Mathematisches Institut der Universität München - B. Pareigis

## 1.) Vortrags- und Vorlesungs-Skripten:

 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 Quanten-Gruppen und nicht-kommutative Geometrie SS 1993 Quantum Groups and Noncommutative Geometry SS 2002 Quantum Groups and Noncommutative Geometry WS 1999 . Chapter 1: Commutative and Noncommutative Algebraic Geometry 09-20-99 . The Principles of Commutative Algebraic Geometry 09-17-99 . Quantum Spaces and Noncommutative Geometry 09-20-99 . Quantum Monoids and their Actions on Quantum Spaces 09-20-99 . Chapter 2: Hopf Algebras, Algebraic, Formal, and Quantum Groups 11-03-99 . Hopf Algebras 09-20-99 . Monoids and Groups in a Category 09-29-99 . Affine Algebraic Groups 11-03-99 . Formal Groups 11-03-99 . Quantum Groups 11-03-99 . Quantum Automorphism Groups 11-03-99 . Duality of Hopf Algebras 11-03-99 . Chapter 3: Representation Theory, Reconstruction and Tannaka Duality 11-03-99 . Representations of Hopf Algebras 11-03-99 . Monoidal Categories 11-03-99 . Dual Objects 11-03-99 . Finite reconstruction 11-03-99 . The coalgebra coend 11-03-99 . The bialgebra coend 11-03-99 . The quantum monoid of a quantum space 11-03-99 . Reconstruction and C-categories 11-03-99 . Chapter 4: The Infinitesimal Theory 12-2-99 . Integrals and Fourier Transforms 12-2-99 . Derivations 12-2-99 . The Lie Algebra of Primitive Elements 12-2-99 . Derivations and Lie Algebras of Affine Algebraic Groups 12-2-99 . Apendix: Toolbox 09-17-99 . Categories 09-17-99 . Functors 09-17-99 . Natural Transformations 09-17-99 . Tensor Products 09-17-99 . Algebras 09-17-99 . Coalgebras 09-17-99 . Bialgebras 09-17-99 . Representable Functors 09-17-99 . Adjoint Functors and the Yoneda Lemma 09-17-99 . Limits and Colimits, Products and Equalizers 11-18-99

## 2.) Veröffentlichungen und Preprints:

### 2.1 Finite Dynamical Systems:

 Boolean Monomial Dynamical Systems ( back) 02.02.2004 (by Omar Colón-Reyes, 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 (655-678) 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 (237-251) 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 two-dimensional 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, 1-12, 2003 (http://www.matha.rwth-aachen.de/publikationen/pumpluen-festschrift/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 algebra-homomorphism $f: A \to B$ 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 algebra-homomorphism $f: A \to B$ \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 (460-494) 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 Yetter-Drinfeld 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 Yetter-Drinfeld 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 Yetter-Drinfeld 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 (321-332) 2000 . Fourier Transforms over Finite Quantum Groups ( back) 11.02.1998 Seminarberichte FB Mathematik, FernUniversität Hagen Bd 63 (561-570) 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. non-commutative and non-cocommutative 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 non-commutative 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$(K) \not||G|$ 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 $H^* = K G = K^{\hat G}$. . Skew-Primitive 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 (219-238) 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 Yetter-Drinfel'd modules is a closed monoidal category. We construct a large family of braided Lie algebras consisting of skew-symmetric endomorphisms of a Yetter-Drinfel'd module with a bilinear form. This generalizes the construction of Lie algebras of classical groups. . On Lie Algebras in the Category of Yetter-Drinfeld Modules ( back) 11.03.1996 Applied Categorical Structures 6 (151-175) 1998 We introduce a concept of Lie algebras in the category of Yetter-Drinfeld 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 Yetter-Drinfeld 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 (139-158) 1997 The category of group-graded 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 (90-154) 1996 - hep-th 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 (Tannaka-Krein) 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 $kG-Mod$ has an additional interesting structure -- the tensor product $V \otimes W$ of two representations is again a representation in a canonical way, $kG-Mod$ 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 $A-Mod$ but also the underlying functor $\omega: A-Mod \to Vek$. 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 $H-Mod$. 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 $\omega: H-D \to D$ 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 $\omega: H-D \to D$. 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 (413-425) 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 Non-Commutative Algebras and Spaces ( back) 08.10.1992 Advances in Hopf algebras. Lecture Notes in Pure and Applied Mathematics Vol. 158, Marcel Dekker (153-186) 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 well-known procedures to obtain bialgebras from endomorphisms of certain objects. First we will construct endomorphism spaces in the category of non-commutative 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 non-commutative spaces and of bialgebras from diagrams of vector spaces, are closely related, and that the case of an endomorphism space of a non-commutative 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 (75-93) 1990 . Twisted Group Rings ( back) 08.05.1989 Comm. in Alg. 17 (2923-2939) 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 Krull-Schmidt theorem in arbitrary categories ( back) 22.06.1994 (by Bodo Pareigis and Helmut Röhrl) Applied Categorical Structures 3 (11-27) 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 Krull-Schmidt Theorem hold. . Left Linear Theories ( back) 29.10.1993 (by Bodo Pareigis and Helmut Röhrl) Applied Categorical Structures 2 (145-171) 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 $L-Mod$, $Mod-L$, and $L-M- Mod$. 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 mathe-lmu.de Nr.14, (26-33) Juni 2006 Probleme beim Auffinden eines gerechten Wahlsystems .

B. Pareigis (pareigis A_T mathematik D_O_T uni-muenchen D_O_T de)    [Letzte Änderung: 08.11.2008]