Meeting
Quaternionic Structures
in
Mathematics and Physics
Rome, Italy
6-10
September
1999
Editors
Stefano Marchiafava

University degli
Studi
di
Roma
"La Sapienza "
Paolo Piccinni
Universita degli
Studi
di
Roma
"La Sapienza "
Massimiliano Pontecorvo
Universita degli
Studi Roma
Tre
Printed with
a
partial support of C.N.R. (Italy)
\fe
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Scientific
wl Singapore • New Jersey
•
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QUATERNIONIC STRUCTURES IN MATHEMATICS AND PHYSICS
Proceedings of the Second Meeting
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This volume is dedicated to the memory of
Andre Lichnerowicz and Enzo Martinelli
FOREWORD
Five years after the meeting "Quaternionic Structures in Mathematics and
Physics", which took place at the International School for Advanced Studies
(SISSA), Trieste, 5-9 September 1994, we felt it was time to have another meeting
on the same subject to bring together scientists from both areas.
The second Meeting on Quaternionic Structures in Mathematics and Physics
was held in Rome, 6-10 September 1999.
Like in 1994, also this time the Meeting opened a semester at The Erwin
Schrodinger International Institute for Mathematical Physics of Vienna, that in
Fall 1999 was dedicated to "Holonomy Groups in Differential Geometry", and
many participants of this ESI program were invited speakers at the Quaternionic
Meeting.
We thank D.V. Alekseevsky, K. Galicki, P. Gauduchon, S. Salamon, members
of the Scientific Committee of this Second Meeting, and all the speakers for their
contribution.
We gratefully acknowledge financial support from Progetto Nazionale di Ricerca
MURST "Proprieta' Geometriche delle Varieta' reali e complesse" (both with
local and national funds), Universita' di Roma "La Sapienza", Universita' di
Roma Tre, Comitato Nazionale per la Matematica - C.N.R We acknowledge
also the hospitality of both Departments of Mathematics of Universities
"
Roma
La Sapienza" and "Roma Tre".
We hearty thank the valuable collaboration of Elena Colazingari in the organi-
zation of the Meeting. We are also grateful to Angelo Bardelloni for the technical
preparation of the electronic version of these Proceedings, to Tiziana Manfroni
and Paolo Marini for setting up the web site and to our colleague Francesco
Pappalardi for TeX-nical support.

Finally, it gives us great pleasure to thank all the participants of the meeting
for their interest and enthusiasm, and of course all the contributors of the present
Proceedings.
Roma, March 2001
Stefano Marchiafava
Paolo Piccinni
Massimiliano Pontecorvo
INTRODUCTION TO THE CONTRIBUTIONS
During the five years, which passed after the first meeting "Quaternionic Struc-
tures in Mathematics and Physics " interest in quaternionic geometry and its appli-
cations continued to increase. A progress was done in constructing new classes of
manifolds with quaternionic structures (quaternionic Kahler, hyper-Kahler, hyper-
complex etc.), studying differential geometry of special classes of such manifolds and
their submanifolds, understanding relations between the quaternionic structure and
other differential-geometric structures, and also in physical applications of quater-
nionic geometry. Some generalizations of classical quaternionic-like structures (like
HKT -structures and hyper-Kahler manifolds with singularities) naturally appeared
and were studied. Some of these results are published in this proceedings.
A new simple and elegant construction of homogeneous quaternionic pseudo-
Kahler manifolds is proposed by V. CORTES. It gives a unified description of
all known homogeneous quaternionic Kahler manifolds as well as new families of
quaternionic pseudo-Kahler manifolds and their natural mirror in the category of
supermanifolds.
Generalizing the Hitchin classification of Sp(l)-invariant hyper-Kahler and quater-
nionic Kahler 4-manifolds, T. NITTA and T. TANIGUCHI obtain a classification of
S'p(l)
n
-invariant quaternionic Kahler metrics on 4n-manifold. All of these metrics
are hyper-Kahler.

I.G. DOTTI presents a general method to construct quaternionic Kahler compact
flat manifolds using Bieberbach theory of torsion free crystallographic groups.
Using the representation theory , U. SEMMELMANN and G. WEINGART find
some Weitzebock type formulas for the Laplacian and Dirac operators on a com-
pact quaternionic Kahler manifold and use them for eigenvalue estimates of these
operators. As an application, they prove some vanishing theorems, for example,
they prove that odd Betti numbers of a compact quaternionic Kahler manifold with
negative scalar curvature vanish.
A hyper-Kahler structure on a manifold M defines a family (J
t
,u>
t
) of complex
symplectic structures, parametrized by t € CP
1
. R. BIELAWSKI gives a general-
ization of the hyper-Kahler quotient construction to the case when a holomorphic
family Gt, t £ CP
1
of complex Lie group is given, such that Gt acts on M as a
group of automorphisms of (J
t
,wt).
The existence of a canonical hyper-Kahler metric on the cotangent bundle T*M
of a Kahler manifold M was proved independently by D.Kaledin and B.Feix. In
the present paper D. KALEDIN presents his proof in simplified form and obtains
an explicit formula for the case when M is a Hermitian symmetric space.
A toric hyper-Kahler manifold is defined as the hyper-Kahler quotient of the
quaternionic vector space IF by a subtorus of the symplectic group Sp(n). H.
KONNO determines the ring structure of the integral equivariant cohomology of a

toric hyper-Kahler manifold.
M. VERBITSKY gives a survey of some recent works about singularities in
hyper-Kahler geometry and their resolution. It is shown that singularities in a
singular hyper-Kahler variety (in the sense of Deligne and Simpson) have a simple
vii
viii INTRODUCTION TO THE CONTRIBUTIONS
structure and admit canonical desingularization to a smooth hyper-Kahler mani-
fold. Some results can be extended to the case of the hypercomplex geometry.
Hypercomplex manifolds (which is the same as 4n-manifolds with a torsion free
connection with holonomy in GL(n,M) ) are studied by H. PEDERSEN. He de-
scribes three constructions of such manifolds: 1) via Abelian monopoles and ge-
odesic congruences on Einstein-Weyl 3-manifolds , 2) as a deformation of Joyce
homogeneous hypercomplex structures on G x T
k
where G is a compact Lie group
and 3) as a deformation of the hypercomplex manifold Vp(M), associated with a
quaternionic Kahler manifold M and an instanton bundle P
—>•
M by the construc-
tion of Swann and Joyce.
M.L. BARBERIS describes a construction of left-invariant hypercomplex struc-
tures on some class of solvable Lie groups. It gives all left-invariant hypercomplex
structures on 4-dimensional Lie groups. Properties of associated hyper-Hermitian
metrics on 4-dimensional Lie groups are discussed.
D.
JOYCE proposes an original theory of quaternionic algebra, having in mind
to create algebraic tools for developing quaternionic algebraic geometry. Appli-
cations for constructing hypercomplex manifolds and study their singularities are
considered.
Properties of hyperholomorphic functions in I

4
are studied by S-L. ERIKSSON-
BIQUE. Hyperholomorphic functions are defined as solutions of some general-
ized Cauchy-Riemann equation, which is defined in terms of the Clifford algebra
C1(M°'
3
)«H©H.
Other more general notion of hyper-holomorphic function on a hypercomplex
manifold M is proposed and discussed in the paper by ST. DIMIEV , R. LAZOV
and N. MILEV.
O.BIQUARD defines and studies quaternionic contact structures on a manifold.
Roughly speaking, it is a quaternionic analogous of integrable CR structures.
Generalizing the ideas of A. Gray about weak holonomy groups, A.SWANN
looks for G-structures which admit a connection with "small" torsion, such that the
curvature of these connections satisfies automatically some interesting conditions,
for example, the Einstein equation.
G. GRANTCHAROV and L. ORNEA propose a procedure of reduction which
associates to a Sasakian manifold S with a group of symmetries a new Sasakian
manifold and relate it to the Kahler reduction of the associated Kahler cone K{S).
The geometry of circles in quaternionic and complex projective spaces are studied
by S. MAEDA and T. ADACHI. The main problem is to find the full system of
invariants of a circle C, which determines C up to an isometry, and to determine
when a circle is closed.
Special 4-planar mappings between almost Hermitian quaternionic spaces are
defined and studied by J. MIKES, J. BELOHL'AVKOV'A and O. POKORN'A.
Some generalization of the flat Penrose twistor space C
2,2
is constructed and
discussed by J. LAWRYNOWICZ and 0. SUZUKI.
M. PUTA considers some geometrical aspects of the left-invariant control prob-

lem on the Lie group Sp(l).
Quaternionic representations of finite groups are studied by G. SCOLARICI and
L. SOLOMBRINO.
Quaternionic and hyper-Kahler manifolds naturally appear in the different phys-
ical models and physical ideas produce new results in quaternionic geometry. For
INTRODUCTION TO THE CONTRIBUTIONS ix
example, Rozanski and Witten introduce a new invariant of hyper-Kahler mani-
fold as the weights in a Feynman diagram expansion of the partition function of
a 3-dimensional physical theory. A variation of this construction, proposed by N.
Hitchin and J. Sawon , gives a new invariant of links and a new relations between
invariants of a hyper-Kahler manifold X, in particular, a formula for the norm of
the curvature of X in terms of some characteristic numbers and the volume of X.
These results are presented in the paper by J. SAWON.
The classical Atiyah-Hitchin-Drinfeld-Manin's monad construction of
anti-self-
dual connections over 5
4
= HP
1
was generalized by M. Capria and S. Salamon to
any quaternionic projective space HP". Using representation theory of compact
Lie groups, Y. NAGATOMO extends the monad construction to any Wolf space
(i.e.
compact quaternionic symmetric space).
A quaternionic description of the classical Maxwell electrodynamics is proposed
in the paper by D. SWEETSER and G. SANDRI.
A. PRASTARO applies his theory of non commutative quantum manifolds to the
category of quantum quaternionic manifolds and discusses the theorem of existence
of local and global solutions of some partial differential equations.
A hyper-Kahler structure on a 4n-manifold M is defined by a torsion free con-

nection V with holonomy group in Sp(n). A natural generalization is a connection
V with a torsion T = (T^) and the holonomy in Sp(n). If the covariant torsion
r = g(T) = (gijT^), where g is the V-parallel metric on M, is skew-symmetric,
then the manifold (M, V,<?) is called hyper-Kahler manifold with a torsion, or
shortly HKT-manifold. If, moreover, the 3-form r is closed, it is called strongly
HKT-manifold. Such manifolds appear in some recent versions of supergravity.
A purely mathematical survey of the theory of HKT-manifolds is given by Y. S.
POON.
Some applications of HKT-geometry to physics is discussed in the paper by G.
PAPADOPOULUS. He describes a class of brain configurations, which are approx-
imations of solutions of 10 and 11 dimensional supergravitation.
A. VAN PROEYEN gives a review of special Kahler geometry (which can be
mathematically denned as the geometry of Kahler manifolds together with a com-
patible, in some rigorous sense, flat connection), its physical meaning and connec-
tions to quaternionic and Sasakian geometry.
Dmitri V. Alekseevsky
Second Meeting on
Quaternionic Structures
in Mathematics and Physics
Roma, Italy
September 6-10, 1999
List of Participants
||Univ. Lecce
~
ABDEL-KHALED Khaled
IAGOSTI
COLAZINGARI Elena Univ. Roma Tre
Helena
@

mat.uniroma3.it
ALEKSEEVSKY Dmitri
~||Sophus Lie Center |[
APOSTOLOVVestislav
||Bulgarian Acad, of Sc. ||
ARMSTRONG John
"||Oxford University ||
BARBERIS Laura
"||FaMAF-Univ. Nac. Cordoba||barberis@mate,uncor.edu
BATTAGLIA Fiammetta
HlUniv. Firenze

BIELAWSKI Roger
HUniversity of Edinburgh ||
BIQUARD Olivier
~|CNRS,
Ecole Polytechnique|[
BONAN Edmond Huniv. Picardie
^|ebonan Qcybercable.fr
BORDONI Manlio
~||Univ. Roma "La Sapienza" ||
BOURGUIGNON Jean-Pierre IHES ||
CHIOSSI Simon G. Univ. Genova
llchiossi® dima.unige.it
ymontroy® wcb.u-net.com
ICHISHOLM Roy Univ. of Kent
CONTESSA Maria Univ. Palermo | |contessa® dipmat.math.unipa.it
| |vicente@ math.uni-bonn.de
CORTES Vincente Univ. Bonn
~|Univ. Cagliari

nidambra@vaxca1 .unica.it
D'AMBRA
Giuseppina
||Bulgarian Acad. Of Sci. ||
DIMIEV Stancho
~|FaMAF-Univ. Nac.
DOTTI Isabel
ERIKSSON-BIQUE Sirkka-Liisa ||Univ. of Joensuu
||Si

^|Univ.Bari
FALCITELLI Maria | |falci@ pascal.dm.uniba.it
FIGUEROA-O'FARRILL Jose Miguel||Univ. of London"

FINO Anna Univ. Torino

nifujiki® math.sci.osaka-u.ac.jp
FUJIKI Akira Osaka Univ.
llUniv. Madrid Carlos III
1|gaeta@roma1
.infn.it
GAETA Giuseppe
GALICKI Krzysztof
Univ. of New Mexico

GAUDUCHON Paul CNRS ypg® math,polytechnique.fr
GOTO Ryushi
Osaka Univ. math.sci.osaka-u.ac.jp
GRANTCHAROVGueo
~|Un.

California at Riverside||,edu
HARUKO Nishi
^Kyushu Univ.
||nishi@ math.kyushu-u.ac.jp
HERRERA Rafael
Yale Univ.
~||~"
HIDEYA Hashimoto ||Nippon Inst, of Technology ||,jp
HIJAZI Oussama LlUniv. Nancy I

IANUS Steriu
Univ. of Bucharest
||ianus@ pompeiu.imar.ro
JOYCE Dominic Oxford Univ.
nidominic.joyce® lincoln.ox.ac.uk
KALEDIN Dmitry
lllndepend.
Univ. of Moscow Ifkaledin® balthi.dnttm.rssi.ru
KOBAK Piotr
yuniwersytet Jagiellonski ||kobak®im.uj.edu.pl
LIST OF PARTICIPANTS
KONDERAK Jerzy
Univ. Bari

KONNO Hiroshi
H|Univ.
ot Tokyo
Hlkonno® ms.u-tokyo.ac.jp
LAKOMA Lenka
||Palacky Univ. Olomouc ]|lenka.lakoma@upol,cz

LAWRYNOWICZ Julian
"llPolish Acad, of Sciences ||
LAZOV Rumyan
||Bulgarian Acad. Sci. |[lazovr@math,bas.bg
MAEDA Sadahiro
Shimane Univ.
Nsmaeda® math.shimane-u.ac.jp
MARCHIAFAVAStefano
"1|Univ. Roma "La Sapienza" |[ .it
MARINOSCI Rosanna
LlUniv. Lecce
MAZZOCCO Renzo
||Univ. Roma "La Sapienza" ||
MIKES Josef
||Palacky Univ. Olomouc |[mikes® risc.upol.cz
MOROIANU Andrei
^|Ecole Polytechnique ||
NAGATOMO Yasuyuki
Univ. of Tsukuba
ninagatomo@ math.tsukuba.ac.jp
NANNICINI Antonella
Univ. Firenze
O'GRADY
Kieran
HUniv. Roma "La Sapienza" ||
OHBA Kiyoshi
Ochanomizu Univ. ||ohba@ math.ocha.ac.jp
ORNEA Liviu
||Univ. of Bucharest |[
PAPADOPOULOS George

LlUniv. of Cambridge, UK ]|
PARTON Maurizio
Univ. Pisa
Hparton@dm,
unipi.it
Hlpastore® pascal.dm.uniba.it
PASTORE Anna Maria Univ. Bari
PEDERSEN Henrik
SDU-Odense Univ.
n|louis.pernas@u-picardie
,fr
PERNAS Louis Univ. de Picardie
HlUniv, Roma "La Sapienza" |[
PICCINNI Paolo
Hlpodesta® alibaba.math.unifi.it
PODESTA' Fabio Univ. Firenze
||Czech Univ. of Agriculturel|
POKORNAaga
PONTECORVO Massimiliano ||Univ. Roma Tre
Hmax® mat.uniroma3.it
~||Un.
California at Riverside||ypoon®math.ucr.edu
POONYatSun
niUniv. Roma "La Sapienza" |[Prastaro@dmmm,uniroma1.it PRASTARO Agostino
H|Un. Pierre et Marie Curiel|rent@math,jussieu.fr
RENTSHLER Rudolf
HISUNY at Stony Brook || ROCEK Martin
RYUSHI Goto Osaka Univ. ||goto@ math.sci.osaka-u.ac.jp
~1|Oxford Univ~
SALAMON Simon

SANDRI Guido Boston Univ.
SAWON Justin ||New College, Oxford Univ. ||
SCOLARICI Giuseppe Univ. Lecce

SEMMELMANN Uwe Univ. of Munich

SOLOMBRINO Luigi Univ. Lecce

SPIRO Andrea Univ. Ancona
||
STANCIU Sonia llmperial College

SWEETSER Douglas
Hsweetser@alum,
mit.edu
TODA Masahito
||Tokyo Metropolitan Univ. ||
VAN PROEYEN Antoine
~1|K.U.
Leuven"
niAntoine.VanProeyen@(ys.kuleuven.ac.be
VANZURA Jiri
"I
I
Acad,
of
Sci.
of CR

| |vanzura@ ipm.cz
VANZUROVAAlena niPalacky Univ.
|
|vanzurov@ risc.upol.cz
VERBITSKY Misha
HlSteklov Institute

WIDDOWS Dominic
"llOxford Univ.

CONTENTS
Foreword v
Introduction to the Contributions vii
List of Participants xi
Hypercomplex Structures on Special Classes of Nilpotent
and Solvable Lie Groups 1
M. L. Barberis
Twistor Quotients of HyperKahler Manifolds 7
R.
Bielawski
Quaternionic Contact Structures 23
0. Biquard
A New Construction of Homogeneous Quaternionic Manifolds
and Related Geometric Structures 31
V. Cortes
Spencer Manifolds 101
St. Dimiev, R. Lazov and N. Milev
Quaternion Kahler Flat Manifolds 117
1.
G. Dotti

Hyperholomorphic Functions in R
4
125
S L.
Eriksson-Bique
A Note on the Reduction of Sasakian Manifolds 137
G. Grantcharov and L. Ornea
A Theory of Quaternionic Algebra, with Applications to
Hypercomplex Geometry 143
D.
Joyce
A Canonical Hyperkahler Metric on the Total Space of
a Cotangent Bundle 195
D.
Kaledin
Equivariant Cohomology Rings of Toric HyperKahler Manifolds 231
H. Konno
xiv CONTENTS
An Introduction to Pseudotwistors Basic Constructions 241
J. Lawrynowicz and 0. Suzuki
Differential Geometry of Circles in a Complex Projective Space 253
S. Maeda and T. Adachi
On Special 4-Planar Mappings of Almost Hermitian
Quaternionic Spaces 265
J. Mikes, J. Belohldvkovd and 0. Pokornd
Special Spinors and Contact Geometry 273
A.
Moroianu
Generalized ADHM-Construction on Wolf Spaces 285
Y. Nagatomo

£p(l)
n
-Invariant Quaternionic Kahler Metric 295
T. Nitta and T. Taniguchi
Brane Solitons and Hypercomplex Structures 299
G. Papadopoulos
Hypercomplex Geometry 313
H. Pedersen
Examplex of Hyper-Kahler Connections with Torsion 321
Y. S. Poon
Theorems of Existence of Local and Global Solutions of PDEs
in the Category of Noncommutative Quaternionic Manifolds 329
A.
Prdstaro
Optimal Control Problems on the Lie Group SP(1) 339
M. Puta
A New Weight System on Chord Diagrams via HyperKahler Geometry 349
J. Sawon
Quaternionic Group Representations and Their Classifications 365
G. Scolarici and L. Solombrino
Vanishing Theorems for Quaternionic Kahler Manifolds 377
U. Semmelmann and G. Weingart
CONTENTS xv
Weakening Holonomy 405
A.
Swann
Maxwell's Vision: Electromagnetism with Hamilton's Quaternions 417
D.
Sweetser and G. Sandri
Special Kahler Geometry 421

A.
Van Proeyen
Singularities in HyperKahler Geometry 439
M. Verbitsky
Second Meeting on
Quaternionic Structures
in Mathematics and Physics
Roma, 6-10 September 1999
HYPERCOMPLEX STRUCTURES ON SPECIAL CLASSES OF
NILPOTENT AND SOLVABLE LIE GROUPS
MARIA LAURA BARBERIS
1. INTRODUCTION
A hypercomplex structure on a manifold M is a family {J
Q
}a=i,2,3 of complex
structures on M satisfying the following relations:
(1.1) Jl = ~I, a = 1,2,3, J
3
=
J
x
J
2
= -J
2
Jx
where / is the identity on the tangent space T
P
M of M at p for all p in M. A
riemannian metric jona hypercomplex manifold (M, {J

Q
}a=i,2,3) is called hyper-
Hermitian when
g(J
a
X,
J
a
Y) = g(X, Y) for all vectors fields X, Y on M, a = 1,2,3.
Given a manifold M with a hypercomplex structure {J
a
}
a
=i,2,3 and a hyper-
Hermitian metric g consider the 2-forms ui
a
, a
—
1,2,3, defined by
uj
a
(X,Y)=g(X,J
a
Y).
The metric g is said to be hyper-Kahler when duj
a
—
0 for a
—
1,2,3.

It is well known (cf. [5]) that a hyper-Hermitian metric g is conformal to a hyper-
Kahler metric g if and only if there exists an exact
1-form
0 € A
X
M such that
(1.2) du
a
= 6Au)
a
, a =1,2,3
where, if g = e
f
g for some / e C°°{M), then 0 = df.
A hypercomplex structure on a real Lie group G is said to be invariant if left
translations by elements of G are holomorphic with respect to J
lt
J
2
and J
3
.
Given g a real Lie algebra, a hypercomplex structure on a is a family {J
a
}a=i,2,3
of endomorphisms of g satisfying the relations (1.1) and the following conditions:
(1.3) iV
a
= 0, a =1,2,3
where / is the identity on g and N

a
is the Nijenhuis tensor corresponding to J
a
:
(1.4) N
a
{x, y) = [J
a
x, J
a
y] - J
a
{[x, J
a
y] + [J
a
x,y\) -
[a;,
y]
for all x,y € g. Clearly, if G is a Lie group with Lie algebra g, a hypercomplex
structure on g induces by left translations an invariant hypercomplex structure on G.
1
2 MARIA LAURA BARBERIS
Two hypercomplex structures {J
Q
}a=i,2,3 and
{J'
a
}
a

=i,2,3
on g are said to be equiv-
alent if there exists an automorphism
<j>
of g such that
4>J
a
=
J'
a
<f>
for a = 1,2,3.
The classification of the four-dimensional real Lie algebras carrying hypercomplex
structures was done in [2], where the equivalence classes of hypercomplex structures
were determined and the corresponding left invariant hyper-Hermitian metrics were
studied. It turns out that all such metrics are conformal to hyper-Kahler metrics (cf.
W).
In the present work we study some remarkable properties of a special hyper-
Hermitian metric which corresponds to a four-dimensional solvable Lie group. We
also sketch a procedure for constructing hypercomplex structures on certain nilpo-
tent and solvable Lie groups, following the lines of [3].
Acknowledgement The author would like to thank the organizers of the meeting
Quaternionic Structures in Mathematics and Physics for their kind invitation to take
part in this event.
2.
A SPECIAL HYPER-HERMITIAN METRIC
Consider the four-dimensional real Lie algebra s = span {ej}j=i, ,i with the fol-
lowing Lie bracket:
[
e

3,e
4
]
= -e
2
, [ei,e
2
]=e
2
, [ei,e,-] = - e
j:
j = 3,4
and let g be the inner product with respect to which {ej}j=i, ,i is an orthonormal
basis.
It follows from [2] that g is hyper-Hermitian. Let
{e
7
}^^ ^
C s* be the dual
basis of {ej}j=i, ,4 and write e
y
- to denote e
l
A e
3
A In this case we have the
following equations for w
a
:
Ul

= -e
12
+ e
34
, u
2
= -e
13
- e
24
, o;
3
= e
14
- e
23
.
To calculate duj
a
, a = 1,2,3, we compute first:
de
1
= 0, de
2
= -e
12
- - e
34
, de
j

= e
lj
, j = 3,4
2 2
to obtain:
-\e™\ duj
2
=
3
-e™, du,
3
= l
so that (1.2) is satisfied for 0 —
—
fe
1
. We can therefore conclude that the left
invariant hyper-Hermitian metric induced by g on the corresponding simply connected
solvable Lie group S is conformally hyper-Kahler. We recall from [2] that g is neither
symmetric nor conformally flat. The Levi-Civita connection V
9
is given as follows:
HYPERCOMPLEX STRUCTURES ON LIE GROUPS
3
v
ei
o,
V
9
v

e
2
0 1
-1 0
V
9
=
-i
«J
1 0
v?
4
=
0
1
V-5 0
0 \\
/
Using these formulas we calculate the curvature tensor R
9
: R?{e\,v) = —V? , for
all v in s and
/
R
9
(e
2
,e
3
)

0 i\
4 o
o |\
•Ye 0
/
i?
9
(e
2
,e
4
) =
\0 &
0
-T
6
9\
J
R
9
(e
3
,
ei
) =
V
o
_7_
16
16

0
J
and after some tedious calculations one can verify that the sectional curvature K
satisfies K <
—
\.
It is possible to show that the connected component of the identity I
0
(S,g) of the
isometry group of (S, g) is a semidirect product S
1
x S. To see this, one shows that
every isometry / of S fixing the identity e of 5 satisfies that (df)
e
is an automorphism
of s. It follows from this fact that Io(S,g) has no discrete co-compact subgroups and
therefore, since S is solvable, S itself does not admit such a discrete subgroup.
3.
HYPERCOMPLEX STRUCTURES ON CERTAIN NILPOTENT AND SOLVABLE LlE
GROUPS
An abelian complex structure on a real Lie algebra g is an endomorphism of g
satisfying
(3.1) J
2
= -/, [Jx, Jy] =
[x,
y],
\/x, y
G
0.

The above conditions automatically imply the vanishing of the Nijenhuis tensor. By
an abelian hypercomplex structure we mean a pair of anticommuting abelian complex
structures. Our main motivation for studying abelian hypercomplex structures comes
from the fact that such structures provide examples of homogeneous HKT-geometries
(where HKT stands for hyper-Kahler with torsion, cf. [8]).
4 MARIA LAURA BARBERIS
It was proved in [1] that if dim
[g,
fl]
< 2 then every hypercomplex structure on g
must be abelian. To complete the classification of the Lie algebras g with dim
[g,
g]
< 2
carrying hypercomplex structures (cf. [1]) it remained to give a characterization in
the case when g is 2-step nilpotent and dim[g,
g]
= 2: this is obtained by taking
m
—
2 in Theorem 3.1 below.
It is a result of [7] that the only 8-dimensional non-abelian nilpotent Lie algebras
carrying abelian hypercomplex structures are trivial central extensions of //-type Lie
algebras. We show in [3] that this does not hold for higher dimensions: there exist
2-step nilpotent Lie algebras which are not of type H carrying such structures.
Let (n, ( , }) be a two-step nilpotent Lie algebra endowed with an inner product
( , ) and consider the orthogonal decomposition n =
3
© 0, where 3 is the center of n
and [0, t] C 3. Define a linear map j : 3

—>•
End (0), z
H->
j
z>
where j
z
is determined
as follows:
(3.2) {j
z
v,w) = (z, [v,w]), Vv,w£t>.
Observe that
j
z
,
z € 3, are skew-symmetric so that z
—>
j
z
defines a linear map j :
3
—•
so(0). Note that Ker(j) is the orthogonal complement of [n, n] in
3.
In particular,
[n,
n]
= 3 if and only if j is injective. Conversely, any linear map j : R
m

—»
so (A;)
gives rise to a 2-step nilpotent Lie algebra n by means of (3.2). It follows that the
center of n is R
m
©(n
ze
mmKer j
z
) and [n,
n]
C R
m
where equality holds precisely when
j is injective. We say that (n, ( , )) is irreducible when 0 has no proper subspaces
invariant by all
j
z
,
z £ 3.
It follows that a two-step nilpotent Lie algebra carrying an abelian complex struc-
ture amounts to a linear map j : 3
—•
u(k) (where dim
t>
= 2k and u(k) denotes the Lie
algebra of the unitary group U(k)). As a consequence of this we obtain the following
result, where we denote by sp(k) the Lie algebra of the symplectic group Sp(k):
Theorem 3.1 ([3]). Every injective linear map j
:

R
m
->•
sp(k)
(TO
< k(2k + 1)) gives
rise to a two-step nilpotent Lie algebra n with dim[n,
n]
= TO carrying an abelian
hypercomplex structure. Conversely, any two step nilpotent Lie algebra carrying an
abelian hypercomplex structure arises in this manner.
Using the same idea as in the above theorem it is possible to construct hypercomplex
structures on certain solvable Lie algebras. In fact, given a two step nilpotent Lie
algebra (n, ( , )) set s = Ra ©n with [a,z] = z, Vz G 3, [a,v] = \v, Vu6 0, where
the inner product on 0 is extended to s by decreeing alt) and (a, a) = 1. This
solvable extension of n has been studied by various authors ([6]). In the special case
when dim3 = 3 (mod 4), dime = 4k and the the endomorphisms
j
z
,
z € 3, defined
as in (3.2), belong to Sp(k), it can be shown that s carries a hypercomplex (hyper-
Hermitian) structure. The procedure is analogous to that in the preceding theorem.
It should be noted that these structures cannot be abelian and the corresponding
metrics are not hyper-Kahler (since they are not flat).
HYPERCOMPLEX STRUCTURES ON LIE GROUPS 5
REFERENCES
1.
M. L. Barberis and I. Dotti Miatello, Hypercomplex structures on a class of solvable Lie groups,
Quart. J. Math. Oxford (2), 47 (1996), 389-404.

2.
M. L. Barberis, Hypercomplex structures on four-dimensional Lie groups, Proc. Amer. Math.
Soc.
125 (4) (1997), 1043-1054.
3.
M. L. Barberis, Abelian hypercomplex structures on central extensions of H-type Lie algebras,
to appear in J. Pure Appl. Algebra.
4.
M. L. Barberis, Homogeneous hyper-Hermitian metrics which are conformally hyper-Kdhler,
preprint.
5.
C. P. Boyer, A note on hyper-Hermitian four-manifolds, Proc. Amer. Math. Soc. 102(1) (1988),
157-164.
6. E. Damek and F. Ricci, Harmonic analysis on solvable extensions of H-type groups, J. Geom.
Anal. 2 (1992), 213-248.
7.
I. Dotti Miatello and A. Fino, Abelian hypercomplex ^-dimensional nilmanifolds, to appear in J.
Global Anal. Geom.
8. G. Grantcharov and Y. S. Poon, Geometry of hyper-Kdhler connections with torsion, preprint,
math.DG/9908015.
FAMAF, UNIVERSIDAD NACIONAL DE CORDOBA, CIUDAD UNIVERSITARIA,
5000
- CORDOBA,
ARGENTINA
E-mail address: barberis8mate.uncor.edu
Second Meeting on
Quaternionic Structures
in Mathematics and Physics
Roma, 6-10 September 1999

TWISTOR QUOTIENTS OF HYPERKAHLER MANIFOLDS
ROGER BIELAWSKI
ABSTRACT.
We generalize the hyperkahler quotient construction to the situation
where there is no group action preserving the hyperkahler structure but for each
complex structure there is an action of a complex group preserving the corre-
sponding complex symplectic structure. Many (known and new) hyperkahler man-
ifolds arise as quotients in this setting. For example, all hyperkahler structures
on semisimple coadjoint orbits of a complex semisimple Lie group G arise as such
quotients of T*G. The generalized Legendre transform construction of Lindstrom
and Rocek is also explained in this framework.
INTRODUCTION
The motivation for this work stems from two problems. The first is the follow-
ing question: when is a complex-symplectic quotient of a hyperkahler manifold hy-
perkahler? A good example is the hyperkahler structure on M = T*G, where G is a
complex semisimple Lie group (found by Kronheimer, cf. [3]). The complex symplec-
tic quotients of M by G are precisely coadjoint orbits of G. These carry hyperkahler
structures by the work of Kronheimer [13], Biquard [5] and Kovalev [12].
The second motivating problem is the generalized Legendre transform (GLT) con-
struction of hyperkahler metrics due to Lindstrom and Rocek [14]. Unlike the ordinary
Legendre transform which produces 4n-dimensional hyperkahler metrics with n com-
muting Killing vector fields, the GLT produces metrics without (usually) any Killing
vector fields. The defining feature of these metrics is that their twistor space admits
a holomorphic projection onto a vector bundle of rank n over CP
1
.
It turns out that in both of these problems there is a group-like object involved,
namely a bundle of complex groups over CP
1
which act fiberwise on the twistor

space Z of a hyperkahler manifold. This action is also Hamiltonian for the twisted
symplectic form of Z. Thus, whenever we have such an action, we can form fiberwise
complex-symplectic quotient of Z giving us (in good cases) a new twistor space.
Similarly, in the case of the GLT, the projection onto the vector bundle V should be
regarded as the moment map for an action of a bundle of abelian groups on Z, which
preserve the twisted symplectic form.
Research supported by an EPSRC Advanced Research Fellowship.
7
8
ROGER BIELAWSKI
We call our bundles of groups over CP
1
twistor groups. The simplest definition of
a twistor group is a group in the category of
spaces
over CP
1
with a real structure.
If a twistor group acts on the twistor space Z of a hyperkahler manifold M, we
can interpret (in most cases) the resulting vector fields on Z as objects on M, namely
either as higher rank Killing spinors (cf. [7]) or, in the E - H formalism (cf. [15]) as
sections of E
<g>
S
2i+1
H (i > 1) satisfying equations analogous to the Killing vector
field equation (case i = 1).
The main purpose of this paper is to introduce the concept of twistor groups and
their actions and to give some interesting examples. We also prove results which can
be viewed as new constructions of hyperkahler manifolds.

1. TWISTOR GROUPS AND THEIR ACTIONS
1.1. Twistor groups. Let X be a complex manifold. A space over X is a complex
space Z together with a surjective holomorphic map {projection)
IT
: Z
—>
X. We
shall say that z —> X is smooth if Z is smooth and n is a submersion.
The category of spaces over X is a category with products (fiber product) and
a final object (x
Id
> x)- ^
n an
y category with such properties we can define a
group as an object Q together with morphisms defining group multiplication, inverse,
and the identity. Thus we define:
Definition 1.1. A group over X is a group in the category of spaces over X.
More explicitly, a group over X is a space Q —'—^ X together with fibrewise
holomorphic maps • : Q x
x
Q
—>
Q [multiplication), i : Q i-» Q [group inverse) and
1 : X
—>
Q [identity section) which commute with 7r and satisfy the group axioms. In
particular, for each x € X (7r
_1
(:r),
•

,i\ _^ , l(aO)
ls a
group.
Remark 1.2. Even if one is interested (as we are) primarily in smooth groups over
X, one cannot avoid the singular ones, since a subgroup of a smooth group can be
singular. In particular the stabilizers of smooth group actions can be singular.
We shall be interested mostly in the case when X — CP
1
and the spaces over
CP
1
come equipped with an antiholomorphic involution [real structure) covering the
antipodal map on CP
1
. The category of spaces with a real structure over CP
1
is also
a category with products and a final object. Therefore we can define:
Definition 1.3. A twistor group is a group in the category of smooth spaces with a
real structure over CP
1
.
Remark 1.4. Although the natural setting is the category of complex spaces rather
than of manifolds, all our examples involve only smooth groups. In addition, the
proofs are either simpler or work only for smooth groups.
Let us give a few examples of twistor groups.
TWISTOR QUOTIENTS
9
Example 1.5. Let G be a complex Lie group equipped with an antiholomorphic in-
volutive automorphism a. Then G x P

1
with the involution (a, a), where a is the
antipodal map, is a twistor group which we shall call a trivial twistor group (with
structure group G) and denote by G.
Example 1.6. Many nontrivial examples arise as twistor subgroups of
G.
For example,
if G acts fibrewise on a space Z with a real structure over CP
1
, then the stabilizer
of any real section of Z is a twistor subgroup of G. In particular, we can take the
adjoint action of G on Z = g ® V, where V is a vector bundle over CP
1
equipped
with a real structure.
Example 1.7. Another important twistor group is constructed as follows. Let G be
reductive and let 6 denote the Lie algebra of the maximal compact subgroup of G.
Let p : su(2)
—•
t be a homomorphism of Lie algebras. For each element z = (a, 6, c),
a
2
+ b
2
+ c
2
= 1, of S
2
~ CP
1

, define a subalgebra n
z
of
JJ
as the sum of negative
eigenspaces of ad(ap(ai)+bp(a2)
+
cp(a3)), where
CTI,
CT
2
,
^3 denote the Pauli matrices.
Now define
AT
as a twistor subgroup of G whose fiber at z is the subgroup of G whose
Lie algebra is N
x
. It is straightforward to observe that the real structure of G acts
on Af. We also observe that each fiber of M is the unipotent radical of the parabolic
subgroup of G determined by p.
Example 1.8. A vector bundle over P
1
equipped with a (linear) real structure is an
abelian twistor group. We observe that a line bundle O(k) is a twistor group if and
only if k is even.
The last example can be generalized as follows. Let Q be any twistor group. For
an open subset U of CP
1
denote by Q

v
the group over U obtained as the restriction
of Q to U. Now suppose that we are given a covering {[/,} of CP
1
, invariant under
the antipodal map and a fibrewise automorphism {faj} of Qu^u, for each nonempty
intersection UiDUj. In addition we suppose that the family of
cj)^
is r-equivariant,
where r is the real structure of Q. Then gluing together Q
Vi
via the 0y gives us a
new twistor group, locally isomorphic to Q. We deduce the following:
Proposition 1.9. Let
Q
be a twistor group. Then the isomorphism classes of twistor
groups locally isomorphic to Q are in bijective correspondence with elements of (non-
abelian) sheaf cohomology group H^CP
1
,A), where A(U) is the group of automor-
phisms ofQxj. •
The subscript K denotes r-invariant elements.
In particular, if G is a complex Lie group with an antiholomorphic automorphism,
then we can consider twistor groups which are locally isomorphic to G. We shall call
such twistor groups locally trivial. We have:
Corollary 1.10. The isomorphism classes of
locally
trivial twistor groups with struc-
ture group G are in 1-1 correspondence with elements of H^CP
1

,0(Aut(G))). O