Classification and Casimir Invariants of Lie-Poisson Brackets

We classify Lie-Poisson brackets that are formed from Lie algebra extensions. The problem is relevant because many physical systems owe their Hamiltonian structure to such brackets. A classification involves reducing all brackets to a set of normal forms, and is achieved partially through the use of Lie algebra cohomology. For extensions of order less than five, the number of normal forms is small and they involve no free parameters. We derive a general method of finding Casimir invariants of Lie-Poisson bracket extensions. The Casimir invariants of all low-order brackets are explicitly computed. We treat in detail a four field model of compressible reduced magnetohydrodynamics.Comment: 59 pages, Elsevier macros. To be published in Physica

Generalized gauge field theories with non-topological soliton solutions

We perform a systematic analysis of the conditions under which \textit{generalized} gauge field theories of compact semisimple Lie groups exhibit electrostatic spherically symmetric non-topological soliton solutions in three space dimensions. By the term \textit{generalized}, we mean that the dynamics of the concerned fields is governed by lagrangian densities which are general functions of the quadratic field invariants, leading to physically consistent models. The analysis defines exhaustively the class of this kind of lagrangian models supporting those soliton solutions and leads to methods for their explicit determination. The necessary and sufficient conditions for the linear stability of the finite-energy solutions against charge-preserving perturbations are established, going beyond the usual Derrick-like criteria, which only provides necessary conditions.Comment: 6 pages, revtex

Quiver Structure of Heterotic Moduli

We analyse the vector bundle moduli arising from generic heterotic compactifications from the point of view of quiver representations. Phenomena such as stability walls, crossing between chambers of supersymmetry, splitting of non-Abelian bundles and dynamic generation of D-terms are succinctly encoded into finite quivers. By studying the Poincar\'e polynomial of the quiver moduli space using the Reineke formula, we can learn about such useful concepts as Donaldson-Thomas invariants, instanton transitions and supersymmetry breaking.Comment: 38 pages, 5 figures, 1 tabl

Fourier-Mukai transforms for coherent systems on elliptic curves

We determine all the Fourier-Mukai transforms for coherent systems consisting of a vector bundle over an elliptic curve and a subspace of its global sections, showing that these transforms are indexed by the positive integers. We prove that the natural stability condition for coherent systems, which depends on a parameter, is preserved by these transforms for small and large values of the parameter. By means of the Fourier-Mukai transforms we prove that certain moduli spaces of coherent systems corresponding to small and large values of the parameter are isomorphic. Using these results we draw some conclusions about the possible birational type of the moduli spaces. We prove that for a given degree $d$ of the vector bundle and a given dimension of the subspace of its global sections there are at most $d$ different possible birational types for the moduli spaces.Comment: LaTeX2e, 21 pages, some proofs simplified, typos corrected. Final version to appear in Journal of the London Mathematical Societ

Protoadditive functors, derived torsion theories and homology

Protoadditive functors are designed to replace additive functors in a non-abelian setting. Their properties are studied, in particular in relationship with torsion theories, Galois theory, homology and factorisation systems. It is shown how a protoadditive torsion-free reflector induces a chain of derived torsion theories in the categories of higher extensions, similar to the Galois structures of higher central extensions previously considered in semi-abelian homological algebra. Such higher central extensions are also studied, with respect to Birkhoff subcategories whose reflector is protoadditive or, more generally, factors through a protoadditive reflector. In this way we obtain simple descriptions of the non-abelian derived functors of the reflectors via higher Hopf formulae. Various examples are considered in the categories of groups, compact groups, internal groupoids in a semi-abelian category, and other ones

On the "three subobjects lemma" and its higher-order generalisations

We solve a problem mentioned in an article of Berger and Bourn: we prove that in the context of an algebraically coherent semi-abelian category, two natural definitions of the lower central series coincide. In a first, "standard" approach, nilpotency is defined as in group theory via nested binary commutators of the form $[[X,X],X]$. In a second approach, higher Higgins commutators of the form $[X,X,X]$ are used to define nilpotent objects. The two are known to be different in general; for instance, in the context of loops, the definition of Bruck is of the former kind, while the commutator-associator filtration of Mostovoy and his co-authors is of the latter type. Another example, in the context of Moufang loops, is given in Berger and Bourn's paper. In this article, we show that the two streams of development agree in any algebraically coherent semi-abelian category. Such are, for instance, all Orzech categories of interest. Our proof of this result is based on a higher-order version of the Three Subobjects Lemma of Cigoli-Gray-Van der Linden, which extends the classical Three Subgroups Lemma from group theory to categorical algebra. It says that any $n$-fold Higgins commutator $[K_1, \dots,K_n]$ of normal subobjects $K_i$ of an object $X$ may be decomposed into a join of nested binary commutators.Comment: 20 pages; revised version, with some simplified proof