
Department of Mathematics, University of Munich
 B. Pareigis
List of all Papers and Scripts by B. Pareigis (MathSciNet References) available on this server
1.) Lecture Notes and Manuscripts
Sind Wahlen undemokratisch?

SS 2006 



Grundregeln der Perspektive

WS 2007 



Kleine Einführung in die Kodierungstheorie

SS 1998 



Ebene Kristallographische Gruppen

SS 2001 



Lineare Algebra für Informatiker

WS 1991/92 
. 
. 
Teil I: Grundlagen, diskrete Mathematik

. 
. 
. 
Einleitung und Inhalt

. 



. 
Kapitel 1:

Grundbegriffe der Mengenlehre

. 



. 
Kapitel 2:

Natürliche Zahlen

. 



. 
Kapitel 3:

Algebraische Grundstrukturen

. 



. 
Kapitel 4:

Kombinatorik und Graphen

. 



. 
Teil II: Lineare Algebra (steht wegen Veröffentlichung in Buchform nicht zur Verfügung. Lineare Algebra für Informatiker. Springer Lehrbuch  Springer Verlag Berlin Heidelberg New York 2000. )

. 
Advanced Algebra

WS 2001/02 



Algebra II (Ring und Modultheorie)

SS 1997 



Algebra II (Körpertheorie)

SS 2003 



Category Theory

SS 2004 



QuantenGruppen und nichtkommutative Geometrie

SS 1993 



Quantum Groups and Noncommutative Geometry

SS 2002 



Quantum Groups and Noncommutative Geometry

WS 1999 
. 
Chapter 1:

Commutative and Noncommutative Algebraic Geometry

092099 



. 

The Principles of Commutative Algebraic Geometry 
091799 



. 

Quantum Spaces and Noncommutative Geometry 
092099 



. 

Quantum Monoids and their Actions on Quantum Spaces 
092099 



. 
Chapter 2: 
Hopf Algebras, Algebraic, Formal, and Quantum Groups 
110399 



. 

Hopf Algebras 
092099 



. 

Monoids and Groups in a Category 
092999 



. 

Affine Algebraic Groups 
110399 



. 

Formal Groups 
110399 



. 

Quantum Groups 
110399 



. 

Quantum Automorphism Groups 
110399 



. 

Duality of Hopf Algebras 
110399 



. 
Chapter 3: 
Representation Theory, Reconstruction and Tannaka
Duality 
110399 



. 

Representations of Hopf Algebras 
110399 



. 

Monoidal Categories 
110399 



. 

Dual Objects 
110399 



. 

Finite reconstruction 
110399 



. 

The coalgebra coend 
110399 



. 

The bialgebra coend 
110399 



. 

The quantum monoid of a quantum space 
110399 



. 

Reconstruction and Ccategories 
110399 



. 
Chapter 4: 
The Infinitesimal Theory 
12299 



. 

Integrals and Fourier Transforms 
12299 



. 

Derivations 
12299 



. 

The Lie Algebra of Primitive Elements 
12299 



. 

Derivations and Lie Algebras of Affine Algebraic Groups 
12299 



. 
Apendix: 
Toolbox 
091799 



. 

Categories 
091799 



. 

Functors 
091799 



. 

Natural Transformations 
091799 



. 

Tensor Products 
091799 



. 

Algebras 
091799 



. 

Coalgebras 
091799 



. 

Bialgebras 
091799 



. 

Representable Functors 
091799 



. 

Adjoint Functors and the Yoneda Lemma 
091799 



. 

Limits and Colimits, Products and Equalizers 
111899 



2.) Publications and Preprints:
2.1 Finite Dynamical Systems:
Boolean Monomial Dynamical Systems ( back)

02.02.2004 



(by Omar ColónReyes, Reinhard Laubenbacher, and Bodo Pareigis)
Preprint 2004
In this paper we focus on the class of nonlinear dynamical systems
$f:\F_2^n \to \F_2^n$ described by special types of polynomials,
namely monomials. That is, we consider systems $f=(f_i)$,
so that each $f_i$ is a polynomial of the form
$\alpha_i x_{i_1} x_{i_2} \cdots x_{i_r}$. This class includes all Boolean networks made up
of AND functions. Associated to a general polynomial system
one can construct its {\em dependency graph} $\mathcal
D(f)$, whose vertices $v_1,\ldots ,v_n$ correspond to the
variables of the $f_i$. There is a directed arrow
$v_i\rightarrow v_j$ if $x_j$ appears in $f_i$. The main results of this paper show that
the cycle structure of the state space $\mathcal S(f)$
can be determined exclusively from the dependency graph
$\mathcal N(f)$, that is, from the structure of the $f_i$.
The key role is played by a numerical invariant associated
to a strongly connected directed graph, that is, a graph in
which there exists a (directed) walk between any two
vertices. For such a graph one can define its {\em loop
number} as the minimum of the distances of two walks from
some vertex to itself. (The number is the same no matter
which vertex is chosen.) It turns out that the dependency
graph of a monomial system can be decomposed into strongly
connected components whose loop numbers determine the
structure of the limit cycles. If the loop number of every
strongly connected component is one, then the state space
has only fixed points as limit cycles, that is, $f$ is a
fixed point system.

. 
Update Schedules of Sequential Dynamical Systems ( back)

19.02.2003 



(by Reinhard Laubenbacher and Bodo Pareigis)
Preprint 2002
Sequential dynamical systems have the property, that the
updates of states of individual cells occur sequentially,
so that the global update of the system depends on the
order of the individual updates. This order is given by an
order on the set of vertices of the dependency graph. It
turns out that only a partial suborder is necessary to
describe the global update. This paper defines and studies
this partial order and its influence on the global update
function.

. 
Decomposition and Simulation of Sequential Dynamical Systems ( back)

22.08.2002 



(by Reinhard Laubenbacher and Bodo Pareigis)
Advances in Applied Mathematics, 30 (655678) 2003
Sequential dynamical systems have been developed as a
basis for a theory of computer simulation. This paper
contains a generalization of this concept. The notion of
morphism of sequential dynamical systems is introduced,
formalizing the concept of simulating one system by
another. Several examples of morphisms are given. Using the
morphism concept, it is shown that every sequential
dynamical system decomposes uniquely into a product of
indecomposable systems.

. 
Equivalence Relations on Finite Dynamical Systems (aka Finite Dynamical Systems) ( back)

11.05.2001 



(by Reinhard Laubenbacher and Bodo Pareigis)
Advances in Applied Mathematics, 26 (237251) 2001
This paper is motivated by the theory of sequential dynamical systems,
developed as a basis for a theory of computer simulation.
We study finite dynamical systems on binary strings, that is, iterates of
functions
from $\{0,1\}^n$ to itself. We introduce several equivalence relations
on systems and study the resulting equivalence classes. The case of
twodimensional systems is studied in detail.

. 
2.2 Hopf Algebras and Quantum Groups:
On Symbolic Computations in Braided Monoidal Categories ( back)

16.12.2002 



Preprint 2003
To appear in: Proceedings of the Conference Hopf Algebras in Noncommutative Geometry and Physics
There are some powerful notations and tools to perform
computations in with tensors, the Sweedler notation for
coalgebras, the Einstein convention to reduce the number of
summation signs in computations with tensors, the Penrose
notation that has been further developed by Joyal and
Street to a graphic calculus in braided monoidal
categories. In 1977 I introduced a method of computation
that looks very much like computation with ordinary
elements or tensors, but can be performed in arbitrary
monoidal categories, by using a Yoneda Lemma like
technique. In the dual of the category of vector spaces
this allows to work with ordinary coalgebras as if they
were algebras. I will show how to expand this technique to
braided monoidal categories, and develop some of the
general rules of computation. As an application I will
derive the well known result that the antipode of a Hopf
algebra in a braided monoidal category is an algebra
antihomomorphism which is expressed by the formulas $S(1) =
1$ and $S(ab) = \langle S(b)S(a),\tau \rangle$.

. 
Tensor Products and Forgetful Functors of Entwined Modules ( back)

03.03.2003 



The Pumplün 70 Festschrift, RWTH Aachen University, Ed. A. Krieg, S. Walcher, 112, 2003
(http://www.matha.rwthaachen.de/publikationen/pumpluenfestschrift/Pareigis.pdf)
Let $A$ and $B$ be a $K$algebras and let $\mcM$ be the
category of $K$modules. It is well known that any
algebrahomomorphism $f: A \to B$ induces a forgetful
functor $U_f: \mcM_B \to \mcM_A$ that commutes with the two
underlying functors $U_A: \mcM_A \to \mcM$ and $U_B: \mcM_B
\to \mcM$ resp. i.e. $U_AU_f = U_B$. Conversely any functor
$\F: \mcM_B \to \mcM_A$ satisfying $U_AU_f = U_B$ is of the
form $\F = U_f$ for a uniquely determined
algebrahomomorphism $f: A \to B$ \cite{PA3}.
Similarly there is a bijection between the monoidal
(tensor) stuctures on $\mcM_A$ and $U_A: \mcM_A \to \mcM$
and the bialgebra structures on $A$.
In this paper we extend these facts to the category $\M$
of entwined modules over an algebra $A$ and a coalgebra
$C$ of an entwined structure $(A,C,\psi)$. This
leads to the definition of morphisms $(f,g):
(A,C,\psi) \to (A',C',\psi')$ where $f: C' \tensor A \to
A'$ is an entwined measuring of algebras and $g: C' \to A'
\tensor C$ is an entwined comeasuring of coalgebras. In the
case of tensor categories and functors on $\M$ we obtain as
diagonal on $A$ an entwined double measuring $\widetilde
\Delta_A: C \tensor C \tensor A \to A \tensor A$ and as
multiplication on $C$ an entwined double comeasuring
$\widetilde \nabla_C: C \tensor C \to A \tensor A \tensor C$
satisfying certain compatibility conditions.

. 
Double Quantum Groups ( back)

11.03.2001 



(by Daniela Hobst and Bodo Pareigis)
J. of Algebra, 242 (460494) 2001
The construction of the Drinfeld double $D(H)$ of a
finite dimensional Hopf algebra $H$ was one of the first
examples of a quasitriangular Hopf algebra whose category
of modules $\M_{D(H)}$ is braided. The braided category of
YetterDrinfeld modules $\mathcal {DY}_H^H$ is the analogue
for infinite dimensional Hopf algebras. It uses a strong
dependence between the $H$module and the $H$comodule
structures.
We generalize this construction to the category $\Mp$ of
entwined modules, that is $A$modules and $C$comodules
over Hopf algebras $A$ and $C$ where the structures are
only related by an entwining map $\psi: C \tensor A \to A
\tensor C$. We show that $\Mp$ is braided iff there is an
$r$map $r: C \tensor C \to A \tensor A$ satisfying
suitable axioms that generalize the axioms of an
$R$matrix. For finite dimensional $C$ there is a
quasitriangular Hopf algebra structure on $\Hom(C,A)$, the
{\em quantum group double}, generalizing the construction
of the Drinfeld double.

. 
Symmetric YetterDrinfeld Categories are Trivial ( back)

27.11.1995 



J. Pure and Appl. Algebra 155 (91) 2001
Let $H$ be a $k$Hopf algebra such that the canonical braiding of the category
$\cal {YD}^H_H$ of YetterDrinfeld modules is a symmetry. Then $H = k$.

. 
Quantum Groups  The Functorial Side ( back)

21.09.2000 



Proceedings of the Conference Categorical Methodes in Algebra and Topology CatMAT 2000, Universität Bremen (321332) 2000

. 
Fourier Transforms over Finite Quantum Groups ( back)

11.02.1998 



Seminarberichte FB Mathematik, FernUniversität Hagen Bd 63 (561570) 1998
In this note we want to clarify the notion of an integral for
arbitrary Hopf algebras that has been introduced a long
time ago. The relation between the integral
on a Hopf algebra and integrals in functional analysis has
only been hinted at in several publications. With the
strong interest in quantum groups, i.e. noncommutative and
noncocommutative Hopf algebras, we wish to show in which
form certain transformation rules for integrals occur in
quantum groups.
Our point of view will be the following. Let $G$ be a
quantum group in the sense of noncommutative algebraic
geometry, that is a space whose function algebra is given
by an arbitrary Hopf algebra $H$ over some base field $K$.
We will also have to use the algebra of linear functionals
$H^* = Hom(H,K)$ with the multiplication induced by the
diagonal of $H$ (called the bialgebra of $G$ in the French
literature). For most of this paper we will assume that $H$
is finite dimensional. Observe that the functions in $H$ do
not commute under multiplication and that they usually have
no general commutation formula.
The model for this setup can be found in functional
analysis. There the group $G$ is a locally compact group,
$H$ the space of representative functions on $G$, and $H^*$
the space of generalized functions or distributions. Then
the functions commute under multiplication.
We will also consider two special examples of our setup.
For an arbitrary finite group $G$ the Hopf algebra $H =
K^G$ is defined to be the algebra of functions on $G$.
Then $H^* = K G$, the group algebra, is the linear dual of
$H$.
If the finite group $G$ is Abelian and if $K$ is
algebraicly closed with char$(K) \notG$
then the corresponding Hopf algebra is as above $H = K^G$
and $H^* = \K G$. By Pontryagin duality there is the group
$\widehat G$ of characters on $G$ such that $H = K^G = K
\widehat G$ and $H^* = K G = K^{\hat G}$. 
. 
SkewPrimitive Elements of Quantum Groups and Braided Lie Algebras ( back)

15.04.1997 



Rings, Hopf algebras, and Brauer groups; lecture notes in pure and applied mathematics Vol. 197, Marcel Dekker (219238) 1998
In the study of Lie groups, of algebraic groups or of
formal groups, the concept of Lie algebras plays a central
role. These Lie algebras consist of the primitive elements.
It is difficult to introduce a similar concept for quantum
groups. Many important quantum groups have braided Hopf
algebras as building blocks. As we will see, most primitive
elements live in these braided Hopf algebras. In previous papers
we introduced the concept of braided Lie
algebras for this type of Hopf algebras. In this paper we
will give a survey of and a motivation for this concept
together with some interesting examples.
We will show that the category of YetterDrinfel'd modules is a closed monoidal
category. We construct a large family of braided Lie algebras consisting of
skewsymmetric endomorphisms of a YetterDrinfel'd module with a bilinear form.
This generalizes the construction of Lie algebras of classical groups.

. 
On Lie Algebras in the Category of YetterDrinfeld Modules ( back)

11.03.1996 



Applied Categorical Structures 6 (151175) 1998
We introduce a concept of Lie algebras in the category of YetterDrinfeld
modules ${\cal {YD}}_K^K$ over a Hopf algebra with bijective antipode over a
field $k$ that generalizes the concepts of ordinary Lie algebras, Lie super
algebras, Lie color algebras, and $(G,\chi)$Lie algebras (as in lie.dvi).
The Lie algebras defined on YetterDrinfeld modules have {\sl partially
defined $n$ary} bracket operations for every $n \in {\bf N}$ and every
primitive $n$th root of unity. They satisfy generalizations of the (anti
)symmetry and Jacabi identities.
Our main aim is to show that these Lie algebras have universal enveloping
algebras which turn out to be Hopf algebras in ${\cal {YD}}_K^K$. Conversely
the set of primitive elements of a Hopf algebra in ${\cal {YD}}_K^K$ is such a
generalized Lie algebra. We also give an example that generalizes the concept
of orthogonal or symplectic Lie algebras.

. 
On Lie Algebras in Braided Categories ( back)

03.04.1995 



Banach Center Publications: Quantum
Groups and Quantum Spaces  Vol.40 (139158) 1997
The category of groupgraded modules over an abelian group $G$ is a monoidal
category. For any bicharacter of $G$ this category becomes a braided monoidal
category. We define the notion of a Lie algebra in this category generalizing
the concepts of Lie super and Lie color algebras. Our Lie algebras have $n$ary
multiplications between various graded components. They possess universal
enveloping algebras that are Hopf algebras in the given category. Their
biproducts with the group ring are noncommutative noncocommutative Hopf
algebras some of them known in the literature. Conversely the primitive
elements of a Hopf algebra in the category form a Lie algebra in the above
sense.

. 
Reconstruction of hidden symmetries ( back)

16.09.1995 



J. Algebra 183 (90154) 1996  hepth 9412085
Groups $G$ are often obtained as groups of symmetries (or automorphisms) of
mathematical structures like a vector space (over a fixed field $k$) or a
diagram of vector spaces, where a symmetry of such a diagram is a family of
automorphisms one for each vector space which are compatible with the linear
maps of the diagram (a natural automorphism). This process of constructing the
group of symmetries is a special case of the notion of (TannakaKrein)
reconstruction.
Conversely given a group $G$ one considers its representations $G \to GL(V)$
in vector spaces $V$ over the field $k$. All representations of $G$ form the
category $kG Mod$ of modules, which we may consider as a diagram of vector
spaces. The category $kGMod$ has an additional interesting structure  the
tensor product $V \otimes W$ of two representations is again a representation
in a canonical way, $kGMod$ is a monoidal category. A special consequence of
reconstruction theory is the fact that $G$ may be recovered as the full group
of those symmetries of this huge diagram which are compatible with the tensor
product. This process seems to be the inverse of the first one. However, in a
more general setting there are subtle deviations. One may reconstruct much
larger groups of symmetries than what one started out with.
More generally we know that algebras $A$, Lie algebras $g$ and Hopf algebras
$H$ can be reconstructed from their categories of modules. For the
reconstruction of an algebra $A$ one actually needs not only the category of
$A$modules $AMod$ but also the underlying functor $\omega: AMod \to Vek$.
Then $A$ (as an algebra) can be reconstructed (up to isomorphism) as
$end(\omega)$, the end of the underlying functor. For the reconstruction of a
Hopf algebra $H$ one needs in addition the monoidal structure of $HMod$. Then
the full Hopf algebra structure can be reconstructed.
For this result one has to consider representations of the given objects
(algebras, groups, Lie algebras, Hopf algebras) in vector spaces.
Representations in categories $D$ of objects with a richer structure like super
vector spaces, $\star$spaces, graded vector spaces, comodules over Hopf
algebras have a different behavior.
We show that one usually reconstructs a much bigger object from $\omega: HD
\to D$ in $D$. In the group case this amounts to additional symmetries which we
call hidden symmetries, in the (Hopf) algebra case the situation is even more
complex but we also talk about hidden symmetries. In certain cases we describe
precisely the additional hidden symmetries by a smash product decomposition of
the reconstructed object.
We control the process of reconstruction by a control category $C$ which
operates on $\omega: HD \to D$. With different choices of the control category
$C$ we obtain different reconstructed objects $U_C$ and study their properties.
We show as main result of this paper that the universal object $U_C$ for a
functor $\omega: B \to A$ tends to decompose into a cosmash product of a Hopf
algebra with a coalgebra. In particular we show the following. If $H$ is a
braided Hopf algebra over a field $k$, $C$ is an $H$comodule coalgebra, and $A
= Vec^H$ is the braided monoidal category of $H$comodules, then the coend of
the functor $\omega: A^C \to A$ is the cosmash product $H \# C$.

. 
On Braiding and Dyslexia ( back)

11.06.1993 



J. of Algebra 171 (413425) 1995
Braided monoidal categories have important applications in knot theory,
algebraic quantum field theory, and the theory of quantum groups and Hopf
algebras. We construct a new class of braided monoidal categories.
Typical examples of braided monoidal categories are the category of modules
over a quasitriangular Hopf algebra and the category of comodules over a
coquasitriangular Hopf algebra. We consider the notion of a commutative algebra
$A$ in such a category. The category of (left and/or right) $A$modules with
the tensor product over $A$ is again a monoidal category which is not
necessarily braided. However, if we restrict this category to a special class
of modules which we call {\sl dyslectic} then this new category of dyslectic
$A$modules turns out to be a braided monoidal category, too. It is a
coreflexive subcategory of all $A$modules.

. 
Endomorphism Bialgebras of Diagrams and of NonCommutative Algebras and Spaces ( back)

08.10.1992 



Advances in Hopf algebras. Lecture Notes in
Pure and Applied Mathematics Vol. 158, Marcel Dekker (153186)
1994
Bialgebras and comodule algebras arise in a very natural way in non
commutative geometry and in representation theory. We want to study some
general principles on how to construct such bialgebras and comodule algebras.
There are two wellknown procedures to obtain bialgebras from endomorphisms of
certain objects. First we will construct endomorphism spaces in the category of
noncommutative spaces. These endomorphism spaces are described through
bialgebras.
Second we find (co)endomorphism coalgebras of certain diagrams of vector
spaces, graded vector spaces, differential graded vector spaces, or others.
Under additional conditions they again will turn out to be bialgebras.
We show that the constructions of bialgebras from noncommutative spaces and
of bialgebras from diagrams of vector spaces, are closely related, and that the
case of an endomorphism space of a noncommutative space is a special case of a
coendomorphism bialgebra of a certain diagram.

. 
Forms of Hopf Algebras and Galois Theory ( back)

24.02.1990 



Topics in Algebra. Banach Center Publ. 26 (7593) 1990

. 
Twisted Group Rings ( back)

08.05.1989 



Comm. in Alg. 17 (29232939) 1989.

. 
2.3 Miscellaneous:
Graphical Calculus Program

31.10.1998 



My LaTeX2e support file grcalc2.sty is
offered here to the public. It is used for drawing graphic diagrams for
computation in rigid or braided monoidal categories. This is the manual for it.

. 
Convexity Theories 0 cont.  Foundations ( back)

01.06.1995 



(by Bodo Pareigis, Dieter Pumplün, and Helmut Röhrl)
Proceedings 2. Gauss Sympos. München 1993, 1995
The main issue of this paper is an axiomatization of the notion of absolutely
convergent series involving a set of summands of fixed (but unrestricted)
infinite cardinality $N$. This notion is used to define the category $N_R
pnSmod^1$ of $R$prenormed $R$semimodules with $N$summation whose
homomorphisms are contractive. Based on this we introduce left $N$convexity
theories $\Gamma$ and the category $\Gamma C$ of left $\Gamma$convex modules.
We show that the closed unit ball functor $N_R pnSmod^1 \to Set$, the forgetful
functor $\Gamma C \to Set$, and the associated $\Gamma$convex module functor
$N_R pnSmod^1 \to \Gamma C$ have left adjoints.

. 
Complements and the KrullSchmidt theorem in arbitrary categories ( back)

22.06.1994 



(by Bodo Pareigis and Helmut Röhrl)
Applied Categorical Structures 3 (1127) 1995
We study direct product decompositions of objects in a finitely complete and
cocomplete category with zero object and certain axioms for a coimage
factorization of morphisms. Direct products $C = A \times B$ can be
characterized by "inner" properties of $C$ and its subobjects $A$ and $B$. We
also show that the Fitting Lemma and the KrullSchmidt Theorem hold. 
. 
Left Linear Theories ( back)

29.10.1993 



(by Bodo Pareigis and Helmut Röhrl)
Applied Categorical Structures 2 (145171) 1994
In this paper we introduce left linear theories of exponent $N$ (a set) on the
set $L$ as maps $L \times L^N \ni (l,\lambda) \to l\cdot \lambda \in L$ such
that for all $l \in L$ and $\lambda, \mu \in L^N$ the relation
$(l\cdot\lambda)\mu = l(\lambda\cdot\mu)$ holds, where $\lambda\cdot\mu \in
L^N$ is given by $(\lambda\cdot\mu)(i) = \lambda(i)\mu, i \in N$. We assume
that $L$ has a unit, that is an element $\delta \in L^N$ with $l\cdot\delta =
l$, for all $l \in L$, and $\delta\cdot\lambda = \lambda$, for all $\lambda \in
L^N$. Next, left (resp. right) $L$modules and $L$$M$bimodules and their
homomorphisms are defined and lead to categories $LMod$, $ModL$, and $LM
Mod$. These categories are algebraic categories and their free objects are
described explicitly. Finally, Hom$(X,Y)$ and $X \otimes Y$ are introduced and
their properties are investigated.

. 
Sind Wahlen undemokratisch? ( back)

20.04.2006 



mathelmu.de Nr.14, (2633) Juni 2006
Probleme beim Auffinden eines gerechten Wahlsystems 
. 
