On central extensions of simple differential algebraic groups

We consider central extensions $Z\hookrightarrow E\twoheadrightarrow G$ in the category of linear differential algebraic groups. We show that if $G$ is simple non-commutative and $Z$ is unipotent with the differential type smaller than that of $G$, then such an extension splits. We also give a construction of central extensions illustrating that the condition on differential types is important for splitting. Our results imply that non-commutative almost simple linear differential algebraic groups, introduced by Cassidy and Singer, are simple.Comment: 13 page

Cellular A1\mathbb A^1-homology and the motivic version of Matsumoto's theorem

We define a new version of $\mathbb A^1$-homology, called cellular $\mathbb A^1$-homology, for smooth schemes over a field that admit an increasing filtration by open subschemes with cohomologically trivial closed strata. We provide several explicit computations of cellular $\mathbb A^1$-homology and use them to determine the $\mathbb A^1$-fundamental group of a split reductive group over an arbitrary field, thereby obtaining the motivic version of Matsumoto's theorem on universal central extensions of split, semisimple, simply connected algebraic groups. As applications, we uniformly explain and generalize results due to Brylinski-Deligne and Esnault-Kahn-Levine-Viehweg, determine the isomorphism classes of central extensions of such an algebraic group by an arbitrary strictly $\mathbb A^1$-invariant sheaf and also reprove classical results of E. Cartan on homotopy groups of complex Lie groups.Comment: v1: 84 pages; v2: 88 pages, abstract added, Section 5.2 revise

Categorical-algebraic methods in non-commutative and non-associative algebra

The objective of this dissertation is twofold: firstly to use categorical and algebraic methods to study homological properties of some of the aforementioned semi-abelian, non-associative structures and secondly to use categorical and algebraic methods to study categorical properties and provide categorical characterisations of some well-known algebraic structures. On one hand, the theory of universal central extensions together with the non-abelian tensor product will be studied and used to explicitly calculate some homology groups and some problems about universal enveloping algebras and actions will be solved. On the other hand, we will focus on giving categorical characterisations of some algebraic structures, such as a characterisation of groups amongst monoids, of cocommutative Hopf algebras amongst cocommutative bialgebras and of Lie algebras amongst alternating algebras

Contributions to the essential dimension of finite and algebraic groups

Essential dimension, introduced by Joe Buhler and Zinovy Reichstein and in its most general form by Alexander Merkurjev is a measure of complexity of algebraic objects such as quadratic forms, hermitian forms, central simple algebras and étale algebras. Informally, the essential dimension of an algebraic object is the number of parameters needed to define it. Often isomorphism classes of objects of some type are in one to one bijection with isomorphism classes of G-torsors. The maximal essential dimension of a G-torsor (called essential dimension of G) gives an invariant of algebraic groups, which will be of primary interest in this thesis. The text is subdivided into four chapters as follows: Chapter I+II: Multihomogenization of covariants and its application to covariant and essential dimension The essential dimension of a linear algebraic group G can be expressed via G-equivariant rational maps phi: A(V) --> A(W), so called covariants, between generically free G-modules V and W. In these two chapters we explore a new technique for dealing with covariants, called multihomogenization. This technique was jointly introduced with Hanspeter Kraft and Gerald Schwarz in an already published paper, which forms the second chapter. Applications of the multihomogenization technique to the essential dimension of algebraic groups are given by results on the essential dimension of central extensions, direct products, subgroups and the precise relation of essential dimension and covariant dimension (which is a variant of the former with polynomial covariants). Moreover the multihomogenization technique allows one to extend a twisting construction introduced by Matthieu Florence from the case of irreducible representations to completely reducible representations. This relates Florence's work on the essential dimension of cyclic p-groups to recent stack theoretic approaches by Patrick Brosnan, Angelo Vistoli and Zinovy Reichstein and by Nikita Karpenko and Alexander Mekurjev. Chapter III: Faithful and p-faithful representations of minimal dimension The study of essential dimension of finite and algebraic groups is closely related to the study of its faithful resp. generically free representations. In general the essential dimension of an algebraic group is bounded above by the least dimension of a generically free representation minus the dimension of the algebraic group. In some prominent cases this upper bound or a variant of it is strict. In this chapter we are guided by the following general questions: What do faithful representations of the least possible dimension look like? How can they be constructed? How are they related to faithful representations of minimal dimension of subgroups? Along the way we compute the minimal number of irreducible representations needed to construct a faithful representation. Chapter IV: Essential p-dimension of algebraic tori This chapter is joint work with Mark MacDonald, Aurel Meyer and Zinovy Reichstein. We study a variant of essential dimension which is relative to a prime number p. This variant, called essential p-dimension, disregards effects resulting from other primes than p. In a recent paper Nikita Karpenko and Alexander Merkurjev have computed the essential dimension of p-groups. We extend their result and find the essential p-dimension for a class of algebraic groups, which includes all algebraic tori and twisted finite p-groups

A Wells type exact sequence for non-degenerate unitary solutions of the Yang--Baxter equation

Cycle sets are known to give non-degenerate unitary solutions of the Yang--Baxter equation and linear cycle sets are enriched versions of these algebraic systems. The paper explores the recently developed cohomology and extension theory for linear cycle sets. We derive a four term exact sequence relating 1-cocycles, second cohomology and certain groups of automorphisms arising from central extensions of linear cycle sets. This is an analogue of a similar exact sequence for group extensions known due to Wells. We also compare the exact sequence for linear cycle sets with that for their underlying abelian groups via the forgetful functor and discuss generalities on dynamical 2-cocycles.Comment: 18 pages, to appear in Homology Homotopy Application