

Acta Polytechnica


https://doi.org/10.14311/AP.2025.65.0500
Acta Polytechnica 65(5):500–514, 2025 © 2025 The Author(s). Licensed under a CC-BY 4.0 licence

Published by the Czech Technical University in Prague

THE SCIENTIFIC LEGACY OF MILOSLAV HAVLÍČEK

Rutwig Campoamor-Stursberg

Universidad Complutense de Madrid, Instituto de Matemática Interdisciplinar, Plaza de Ciencias 3,
E-28040 Madrid, Spain
correspondence: rutwig@ucm.es

Abstract. We review the main research achievements of Miloslav Havĺıček in algebraic methods
in Quantum Theory, his extensive work on realisations of Lie algebras and superalgebras and their
representation theory, quantum groups and differential equations.

Keywords: Realisations of Lie algebras, Lie gradings, quantum groups, differential equations, quantum
mechanics.

1. Introduction. academic and
professional trajectory

It is certainly not an easy task to condense the life
and work of Miloslav Havĺıček (1938–2024) to few
pages, nor is to emphasise his influence at the Faculty
of Nuclear Sciences and Physical Engineering of the
Czech Technical University in Prague and the Faculty
of Mathematics and Physics at Charles University,
where he has, for many decades, trained dozens of
leading specialists, firmly contributing to the solid
reputation of these institutions as centres of excel-
lence in mathematical physics. It is our purpose to
recall some of the main achievements of Miloslav in
algebraic methods in quantum theory, emphasising
various problems that have determined and motivated
important research topics that are still of current in-
terest and application.

Havĺıček’s academic and professional trajectory has
been marked for many years by the difficulties and
mistrust of the government and university officials,
plenty of bitter experiences and privations, as well
as some endearing anecdotes, like the confiscation of
his doctoral dissertation by Soviet custom officers due
to the extremely suspicious name of “Li” in the ti-
tle (with Li being the Russian form of Sophus Lie’s
surname, written and spelled as the Chinese surname
Li), in a time where the confrontation of the People’s
Republic of China and the Soviet Union was at its
highest point. This arbitrary intervention did, how-
ever, not prevent Miloslav from demonstrating his
exceptional scientific ability during the examination.

Despite all obstacles, Miloslav never lost faith that
education should be devoid of any political interven-
tionism and favoritism, and based solely on personal
qualifications. These were the precepts he maintained
while being the Dean of the Faculty, as well as a mem-
ber of the various scientific associations to which he
belonged, such as the International Association of
Mathematical Physics (IAMP) or the Union of Czech
Mathematicians and Physicists (JČMF). Miloslav also
served as Scientific Council Member of the ČVUT for

the terms 1990–1994 and 2000–2006, of the Faculty
of Mathematics and Physics at Charles University
for the terms 1999–2002 and 2002–2005, and of the
Institute of Informatics of the Czech Academy of Sci-
ences (AV ČR) between 2001 and 2005, also acting as
Vice-Chair of the Academic Assessment Board at this
institution between 1993 and 1999. His efforts, devo-
tion, and altruistic dedication were finally recognised
officially in 1998, when Miloslav was awarded with
the Prize of the Minister of Education (1st degree).
Rigour and scientific excellence were also the guiding
principles in the foundation of the Doppler Institute in
1993, where Miloslav’s efforts, among other colleagues,
managed to finally found a research institute that
crystallised the high scientific level of the seminars
they had led for many years.

To offer a glimpse into the captivating and hum-
ble personality of Miloslav, who always avoided the
spotlight and the pursuit of fame, let us recall his
response when asked about his experiences as Dean
of the faculty in the terms 1990–1994 and 2000–2006,
Havĺıček answered, “The faculty actually runs itself”,

As a teacher, he was well-known to be strict and
demanding, but impartial and fair. He was perfectly
aware that, in order to create a really selective, repre-
sentative, and competitive scientific school, rigor was
the first commandment. In his own words, on the
occasion of the 50th anniversary of FJFI, he stated:
“We don’t pretend to be an easy school, but for the
effort, we offer individual attention, quick involvement
in faculty research teams and an emphasis on our
students’ own creative work. These students are then
able to establish themselves at the top workplaces
in the Czech Republic and abroad”. At the same
occasion, and concerning the social relevance of the
rôle that higher education had to play in the devel-
opment of the country, deprived from any political
fanatism, he declared that “You may be surprised,
but applied natural sciences penetrate into virtually
all components of human activities, from theoretical
physics and research into the construction of matter
through medicine, energy, environmental protection,

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vol. 65 no. 5/2025 The scientific legacy of Miloslav Havĺıček

monument research, food science and many others.
We have a good historical basis, excellent scientific
and technological level and capable people, our stu-
dents are among the elite of the Czech intelligentsia –
what other field can say this?”

Despite the fact that Havĺıček was by no means con-
sidered a political activist “by the book”, a fact which
would cause difficulties for him until 1989, but without
bending either his spirit or convictions, he was allowed
to defend the DrSc. Dissertation in the Laboratory of
Theoretical Physics at JINR in Dubna, showing that
his work had truly impressed the authorities there, as
well as in Prague. In this context, it is not surprising,
that, despite his political convictions, Miloslav was
awarded the First Prize for Theoretical Physics at
the JINR in Dubna in 1985, jointly with Jǐŕı Blank,
Jaroslav Dittrich, Pavel Exner and Genrikh Ivanovich
Kolerov, in recognition of their important work on
operator methods in quantum physics, with Miloslav’s
contributions to realisations of Lie algebras being part
of the awarded collection of papers. The Disserta-
tion, dealing with canonical realisations of classical
Lie algebras [1], was brilliantly defended in Dubna in
1979. The opponents of the thesis were three highly
reputed and influential scientists, Vladas Vladovich
Vanagas from the Lithuanian Academy of Sciences,
Alexander Aleksandrovich Kirillov from the Moscow
State University, and Yakov Abramovich Smorodinsky
from JINR, all of them world leading experts in group
theoretical methods in physics.

The wide range of Miloslav’s scientific interests is
well reflected by enumerating his (main) collaborators,
also illustrating that, despite the disparity of specific
subjects, the nucleus was always related to his passion
for group theory. Among the collaborators whose main
affiliation belongs to an establishment in Prague1, we
mention (given in alphabetical order): Martin Ba-
covský, Miroslav Bednář, Jǐŕı Blank, Čestmı́r Burd́ık,
Goce Chadzitaskos, Pavel Exner, Jǐŕı Hořeǰśı, Jan
Kotrbatý, Ondřej Navrátil, Edita Pelantová, Severin
Pošta, Jǐŕı Tolar, Ivan Úlehla, and Jan Votruba.

Besides them, Miloslav actively worked with Jǐŕı
Patera and Pavel Winternitz from the Centre de
Recherches Mathématiques in Montréal (Canada),
Anatoly Ul’yanovich Klimyk from the Institute of The-
oretical Physics in Kiev (Ukraine), Raisa Moiseevna
Asherova, Yuri Fedorovich Smirnov, and Valeriy Niko-
laevich Tolstoy from the Skobeltsyn Institute of Nu-
clear Physics at the Lomonosov Moscow State Uni-
versity, as well as Wolfgang Lassner from the Leipzig
University (at the times of the German Democratic
Republic, Karl-Marx Universität) and Patrick J. Moy-
lan from the University of Pennsylvania in Abington
(USA).

These names by no means exhaust the collaboration
of Havĺıček with other scholars. He was always willing
to help and counsel, and has motivated many people

1Or whose principal affiliation was related to Prague when
the publications appeared.

to pursue their researches and to clear up doubts or
apparently unsurmountable difficulties. Anecdotally,
I remember Miloslav’s astonishing ability to find rare
or untraceable references, like a series of articles in
hard-to-find Soviet journals that I had, after several
desperate attempts in many places, asked him for
during a visit back in 2006, and that he managed to
procure within a few days.

2. The infinite dimensional Lie
algebra A(P, S)

The topic of Miloslav’s CSc. thesis [2], defended in
December 1968, benefitted from a productive collabo-
ration (and lifetime friendship) with Jan Votruba, son
of the eminent physicist and pioneer of modern theoret-
ical physics in Czechoslovakia, Václav Votruba, with
both Votrubas being undoubtedly very influential in
the formation of Miloslav’s scientific interests. These
works already show the deep interest that Miloslav
took in combining the algebraic/geometric formalism
with real physical phenomena and modelization, a sub-
ject that had begun to be considered systematically in
the mid 1960’s (see e.g. [3]), based on the pioneering
work by Giulio Racah in the 1950s [4]. From these
initial articles referring to this subject, that we enu-
merate below, many new ideas and problems emerged,
that would constitute one of the main activity areas
of Miloslav for the rest of his scientific life:
• M. Havĺıček, J. Votruba. On the physical rep-

resentations of an infinite-dimensional lie algebra.
Czechoslovak Journal of Physics B 16(8):631–642,
1966. https://doi.org/10.1007/BF01689564

• M. Havĺıček, J. Votruba. The tensor product of one-
particle representations of an infinite-dimensional
Lie algebra. Czechoslovak Journal of Physics B
17(10):809–821, 1967. https://doi.org/10.1007/
BF01691631

• J. Votruba, M. Havĺıček. On the representation of
infinite-dimensional Lie algebra A(P, S). In High
Energy Physics and Theory of Elementary Particles,
pp. 330–335. International School of Theoretical
Physics, Naukova Dumka, Kyiv, 1966

• J. Votruba, M. Havĺıček. Mass formulas for me-
son nonets. Czechoslovak Journal of Physics B
19(6):721–729, 1969. https://doi.org/10.1007/
BF01697128

• M. Havĺıček. Representations of an algebra of
the Gell-Mann-Dashen type. Communications in
Mathematical Physics 13(1):73–80, 1969. https:
//doi.org/10.1007/BF01645272

• M. Havĺıček. About one class of representations of
the Lie algebra. Communications in Mathematical
Physics 20(2):130–142, 1971. https://doi.org/
10.1007/BF01646532

These papers extended earlier work by Formánek on
the nontrivial coupling A(P, S) of internal and space-

501

https://doi.org/10.1007/BF01689564
https://doi.org/10.1007/BF01691631
https://doi.org/10.1007/BF01691631
https://doi.org/10.1007/BF01697128
https://doi.org/10.1007/BF01697128
https://doi.org/10.1007/BF01645272
https://doi.org/10.1007/BF01645272
https://doi.org/10.1007/BF01646532
https://doi.org/10.1007/BF01646532


Rutwig Campoamor-Stursberg Acta Polytechnica

time symmetries [11, 12], combining the Poincaré
algebra with SU(2) in new representations. Specif-
ically, the Lie algebra A(P, SU(2)) proposed in this
extension has the basis:

[Lµν , Lρλ] = i (gµρLνλ − gνρLµλ

+gµλLρν − gνλLρµ),
[Pµ, Lρλ] = i (gρµPν − gνµPρ),
[Pµ, Pν ] = 0,[

Pµ, T
(1)
ρ1···ρm

]
= − αiT (2)

µρ1···ρm ,[
Pµ, T

(2)
ρ1···ρm

]
= αiT (1)

µρ1···ρm ,[
Pµ, T

(3)
ρ1···ρm

]
= 0,[

Lµν , T
(i)
ρ1···ρm

]
= i

m∑
j=1

(
gνρj T

(i)
ρ1···ρj−1µ···ρm

−gµρj T
(i)
ρ1···ρj−1ν···ρm

)
,[

T
(i)
ρ1···ρm , T

(k)
ρ1···ρm

]
= iϵikℓT

(ℓ)
µ1···µnρ1···ρm ,

where Pµ, Lνρ, µ, ν, ρ = 0 . . . 3; i, k, ℓ = 1, 2, 3.
It should be noted that structures of this type were

later obtained and considered independently in the
context of Kac–Moody algebras, their generalizations
and extensions, as well as other infinite-dimensional
algebraic structures.

The representations of this algebra, specifically the
irreducible Hermitean representations (IHR in short),
also considering the special case of one isomultiplet,
were constructed and studied in detail. In addition,
the mass formulae (depending on the third compo-
nent of the isotopic spin) and the spin-spectrum were
analysed, hence extending considerably an earlier sym-
metry scheme for particles proposed by Votruba and
Havĺıček [13]. In connection with the classification of
one-particle states into S-multiplets (i.e. one-particle
representations), the tensor product of two such rep-
resentations and the corresponding reduction problem
were considered. Here, an interesting phenomenon
emerged: while in the case of irreducible Hermitean
representations with a single value of TiTi, the anal-
ogy of SU(2) and A(P2, SU(2)) was almost complete
(from the perspective of the pure SU(2) classification),
for the tensor products the behaviour showed impor-
tant differences, and a new series of IHR appeared,
that determined an action on Hilbert spaces decom-
posable as direct sums of a (finite) discrete number of
irreducible spaces associated to the Poincaré algebra,
leading to non-linear mass formulae.

In the context of infinite dimensional algebras, and
motivated by a problem of Dashen and Gell-Mann
arising in the application of current algebras to ele-
mentary particle physics, Havĺıček proposed an alter-
native construction, starting from centreless Lie alge-
bras of finite dimension, and studying a certain class
(called F -representations) of this infinite-dimensional
object in a Hilbert space, considering those operators

represented linearly and possessing a common dense
domain in the Hilbert space, subjected to certain lin-
ear constraints. In a more general frame, operator
representations of pairs (g′, g) in Hilbert spaces were
studied in connection with the integrability property
with respect to a distinguished subalgebra, under cer-
tain conditions. A notion of irreducibility for such
representations was introduced, for which a version
of the Schur lemma was obtained; specifically, that
a symmetric operator that commutes with the repre-
sentation and leaves a dense space D in the domain
invariant is necessarily a scalar.

3. Realisations of Lie algebras
During his stay at the JINR in Dubna (1974–1977),
Miloslav met Wolfgang Lassner, a physicist from the
Karl Marx Universität Leipzig (GDR), who had writ-
ten his CSc. thesis on the connected subgroups of the
Poincaré group [14], being simultaneously an expert in
symbolic computation and algorithm programming2.
Sharing common interests, jointly with Pavel Exner,
with whom Miloslav had already studied some aspects
of canonical matrix realisations3, they combined their
expertise for an ambitious problem, still offering new
challenges even today: the realisations of Lie algebras
by creation and annihilation operators.

The problem of realisations of Lie algebras arises
naturally in many physical applications, such as solu-
tions of nuclear models, collective motions, many-body
problems, and dynamical algebras in the context of
particle physics, where the generators are identified
with differential operators with respect to quantum-
mechanical variables qi and pi. Though several results
were available in the mid 1970s, no systematic and
profound analysis of the problem had been yet consid-
ered. The subject of Havĺıček’s DrSc. dissertation was
exactly this problem: a methodic approach to classify
and construct realisations of the classical Lie algebras
in the Weyl algebra and some of its extensions. These
works, carried out between 1974 and 1977 [1], also
considered the problem of computing the eigenvalues
of the corresponding (generalised) Casimir operators
on these realisations, expanding on work by other
authors on the eigenvalue spectra of Casimir opera-
tors on irreducible representations (see e.g. [16] and
references therein)4. This work significantly extended
the known knowledge about the properties of canoni-
cal realisations of important families of classical Lie
algebras, and served as a natural basis for generali-
sations to other types of Lie algebras that have been
widely considered in the literature. The dissertation

2Actually, Lassner presented his Dissertation B on a topic
also deeply related to realisations of Lie algebras [15].

3One of the first publications on the subject was the o(m, n)
case, presented at an international symposium on high energy
physics and elementary particles held in Varna (Bulgaria) in
1974.

4The length of the thesis is 112 pages, with 111 references
that comprised most of the work on realisations published until
1979.

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vol. 65 no. 5/2025 The scientific legacy of Miloslav Havĺıček

comprises the material of the main articles concerning
the Lie algebra case and appeared in these years5:

• M. Havĺıček, P. Exner. On the minimal canonical
realizations of the Lie algebra Oc(n). Annales de
l’Institut Henri Poincaré Section A 23(4):313–333,
1975

• M. Havĺıček, P. Exner. Matrix canonical realizations
of the Lie algebra o(m, n). I. Basic formulae and
classification. Annales de l’Institut Henri Poincaré
Section A 23(4):335–347, 1975

• M. Havliček, W. Lassner. Canonical realizations
of the Lie algebras gl(n,R) and sl(n,R). I. Formu-
lae and classification. Reports on Mathematical
Physics 8(3):391–399, 1975. https://doi.org/10.
1016/0034-4877(75)90081-6

• M. Havliĉek, W. Lassner. Canonical realiza-
tions of the Lie algebras gl(n,R) and sl(n,R).
II. Casimir operators. Reports on Mathematical
Physics 9(2):177–185, 1976. https://doi.org/10.
1016/0034-4877(76)90053-7

• P. Exner, M. Havĺıček, W. Lassner. Canonical
realizations of classical Lie algebras. Czechoslo-
vak Journal of Physics B 26(11):1213–1228, 1976.
https://doi.org/10.1007/BF01589833

• M. Havĺıček, W. Lassner. Canonical realizations of
the Lie algebra sp(2n,R). International Journal of
Theoretical Physics 15(11):867–876, 1976. https:
//doi.org/10.1007/BF01807449

• M. Havĺıček, W. Lassner. On the “near to
minimal” canonical realizations of the Lie alge-
bra Cn. International Journal of Theoretical Physics
15(11):877–884, 1976. https://doi.org/10.1007/
BF01807450

• M. Havĺıček, W. Lassner. Matrix canonical re-
alizations of the Lie algebra u(p, q). Reports on
Mathematical Physics 12(1):1–8, 1977. https://
doi.org/10.1016/0034-4877(77)90040-4

• M. Havĺıček, P. Exner. Matrix canonical realizations
of the Lie algebra o(m, n). II. Casimir operators.
Czechoslovak Journal of Physics B 28(9):949–962,
1978. https://doi.org/10.1007/BF01596007

• P. Exner, M. Havlicek, W. Lassner. Boson repre-
sentations of classical Lie algebras. In International
Conference on Operator Algebras, Ideals and their
Applications in Theoretical Physics, pp. 277–278.
1977

• Č. Burd́ık, M. Havĺıček. Boson realizations of semi-
simple Lie algebras. In P. Winternitz, J. Harnad,
C. S. Lam, J. Patera (eds.), Symmetry in Physics,
vol. 34 of CRM Proceedings & Lecture Notes, pp.
87–98. American Mathematical Society, 2004

5The corresponding preprints at JINR, as far as they could
be found, are: E2-8089, E2-8533, E2-8700, E2-8646, E2-8842,
E2-9160, E2-9617, E2-9161.

The three main aspects of the generalised approach
and classification of realisations can be summarised
as follows:
(a) New recursion formulae determining realisations

with special properties for all types of classical com-
plex Lie algebras and their consecuenques.

(b) Analogous recursion formulae for the following
types of (noncompact) real forms: sl(nR), su(p, q),
o(p, q), sp(2n,R).

(c) Comparison with minimal realisations and reali-
sations of Gel’fand–Kirillov type [28, 29].
The embeddings of the classical series An, Bn, Cn,

and Dn in the Weyl algebra W2N(d) were revisited and
expanded, taking into account the Schurean property,
and the number of realisation parameters determined
for each class of realisations. Of special interest are
the minimal and maximal number values such that
the resulting realisations are nontrivial, with special
emphasis on the Gel’fand–Kirillov conjecture. Within
the range N(d) = 1, · · · , n, the following numbers of
pairs were found for the classical series:

An Bn Cn Dn

N(d) (2n+1−d)d
2 d(2n − d) d(2n − d) d(2n − d − 1)

The problem of realisations in the quotient division
ring of the Weyl algebras was also satisfactorily com-
pleted, and by denoting n0 the minimal number of
canonical pairs leading to nontrivial realisations, and
nk the maximal integer such that all realisations in
W2(n0+nk) are Schurean, the following values were
computed:

An Bn(n > 1) Cn Dn(n > 2)
n0 n 2n − 2 n 2n − 3
nk 0 1 n − 1 1

The nucleus of the work, however, concerns studying
the set of skew-Hermitean and Schurean6 realisations
for the real Lie algebras in (b). To this extent, the
usual approach is enriched by considering further suit-
able embeddings of the Weyl algebra into more general
algebraic structures, that allow for a greater gener-
ality of the operators realizing the Lie algebras, and
avoiding the degeneracy concerning the number of re-
sulting independent Casimir operators. However, the
removal of degeneracy does not necessarily imply the
preservation of the Schurean property, and obtaining
the criteria and explicit construction procedures for
non-degenerate Schurean realisations was posed. In
the first papers with Pavel Exner on this topic, the ex-
istence problem was positively answered for the class
o(n, m) of pseudo-orthogonal Lie algebras by means
of canonical matrix realisations, i.e. by realisations in
the matrix Weyl algebra W2n,M = W2n ⊗ Mat(M,C),
also considering the case of minimal canonical realisa-
tions for the complexification o(n,C). The analogous

6In other words, such that the Casimir operators are realised
by scalar multiples of the identity.

503

https://doi.org/10.1016/0034-4877(75)90081-6
https://doi.org/10.1016/0034-4877(75)90081-6
https://doi.org/10.1016/0034-4877(76)90053-7
https://doi.org/10.1016/0034-4877(76)90053-7
https://doi.org/10.1007/BF01589833
https://doi.org/10.1007/BF01807449
https://doi.org/10.1007/BF01807449
https://doi.org/10.1007/BF01807450
https://doi.org/10.1007/BF01807450
https://doi.org/10.1016/0034-4877(77)90040-4
https://doi.org/10.1016/0034-4877(77)90040-4
https://doi.org/10.1007/BF01596007


Rutwig Campoamor-Stursberg Acta Polytechnica

construction for the pseudo-unitary group u(r, s) was
written in collaboration with Wolfgang Lassner.

Also in collaboration with Lassner, Miloslav con-
sidered the problem for the canonical embedding
gl(n,R) ⊃ sl(n,R), but focusing on embeddings into
a Weyl algebra with an adequate number of canon-
ical pairs. A family of (d + 1)-parametric classes
with d ≤ n − 1 in the Weyl algebra in dependence
of N(d) = (2n−d−1)d

2 pairs was obtained, having the
following salient properties:
(1.) The realisations are skew-Hermitean.
(2.) Casimir operators are realised as scalars

(Schurean).
(3.) Non-equivalence of realisations is determined up

to endomorphisms in the Weyl algebra.
(4.) The obtained realisations in the (d + 1)-class

admit Casimir operators whose eigenvalues can be
expressed polynomially in terms of (d+1) symmetric
functions of the parameters.

The corresponding realisations for the special Lie al-
gebra sl(n,R) were recovered obtained by restriction,
still preserving the above-mentioned structural prop-
erties. In this context, it is worthy to recall how
canonical realisations Fµν of gl(n − 1,R) in W2m with
commutation relations:

[Fµν , Fρσ] = δνρFρσ − δµσFρν

were expanded to an α-parameterised family of re-
alisations of gl(n,R) in W2n−2+2m, defining the new
operators as:

Eµν = qµpν + Fµν + 1
2δµν1,

Enµ = − pµ,

Eµn = qµ

(∑
ν

qνpν + n

2 − iα
)

+
∑

ν

qνFµν ,

Enn = −
∑

ν

qνpν −
(

n − 1
2 − iα

)
1, α ∈ C.

(1)

The real importance of these formulae, which have
implications far beyond of what was explicitly stated
in these papers, was only recognised much later in
the context of quasi-exactly solvable models (see
e.g. [30, 31]).

The symplectic algebra, appearing naturally in the
context of canonical transformations as a subgroup
of the corresponding inhomogeneous group, as well
as in other physical contexts such as dynamical alge-
bras in n-dimensional harmonic oscillators, prior to
their application in nuclear collective motions [32, 33],
was the last class to be considered to complete the
discussion on the classical series. Here, it was ob-
served that certain one-parameter sets of realisations
of sp(2n,R) could be obtained by means of the pre-
viously studied one-parameter set of minimal realisa-
tions for gl(2n,R), leading to an extremely relevant

formula for constructing canonical realisations of arbi-
trary finite-dimensional Lie algebras. Several of these
realisations were later used for studying the metaplec-
tic group; specifically, the quantisation schemes invari-
ant under metaplectic transformations, of relevance in
the context of wave optics with aberration [34]. Other
applications of these results were found in the algo-
rithmic construction of invariants of inhomogeneous
Lie algebras obtained by contraction of symplectic
groups [35].

Some of the generic results concerning the realisa-
tions of the mentioned classes of Lie algebras can be
summarised in table form as follows:

gl(n,R) u(p, q) o(m, n) sp(2n,R)
Alg. W2N(d) W

N(d)
2N(d),M(d) W2N(d),M(d) W2N(d)

d 1, · · · , n − 1 1, · · · , 2q − δpq 1, · · · , n 1, · · · , n

N(d) (2n−d−1)d
2

(2p+2q−d−1)d
2 d(m + n − d − 1) d(2n − d)

M(d) − ξ0 ξ0 −

where W
N(d)
2N(d),M(d) denotes a certain localisation of

the Weyl algebra (i.e. rational functions of canonical
pairs) and the precise value of ξ0 depends on the value
of d as well as those of (p, q) and (m, n).

As commented before, the Equation (1) contains
much more information than suspected, although this
fact is not easily recognisable. Actually, to a certain
extent, it seems natural to extrapolate the given re-
alisations to the combined case of differential and
matrix representations, hence leading naturally to the
notion of mixed realisations or matrix differential op-
erators. This was actually the ansatz considered by
Yuri F. Smirnov and Alexander V. Turbiner in [30],
where, starting from the usual vector field realisation
Êµν = xµ

∂
∂xν

and a representation Mρσ by (unspec-
ified) operators that commute with the realisation,
mixed realisations of gl(n + 1,R) were obtained with
a striking similarity to Equation (1)7:

Eµν = Êµν + Mµν ,

Enµ = ∂

∂xµ
,

Eµn = xµ

(
k −

∑
ν

xν
∂

∂xν

)
−
∑

ν

xνFµν ,

Enn = k −
∑

ν

xν
∂

∂xν
, k ∈ R.

(2)

To this day, it remains unknown to which extent
Miloslav had already considered the possibility of
mixed realisations, as no explicit hint could be found
in either the Dubna preprints or the published versions.
However, it was clear to many people, among them
to Israel Moiseevich Gel’fand, that Miloslav knew
more than he had shared; a suspicion that has been
later confirmed through personal conversations with
him8. Humble as he was, Miloslav never claimed any

7Up to an inessential change of the sign in the Eµn genera-
tors.

8See the introduction of [30] for a first-hand account.

504


vol. 65 no. 5/2025 The scientific legacy of Miloslav Havĺıček

credit for the obtainment of the algebra gl(n+1,R) of
matrix differential operators, though strong hints in
that direction that can be extracted after a detailed
and quite laborious analysis of the papers lead us to
suspect that this construction was to a certain extent
familiar to him.

4. Realisations of Lie
superalgebras

As a natural extension of the work already achieved
for Lie algebras, relevant physical applications led
Miloslav and collaborators to consider the analogous
problem for the case of Lie superalgebras [36], also
combined with aspects of the general representation
theory, where several important structural results
could be obtained. Among these works, from which
some of the most relevant are listed below, the first
were written in collaboration with Lassner and Exner,
where they extended the techniques that they had de-
veloped for the case of Lie algebras, leading to a series
of papers with Jǐŕı Blank (1939–1990) as the main
collaborator:
• J. Blank, M. Havĺıček, M. Bednář, W. Lass-

ner. Canonical representations of the Lie su-
peralgebra osp(1, 4). Czechoslovak Journal of
Physics B 31(11):1286–1301, 1981. https://doi.
org/10.1007/BF01603588

• J. Blank, M. Havĺıček, P. Exner, W. Lassner. Boson-
fermion representations of Lie superalgebras: The
example of osp(1, 2). Journal of Mathematical
Physics 23(3):350–353, 1982. https://doi.org/
10.1063/1.525373

• J. Blank, M. Havĺıček. Irreducible *-representations
of Lie superalgebras B(0, n) with finite-degenerated
vacuum. General considerations. Tech. Rep. E2-85-
112, JINR Dubna, 1985

• J. Blank, M. Havĺıček. Irreducible *-representations
of Lie superalgebras B(0, n) with finite-degenerated
vacuum. Results for B(0, 1). Tech. Rep. E2-85-160,
JINR Dubna, 1985

• Y. F. Smirnov, V. N. Tolstoi, A. A. Sakharuk, et al.
The Dyson type boson realizations for representa-
tions of the semisimple Lie algebras and superal-
gebras. In Group Theoretical Methods in Physics,
vol. 1–3, pp. 67–76. Harwood Academic Publishers,
1982

• J. Blank, M. Havĺıček. Irreducible *-representations
of Lie superalgebras B(0, n) with finite-degenerated
vacuum. Journal of Mathematical Physics
27(12):2823–2831, 1986. https://doi.org/10.
1063/1.527257

• J. Blank, M. Havĺıček. Irreducible *-representations
of the Lie superalgebras B(0, n) with finite-
degenerated vacuum. II. Journal of Mathematical
Physics 29(3):546–559, 1988. https://doi.org/
10.1063/1.528048

• J. Blank, M. Havĺıček. On the tensor
product of supersingleton representations of
osp(1, 2n). Czechoslovak Journal of Physics B
39(11):1192–1207, 1989. https://doi.org/10.
1007/BF01605320

• J. Blank, M. Havĺıček. On the tensor product
of supersingleton representations of Lie superal-
gebras osp(1, 2n). In Selected Topics in Quantum
Field Theory and Mathematical Physics, pp. 190–
196. World Scientific, 1989

Among the important structural results obtained
for B(0; n)-type superalgebras, we emphasise a new
method for constructing infinite-dimensional represen-
tations of superalgebras, as well as their description in
terms of creation-annihilation operators of para-Bose
systems with n degrees of freedom. A lot of attention
was also devoted to the *-representations of Lie super-
algebras of type B(0; n) and real form osp(1; 2n)9, and
the tensor product decomposition of metaplectic repre-
sentation σn. In particular, a precise decomposition of
the *-representation σ̂n of the orthosymplectic super-
algebra osp(1; 2n) related to the Kronecker product
of two supersingleton representations was obtained.
These results allowed a considerable simplification in
the description of generators and relations in the odd
sector.

Concerning the *-representations of osp(1; 2n), it
was observed that the commutation relations of the
odd generators of the algebra coincide with those
derived by Y. Ohnuki and S. Kamefuchi in 1982,
considering parastatistics of order 2, for construct-
ing the reduction of the tensor product of generic
star representations of osp(1; 2n). Calling P

(n)
M the

representation of sp(n,R) obtained by restriction
to the even sector of the orthosymplectic superal-
gebra, several intriguing properties were deduced.
In particular, it was shown that each representa-
tion P

(n)
M generically admits two non-equivalent *-

extensions to the pseudounitary group SU(n, n). For
the special value n = 2, the procedure provided ex-
plicit forms for infinite-dimensional representations
of osp(1; 4), previously considered and classified by
Heidenreich [46].

5. Quantum groups
With the emergence of quantum groups in the late
1980s and the beginning of 1990s, Havĺıček’s attention
was focused on this important topic, where he collab-
orated with several colleagues in Prague and abroad,
mainly with Anatoly U. Klimyk in Kiev, a great spe-
cialist in structure and representation theory of Lie
(super)algebras, as well as Edita Pelantová and Sev-
erin Pošta. These works were primarily devoted to
a concise, systematic and deep structural study of

9Ω(z∗) = Ω(z)∗ adjoint operation adjoint of linear differen-
tial operator.

505

https://doi.org/10.1007/BF01603588
https://doi.org/10.1007/BF01603588
https://doi.org/10.1063/1.525373
https://doi.org/10.1063/1.525373
https://doi.org/10.1063/1.527257
https://doi.org/10.1063/1.527257
https://doi.org/10.1063/1.528048
https://doi.org/10.1063/1.528048
https://doi.org/10.1007/BF01605320
https://doi.org/10.1007/BF01605320


Rutwig Campoamor-Stursberg Acta Polytechnica

Uq(gl(n,C)) and Uq(som), although other topics con-
cerning the general structure of quantum groups were
considered as well:
• Č. Burd́ık, M. Havĺıček, A. Vančura. Irre-

ducible highest weight representations of quantum
groups Uq(gl(n,C)). Communications in Mathe-
matical Physics 148(2):417–423, 1992. https://
doi.org/10.1007/BF02100869

• M. Havĺıček, E. Pelantová, A. Klimyk. Nonstandard
Uq(so3) and Uq(so4): Tensor products of represen-
tations, oscillator realizations and roots of unity.
Czechoslovak Journal of Physics 47(1):13–16, 1997.
https://doi.org/10.1023/A:1021431709238

• M. Havĺıček, E. Pelantová. Santilli-Fairlie algebra
Uq(so3): Tensor products of representations, oscilla-
tor realizations and roots of unity. Hadronic Journal
20(6):603–614, 1997

• M. Havĺıček, S. Pošta, A. U. Klimyk. Repre-
sentations of the cyclically symmetric q-deformed
algebra Uq(so3). Czechoslovak Journal of
Physics 48(11):1347–1353, 1998. https://doi.
org/10.1023/A:1021692803323

• M. Havĺıček, A. U. Klimyk, E. Pelantová. Repre-
sentations of the q-deformed algebra Uq(so4) for q
a root of unity. Methods in Functional Analysis and
Topology 4(3):39–44, 1998

• M. Havĺıček, A. U. Klimyk, S. Pošta. Rep-
resentations of the cyclically symmetric q-
deformed algebra soq(3). Journal of Mathematical
Physics 40(4):2135–2161, 1999. https://doi.org/
10.1063/1.532856

• M. Havĺıček, A. U. Klimyk, S. Pošta. Representa-
tions of the q-deformed algebra Uq(iso2). Journal of
Physics A: Mathematical and General 32(25):4681,
1999. https://doi.org/10.1088/0305-4470/32/
25/310

• M. Havĺıček, S. Pošta, A. U. Klimyk. Representa-
tions of the q-deformed algebra soq(2, 1). In Sym-
metry in Nonlinear Mathematical Physics, vol. 30
of Proceedings of Institute of Mathematics of NAS
of Ukraine, pp. 280–287. 2000

• M. Havĺıček, A. U. Klimyk, S. Pošta. Central
elements of the algebras U ′

q(som) and Uq(isom).
Czechoslovak Journal of Physics 50(1):79–84, 2000.
https://doi.org/10.1023/A:1022825031633

• M. Havĺıček, A. U. Klimyk, S. Pošta. Clas-
sification of representations of the alge-
bra Uq′(so3) through examples. Czechoslovak
Journal of Physics 50(11):1235–1238, 2000.
https://doi.org/10.1023/A:1022804806462

• M. Havĺıček, S. Pošta, A. U. Klimyk. Some
basic properties of nonstandard deformations
U ′

q(so3), U ′
q(so4). In S. T. Ali, H.-D. Doebner,

M. Keyl, R. Werner (eds.), Trends in Quantum
Mechanics: Proceedings of the International Sympo-
sium Goslar, Germany, pp. 10–17. 1999

• M. Havĺıček, S. Pošta. On the classification of
irreducible finite-dimensional representations of
U ′

q(so3) algebra. Journal of Mathematical Physics
42(1):472–500, 2001. https://doi.org/10.1063/
1.1328078

• M. Havĺıček, A. U. Klimyk, S. Pošta. Representa-
tions of the q-deformed algebra U ′

q(so4). Journal
of Mathematical Physics 42(11):5389–5416, 2001.
https://doi.org/10.1063/1.1402631

• M. Havĺıček, S. Pošta, A. U. Klimyk. Classifi-
cation of representations of the algebra Uq′(so3)
through examples II. Physics of Atomic Nu-
clei 64(12):2151–2155, 2001. https://doi.org/10.
1134/1.1432917

• Č. Burd́ık, M. Havĺıček, O. Navrátil, S. Pošta. Ideals
of the enveloping algebra U(osp(1, 2)). Journal of
Generalized Lie Theory and Applications 2(3):132–
136, 2008

• R. M. Asherova, Č. Burd́ık, M. Havĺıček, et al.
q-analog of Gel’fand-Graev basis for the noncom-
pact quantum algebra Uq(u(n, 1)). Symmetry,
Integrability and Geometry: Methods and Appli-
cations 6:10, 2010. https://doi.org/10.3842/
SIGMA.2010.010

• M. Havĺıček, S. Pošta. Central elements of
quantum deformations. AIP Conference Proceed-
ings 1307(1):125–130, 2010. https://doi.org/10.
1063/1.3527408

• M. Havĺıček, S. Pošta. Center of quantum al-
gebra U ′

q(so3). Journal of Mathematical Physics
52(4):043521, 2011. https://doi.org/10.1063/1.
3579992

These articles, of great technical complexity, ex-
panded upon and generalised various results, and
clarified many questions that have, until then, been
only partially studied or understood. An exhaustive
study of the standard Uq(so(n)) and nonstandard
U ′

q(so(n)) quantum algebras for n = 2, 3, 4 and their
representations began. Being rather difficult to con-
dense the depth of these works to a few lines, we
merely enumerate some of the salient structural re-
sults. One remarkable achievement was the construc-
tion of tensor products in the Santilli-Fairlie algebra
Uq(so(3)), which does not possess a Hopf algebra struc-
ture, a property that is central for the description of
tensor products in the context of quantum groups.
Nevertheless, a procedure for constructing a tensor
product was given, together with an explicit method
for the product of two irreducible representations of
Uq(so(3)) for positive q. Moreover, for the case, when
q is a root of unity, irreducible representations were
constructed. In order to cover the remaining cases,
an algebra homomorphism Ψ from the nonstandard
algebra Uq(so(3)) to the extension Ûq(sl(2)) of the
Hopf algebra Uq(sl(2)) was introduced. The composi-
tion of Ψ with irreducible representations of Ûq(sl(2))
led to (not necessarily irreducible) representations of

506

https://doi.org/10.1007/BF02100869
https://doi.org/10.1007/BF02100869
https://doi.org/10.1023/A:1021431709238
https://doi.org/10.1023/A:1021692803323
https://doi.org/10.1023/A:1021692803323
https://doi.org/10.1063/1.532856
https://doi.org/10.1063/1.532856
https://doi.org/10.1088/0305-4470/32/25/310
https://doi.org/10.1088/0305-4470/32/25/310
https://doi.org/10.1023/A:1022825031633
https://doi.org/10.1023/A:1022804806462
https://doi.org/10.1063/1.1328078
https://doi.org/10.1063/1.1328078
https://doi.org/10.1063/1.1402631
https://doi.org/10.1134/1.1432917
https://doi.org/10.1134/1.1432917
https://doi.org/10.3842/SIGMA.2010.010
https://doi.org/10.3842/SIGMA.2010.010
https://doi.org/10.1063/1.3527408
https://doi.org/10.1063/1.3527408
https://doi.org/10.1063/1.3579992
https://doi.org/10.1063/1.3579992


vol. 65 no. 5/2025 The scientific legacy of Miloslav Havĺıček

Uq(so(3)). Analysing in detail the decomposition into
irreducible components, all IR of Uq(so(3)) when q is
not a root of unity could be determined.

Another difficult problem addressed in these arti-
cles was the analysis of the centre of the nonstan-
dard quantum algebra U ′

q(so(3)) (earlier proposed by
Gavrilik and Klimyk in [65] in order to construct
representations by operators acting according to the
Gel’fand-Tsetlin pattern). For the case of roots of
unity, it was shown that there are four generators of
the center, which depend on the order of q and whose
polynomial dependence was explicitly indicated, while
in the non-root of unity case, the centre was shown
to be a polynomial algebra in one variable. These
results were relevant for the general case of the central
elements of U ′

q(som) and U ′
q(isom), allowing also to

find relations with the case of U ′
q(so3). In this context,

Miloslav and S. Pošta showed how the so-called Dia-
mond lemma, a quite unknown result in ring theory,
could be successfully applied to the Poincaré–Birkhoff–
Witt property in quantum algebras, allowing to derive,
in simple and elegant way, certain facts concerning
their centre.

Besides these exhaustive results, a quantum ana-
logue of Gel’fand–Graev bases for the noncompact
quantum algebra Uq(u(n, 1)) was also obtained, and
the Hermitean irreducible representations correspond-
ing to a discrete series were computed. These re-
sults were shown to be crucial for the computation
of pairings on Mickelsson algebras, and hence for the
computation of inverse Shapovalov forms [66].

6. Lie gradings
The notion of gradation in Lie algebras is an old one,
and besides the well known Cartan decomposition,
other gradings have already been used in the liter-
ature (e.g. in the superalgebra context), although
the problem had not been considered generically. In
1989, Hans Zassenhaus and Jǐŕı Patera began such
a systematic classification of gradings [67], introduc-
ing the notion of fine grading and giving a criterion
for this property. The classification problem, however,
remained mainly untouched for some years due to the
death of Zassenhaus in 1991, and it was not until the
mid 1990s, when Edita Pelantová and Miloslav joined
Patera to complete the task, with Jǐŕı Tolar joining
the group later. These works not only finished the
program established by Patera and Zassenhaus, but
also led to new techniques that have found wide appli-
cations, both in the Lie theory and physical problems.
We enumerate Miloslav’s principal articles concerning
this subject:
• M. Havĺıček, J. Patera, E. Pelantová. On the max-

imal Abelian subgroups of the diagonalizable au-
tomorphisms of simple classical Lie algebras. In
H. D. Doebner, P. Nattermann, W. Scherer (eds.),
Group Theoretical Methods in Physics, Proceedings
of XXI International Colloquium on Group The-

oretical Methods in Physics, pp. 116–120. World
Scientific, Singapore, 1997

• M. Havĺıček, J. Patera, E. Pelantová. On
the fine gradings of simple classical Lie alge-
bras. International Journal of Modern Physics A
12(1):189–194, 1997. https://doi.org/10.1142/
S0217751X97000268

• M. Havĺıcek, J. Patera, E. Pelantova. On Lie
gradings II. Linear Algebra and its Applica-
tions 277(1–3):97–125, 1998. https://doi.org/10.
1016/S0024-3795(97)10039-8

• M. Havĺıček, J. Patera, E. Pelantová. On Lie grad-
ings III. Gradings of the real forms of classical Lie
algebras. Linear Algebra and its Applications 314(1–
3):1–47, 2000. https://doi.org/10.1016/S0024-
3795(00)00099-9

• J. Patera, M. Havĺıček, E. Pelantová, J. Tolar. On
fine gradings and their symmetries. Czechoslovak
Journal of Physics 51(4):383–391, 2001. https://
doi.org/10.1023/A:1017501925328

• M. Havĺıček, J. Patera, E. Pelantová, J. Tolar. Au-
tomorphisms of the fine grading of sl(n,C) asso-
ciated with the generalized Pauli matrices. Jour-
nal of Mathematical Physics 43(2):1083–1094, 2002.
https://doi.org/10.1063/1.1430046

• M. Havĺıček, E. Pelantová, J. Patera, J. Tolar.
Distinguished bases of sl(n,C) and their symme-
tries. In Quantum Theory and Symmetries, pp. 366–
370. World Scientific, 2002. https://doi.org/10.
1142/9789812777850_0043

• M. Havĺıček, J. Patera, E. Pelantová, J. Tolar. On
Pauli graded contractions of sl(3,C). Journal of
Nonlinear Mathematical Physics 11(1):37–42, 2004.
https://doi.org/10.2991/jnmp.2004.11.s1.4

• M. Havĺıček, E. Pelantová, J. Tolar. On represen-
tations of sl(n,C) compatible with a Z2-grading.
Acta Polytechnica 50(5):30–39, 2010. https://doi.
org/10.14311/1261

These works in particular use the classification of
the (maximal) commutative subgroups in the auto-
morphism group of gl(n,C), with special emphasis on
the inner and outer automorphisms, as fine gradings
can be classified, up to equivalence, by the maximal
Abelian groups of diagonalisable automorphisms of
the Lie algebra (called MAD-groups)10. As a first
decisive advance towards the classification, a detailed
procedure to determine fine gradings for the class of
classical Lie algebras was given, and an equivalence
criterion for fine gradings based on labeled graphs was
proposed. It was soon recognised that fine gradings
constitute a powerful tool for the structural study
of solvable (respectively nilpotent) Lie algebras [78],
and that the contraction and deformation theories of

10From a different point of view, the problem of Abelian sub-
groups had also been considered by Suprunenko and Tyshkevich
in 1966 [77].

507

https://doi.org/10.1142/S0217751X97000268
https://doi.org/10.1142/S0217751X97000268
https://doi.org/10.1016/S0024-3795(97)10039-8
https://doi.org/10.1016/S0024-3795(97)10039-8
https://doi.org/10.1016/S0024-3795(00)00099-9
https://doi.org/10.1016/S0024-3795(00)00099-9
https://doi.org/10.1023/A:1017501925328
https://doi.org/10.1023/A:1017501925328
https://doi.org/10.1063/1.1430046
https://doi.org/10.1142/9789812777850_0043
https://doi.org/10.1142/9789812777850_0043
https://doi.org/10.2991/jnmp.2004.11.s1.4
https://doi.org/10.14311/1261
https://doi.org/10.14311/1261


Rutwig Campoamor-Stursberg Acta Polytechnica

Lie algebras could be approached alternatively with
these techniques, leading to new and interesting ap-
plications [79, 80]. In this context, the gl(n,C)-fine
gradings were analysed in detail in connection with
sl(n,C)-contractions. In particular, for the fine grad-
ings of the latter algebra, it was observed that the
group SL(2,Zn) plays an analogue rôle to the Weyl
group for the canonical Cartan grading.

Once the classical case systematised, the next diffi-
cult step consisted in extending these results to the
real forms of the classical Lie algebras. To this extent,
the maximal abelian diagonalisable groups of auto-
morphisms for the real forms were studied, and four
types of matrix subgroups of GL(n,R) defined. This
allowed them to transfer the classification problem of
non-conjugate maximal abelian diagonalisable groups
for real forms to the description of equivalence classes
within the four types of matrix subgroups, hence en-
abling them to establish a precise classification.

Several other questions related to fine gradings were
considered, such as the symmetries associated with
the gradings and the class of fine gradings related to
the n-dimensional Pauli matrices, which had found
several intriguing applications. In this context, par-
ticular attention was given to the Pauli grading of
sl(3,C), where it was observed that the symmetry
transformations of a given grading are in a one-to-one
correspondence with a certain quotient group related
to automorphisms that permute the grading blocks.
For the case of sl(3,C), the Jacobi identity associated
with a graded contraction therefore led to a system
of quadratic equations for the contraction parameters,
the symmetries of which were related to Z2×SL(2,Z3).
These nonlinear equations were also analysed for their
solutions. It should be mentioned that the compati-
bility of representations with Z2-gradings, which have
many potential applications to the case of Lie super-
algebras, was also studied in detail for several Lie
algebras.

7. Differential equations
Miloslav’s work on differential equations can be di-
vided into two separate subjects: on the one hand, the
computation of spectra and criteria for determining
the eigenvalues of the Schrödinger equation, and, on
the other hand, the problem of existence and obtain-
ing nonlinear superposition principles for (systems)
of differential equations. The works on spectra were
done in collaboration with Ivan Úlehla and Jǐŕı Hořeǰśı,
reputed specialists in quantum theory, while those on
superposition rules were written with Severin Pošta
and Pavel Winternitz, expanding previous work on
the topic [81, 82]:
• I. Úlehla, M. Havĺıček. New method for computation

of discrete spectrum of radical Schrödinger operator.
Aplikace matematiky 25(5):358–372, 1980

• I. Úlehla, M. Havĺıček, J. Hořeǰsi. Eigenvalues of the
Schrödinger operator via the Prüfer transformation.

Physics Letters A 82(2):64–66, 1981. https://doi.
org/10.1016/0375-9601(81)90938-5

• M. Havĺıček, S. Pošta, P. Winternitz. Nonlinear
superposition formulas based on imprimitive group
action. Journal of Mathematical Physics 40(6):3104–
3122, 1999. https://doi.org/10.1063/1.532749

• M. Havĺıček, S. Pošta, P. Winternitz. Superposition
formulas based on nonprimitive group action. In
A. Coley, D. Levi, R. Milson, et al. (eds.), Bäcklund
and Darboux Transformations. The Geometry of
Solitons, vol. 29 of CRM Proceedings & Lecture
Notes, pp. 225–231. American Mathematical Society,
2001
In the first of the articles with Úlehla, a new method

for computing the discrete spectrum of a one-particle
Schrödinger operator under spherical symmetry and
appropriate potentials such that the spectrum is finite
was proposed, by means of a transformation of the
radial Schrödinger equation into a first-order ODE for
the function z(x, χ):

dz

dx
= (ℓ + 1) cos2 z − 1

ℓ + 1(v(x, χ) + χ2) sin2 z,

with the number of eigenvalues being determined by
the condition z(∞, 0) = (2k + 1) π

2 , such that the dis-
continuity points coincide with the eigenvalues (hence
constituting a kind of analogue of the Levinson theo-
rem for this function). The procedure contemplated
the replacement of the usual Ritz variational method
by direct integration techniques. These discontinuities
were obtained by successive integration, with accuracy
increased for truncations at suitable values depending
on the potential.

In a following paper, jointly with Jǐŕı Hořeǰśı, the
eigenvalue problem for central potential was consid-
ered from the perspective of a modified Prüfer trans-
formation, giving two theorems that established a the-
oretical framework for numerical eigenvalue computa-
tion for the case of rapidly decreasing and confining
central potential, considered unified for the first time.
This approach was later extended in [87] to long-range
potentials, including the Coulomb case.

The papers on superposition, which is very deeply
connected with the classification of realisations of Lie
algebras by vector fields, continue the extensive work
of Pavel Winternitz on the subject, focusing on im-
primitive group actions on manifolds, specifically for
the special linear groups SL(n,C), making use of the
realisations obtained before by Havĺıček and Lassner.
It was shown that such decomposable systems can
be studied with indecomposable systems serving as
building blocks. The cases for n = 3, 4 were computed
explicitly, including the superposition rules for the
corresponding ODE systems, providing also a descrip-
tion for the general case of SL(n,C), including some
relations to soliton equations and Bäcklund transfor-
mations. More specifically, when compared to the
well-known situation with SL(2,R) and SL(2,C), it

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was concluded that the equations derived from the
imprimitive action of SL(n,C) should appear as Bäck-
lund transformations for integrable systems defined
on flag manifolds.

8. Representation theory
Havĺıček’s extensive work on the realisation problem
of Lie algebras and superalgebras also had many im-
portant implications for their representation theory,
which led to a first series of papers on highest weight
representations, written in collaboration with Pavel
Exner and Čestmı́r Burd́ık, and published at the be-
ginning 1980s, as well as some applications to the
Poincaré group, in collaboration with Jan Kotrbatý,
Severin Pošta and Patrick Moylan:
• Č. Burd́ık, M. Havĺıček, P. Exner. Highest-weight

representations of the sl(n + 1, C) algebras: Maxi-
mal representations. Journal of Physics A: Math-
ematical and General 14(5):1039, 1981. https:
//doi.org/10.1088/0305-4470/14/5/023

• Č. Burd́ık, P. Exner, M. Havĺıček. Highest-weight
representations of sl(2, C) and sl(3, C) via canonical
realizations. Czechoslovak Journal of Physics B
31(5):459–469, 1981. https://doi.org/10.1007/
BF01596411

• Č. Burd́ık, P. Exner, M. Havĺıček. A com-
plete set of irreducible highest-weight represen-
tations for sl(3, C). Czechoslovak Journal of
Physics B 31(11):1201–1206, 1981. https://doi.
org/10.1007/BF01603579

• M. Havĺıček, P. Moylan. An embedding of the
Poincaré Lie algebra into an extension of the
Lie field of SO0(1, 4). Journal of Mathemati-
cal Physics 34(11):5320–5332, 1993. https://doi.
org/10.1063/1.530307

• S. Pošta, M. Havĺıček. Note on Verma bases for
representations of simple Lie algebras. Acta Poly-
technica 53(5):450–456, 2013. https://doi.org/
10.14311/AP.2013.53.0450

• M. Havĺıček, J. Kotrbatý, P. Moylan, S. Pošta. Con-
struction of representations of Poincaré group us-
ing Lie fields. Journal of Mathematical Physics
59(2):021702, 2018. https://doi.org/10.1063/1.
4993153

• M. Havĺıček, J. Kotrbatý, P. Moylan, S. Pošta.
(Heisenberg-)Weyl algebras, Segal-Bargmann trans-
form and representations of Poincaré groups. Jour-
nal of Physics: Conference Series 1194(1):012043,
2019. https://doi.org/10.1088/1742-
6596/1194/1/012043

Using previous results concerning the canonical bo-
son representations of sl(n + 1,C), for every given
weight m, so-called maximal representations Dm were
constructed, which contained an irreducible represen-
tation with m as the highest weight. Conditions for
the irreducibility of the maximal representations were

analysed, and a comparison with the standard con-
struction of representations performed. An advantage
of the new procedure was the explicitness of the con-
struction, which proved valuable for applications. In
this context, the cases of sl(2,C) and sl(3,C) were
analysed in further detail, and irreducibility crite-
ria for some types of infinite-dimensional representa-
tions were obtained by means of the canonical boson
realisations. This allowed to derive a complete de-
scription of the irreducible highest-weight representa-
tions of sl(3,C), which were shown to be partitioned
into five disjoint classes, one corresponding to finite-
dimensional representations, and the four remaining
to infinite-dimensional representations:

ΩF = {Λ : Λi ∈ N, i = 1, 2},

Ω1 = {Λ : Λ1 ∈ N, Λ2 /∈ N},

Ω2 = {Λ : Λ1 /∈ N, Λ2 ∈ N},

Ω12 = {Λ : Λi /∈ N, i = 1, 2; 1 + Λ1 + Λ2 ∈ N},

Ωmax = {Λ : Λi /∈ N, i = 1, 2; 1 + Λ1 + Λ2 /∈ N},

where ΩF is well known from the work of
I. M. Gel’fand and M. L. Tsetlin, and Ω1, Ω2, and
Ωmax have already been obtained in previous work.
The remaining class Ω12 was obtained, completing the
classification of infinite-dimensional highest weight
representations. Beyond the explicit results, these pa-
pers contain precise indications for higher ranks, also
indicating the great complexity in deriving explicit
expressions for the corresponding classes.

Another remarkable contribution to representation
theory was made in collaboration with Severin Pošta.
This work provided an alternative construction of
the Verma basis of the enveloping algebra and finite-
dimensional representations of the An Lie algebras, as
well as providing a more compact proof of the so-called
Verma inequalities [95].

In the context of specific applications of the
Poincaré group, Miloslav wrote three papers with
Patrick Moylan that extended previous (and appar-
ently unpublished [96]) work. The first one, from
1993, showed that the principal series of unitary
ray representations of SO0(1, 4) is related, via the
*-isomorphism between algebraic extensions of the Lie
fields of SO0(1, 4) and the Poincaré group, to real
mass, and positive and negative energy representa-
tions of the Poincaré Lie algebra with an arbitrary
spin. This approach is deeply related to the embedding
problem of Lie algebras into their universal enveloping
algebra, leading to representations by higher order
differential operators, with important implications
concerning quasi-exactly solvable systems in quantum
mechanics [97].

In the second one (jointly with Jan Kotrbatý and
Severin Pošta), which is one of the last papers of
Havĺıček, the representations of the Poincaré group in
n = 2, 3, 4 dimensions were realised on Hilbert spaces
associated with the space of the Schrödinger represen-
tation of Weyl algebras. The localisation of universal

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https://doi.org/10.1007/BF01596411
https://doi.org/10.1007/BF01603579
https://doi.org/10.1007/BF01603579
https://doi.org/10.1063/1.530307
https://doi.org/10.1063/1.530307
https://doi.org/10.14311/AP.2013.53.0450
https://doi.org/10.14311/AP.2013.53.0450
https://doi.org/10.1063/1.4993153
https://doi.org/10.1063/1.4993153
https://doi.org/10.1088/1742-6596/1194/1/012043
https://doi.org/10.1088/1742-6596/1194/1/012043


Rutwig Campoamor-Stursberg Acta Polytechnica

enveloping algebras is used as the main technique, de-
termining first the algebraic relations between genera-
tors of the Poincaré algebra and Heisenberg pairs, in
analogy with the Gel’fand–Kirillov conjecture. From
these, the realisation of the representations is deduced.
It is shown that the action of the basis elements
can be integrated to strongly continuous unitary one-
parameter subgroups on the Hilbert space, from which
a further integration procedure leads to a representa-
tion of the corresponding Poincaré group. The equiva-
lence of these unitary irreducible representations with
those arising from the standard approach using the
Wigner-Mackey theory was also proved.

9. A fundamental reference for
quantum mechanics

In 1993, the textbook “Linear Operators in Quantum
Physics” by J. Blank, P. Exner and Miloslav finally
appeared, after an adventurous and long genesis [98].
The idea of preparing a book on the fundamental
aspects of Quantum Mechanics can be traced back
to 1973, when the authors, based on their lectures
on the topic at the Charles University and the Czech
Technical University, prepared a series of lecture notes,
that where revised, compiled and given a textbook
form in the late 1980s. The publication, initially
scheduled for 1989, was delayed by several years, due
to the political and economic changes that shook the
country. Unfortunately, prior to its publication, Jǐŕı
Blank died in 1990, and could not see the great success
that the text would have, still being one of the central
references in courses on Quantum Mechanics even
today.

The book explained in detail the theory of linear
operators on Hilbert spaces and their application in
quantum theory. With an initial survey on fundamen-
tals of linear algebra, topology and functional analysis,
as well as measure theory, the main facts on Hilbert
spaces and (bounded and unbounded) operators were
presented. Physical applications of quantum theory
with special emphasis on non-relativistic quantum me-
chanics, the second quantisation, and the scattering
theory were developed in detail, enriched with many
supplementary material and numerous exercises.

The relevance of this text was, however, not appre-
ciated abroad until its first English edition appeared
in 1994 at the American Institute of Physics [99]. Al-
though extensive modifications were required due to
technical and editorial reasons, the spirit of the first
edition remained untouched, and both versions, al-
beit their differences, can be seen as uniform. The
reception was very favourable, as can be seen in the
different reviews of the book at Zentralblatt der Math-
ematik and Mathematical Reviews, among others11,
and the authors were awarded the Rector’s Prize in
1995 for their important contribution. Some years

11See e.g. M. Gorzelańczyk Zbl 0873.46038 or H. Baumgärtel
MR1275370.

later, a second edition with the addition of new topics
was published in 2008, also being welcomed by the
physical community, receiving hundreds of citations
in other textbooks and research articles.

10. Conclusion
These brief summaries of extensive subjects do not
exhaust Havĺıček’s activity, as he also wrote other
papers on quite a number of specific questions, such
as the integrability properties of representations of
semisimple Lie algebras in Hilbert spaces [100], the de-
scription of unstable systems [101], or about the notion
of quantum-mechanical pseudo-Hamiltonians [102], to
which a large number of contributions to national and
international conferences must be added.

To sum it up, Miloslav’s work has covered almost
any of the physical problems that uses group the-
ory and algebraic formalism as a fundamental tool,
providing new insights and motivating new notions
and techniques that have since been proved to be very
fruitful. Certainly, there are still a lot of new problems
that arise from his work, as the completion of the clas-
sification of canonical realisations for the exceptional
complex and compact Lie algebras of rank d > 2 (with
the case of G2 having been partially classified in [103]),
which is certainly a difficult question, specially for the
case of E8, due to the intricate structural particu-
larities of these algebras. Another relevant problem
concerns the mixed realisations, that so far have only
been obtained for the (special) general linear algebra
sl(n + 1,R) and gl(n + 1,R), but that are likely to
be extended to other algebras of the classical series
by subtle adaptation of Miloslav’s methods, and even
considered for the class of inhomogeneous and other
distinguished subalgebras by means of the contraction
formalism, with many conceivable applications [104].
There still remains a lot of material to be properly
understood in the series of papers on realisations of
Lie algebras, that could potentially be also generalised
to the superalgebra case and even beyond.

There is no doubt that Havĺıček’s work has deeply
influenced the younger generations of physicists in the
Czech Republic and abroad, as well as many specialists
working on group theory from the physical perspective,
such as the writer of these lines, inspiring further
investigations in several of the problems where he
proposed an elegant and effective ansatz.

We would like to finish this short reminiscence with
a sentence that, to the author’s understanding, per-
fectly summarises the influence of Havĺıček’s work and
personality in Prague and abroad: “Miloslav̊uv odkaz
žije dál”.

Acknowledgements
Miloslav Znojil, Pavel Exner, Edita Pelantová and Libor
Šnobl are gratefully acknowledged for valuable comments
that have improved the article. The author is specially
indebted to Jindra Niederlová and Alexander V. Turbiner
for providing some personal information about Miloslav

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that has helped to clarify some points in the exposition,
as well as for motivating a written version of the talk.
The author also acknowledges the Dean’s Office of the
Faculty of Mathematics and Physics, Charles University,
the Institute of Informatics of the Academy of Sciences
of the Czech Republic and the Archives of the Czech
Technical University for providing updated information
on Havĺıčeks professional trajectory.

During the preparation of this work, the author received
partial support from the Agencia Estatal de Investigación
(Spain) under the grant PID2023-148373NB-I00 funded
by MCIN/AEI/10.13039/501100011033/FEDER, UE.

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	Acta Polytechnica 65(5):500–514, 2025
	1 Introduction. academic and professional trajectory
	2 The infinite dimensional Lie algebra A(P,S)
	3 Realisations of Lie algebras
	4 Realisations of Lie superalgebras
	5 Quantum groups
	6 Lie gradings
	7 Differential equations
	8 Representation theory
	9 A fundamental reference for quantum mechanics
	10 Conclusion
	Acknowledgements
	References