Generalized Robba rings

We prove that any projective coadmissible module over the locally analytic distribution algebra of a compact $p$-adic Lie group is finitely generated. In particular, the category of coadmissible modules does not have enough projectives. In the Appendix a "generalized Robba ring" for uniform pro-$p$ groups is constructed which naturally contains the locally analytic distribution algebra as a subring. The construction uses the theory of generalized microlocalization of quasi-abelian normed algebras that is also developed there. We equip this generalized Robba ring with a self-dual locally convex topology extending the topology on the distribution algebra. This is used to show some results on coadmissible modules.Comment: with an appendix by Peter Schneider; revised; new titl

Higher Yang-Mills Theory

Electromagnetism can be generalized to Yang-Mills theory by replacing the group U(1)$ by a nonabelian Lie group. This raises the question of whether one can similarly generalize 2-form electromagnetism to a kind of "higher-dimensional Yang-Mills theory". It turns out that to do this, one should replace the Lie group by a "Lie 2-group", which is a category C where the set of objects and the set of morphisms are Lie groups, and the source, target, identity and composition maps are homomorphisms. We show that this is the same as a "Lie crossed module": a pair of Lie groups G,H with a homomorphism t: H -> G and an action of G on H satisfying two compatibility conditions. Following Breen and Messing's ideas on the geometry of nonabelian gerbes, one can define "principal 2-bundles" for any Lie 2-group C and do gauge theory in this new context. Here we only consider trivial 2-bundles, where a connection consists of a Lie(G)-valued 1-form together with an Lie(H)-valued 2-form, and its curvature consists of a Lie(G)-valued 2-form together with a Lie(H)-valued 3-form. We generalize the Yang-Mills action for this sort of connection, and use this to derive "higher Yang-Mills equations". Finally, we show that in certain cases these equations admit self-dual solutions in five dimensions.Comment: 20 pages LaTeX with XY-pic figure

Formal groups and unipotent affine groups in non-categorical symmetry

As is well known, in characteristic zero, the Lie algebra functor gives two category equivalences, one from the formal groups to the finite-dimensional Lie algebras, and the other from the unipotent algebraic affine groups to the finite-dimensional nilpotent Lie algebras. We prove these category equivalences in a quite generalized framework, proposed by Gurevich [D.I. Gurevich, The Yang–Baxter equation and generalization of formal Lie theory, Soviet Math. Dokl. 33 (1986) 758–762] and later by Takeuchi [M. Takeuchi, Survey of braided Hopf algebras, in: N. Andruskiewitsch, et al. (Eds.), New Trends in Hopf Algebra Theory, in: Contemp. Math., vol. 267, Amer. Math. Soc., Providence, RI, 2000, pp. 301–324], of vector spaces with non-categorical symmetry. We remove the finiteness restriction from the objects, by using the terms of Hopf algebras and Lie coalgebras

Quasi-Coxeter categories and a relative Etingof-Kazhdan quantization functor

Let g be a symmetrizable Kac-Moody algebra and U_h(g) its quantized enveloping algebra. The quantum Weyl group operators of U_h(g) and the universal R-matrices of its Levi subalgebras endow U_h(g) with a natural quasi-Coxeter quasitriangular quasibialgebra structure which underlies the action of the braid group of g and Artin's braid groups on the tensor product of integrable, category O modules. We show that this structure can be transferred to the universal enveloping algebra Ug[[h]]. The proof relies on a modification of the Etingof-Kazhdan quantization functor, and yields an isomorphism between (appropriate completions of) U_h(g) and Ug[[h]] preserving a given chain of Levi subalgebras. We carry it out in the more general context of chains of Manin triples, and obtain in particular a relative version of the Etingof-Kazhdan functor with input a split pair of Lie bialgebras. Along the way, we develop the notion of quasi-Coxeter categories, which are to generalized braid groups what braided tensor categories are to Artin's braid groups. This leads to their succint description as a 2-functor from a 2-category whose morphisms are De Concini-Procesi associahedra. These results will be used in the sequel to this paper to give a monodromic description of the quantum Weyl group operators of an affine Kac-Moody algebra, extending the one obtained by the second author for a semisimple Lie algebra.Comment: 63 pages. Exposition in sections 1 and 4 improved. Material added: definition of a split pair of Lie bialgebras (sect. 5.2-5.5), 1-jet of the relative twist (5.20), PROP description of the Verma modules L_-,N*_+ (7.6), restriction to Levi subalgebras (8.4), D-structures on Kac-Moody algebras (9.1

On the Passi and the Mal'cev functors

The author has shown that the category of analytic contravariant functors on $\mathbf{gr}$, the category of finitely-generated free groups, is equivalent to the category of left modules over the PROP associated to the Lie operad, working over $\mathbb{Q}$. This exploited properties of the polynomial filtration of the category of contravariant functors on $\mathbf{gr}$. The first purpose of this paper is to strengthen the corresponding result for covariant functors on $\mathbf{gr}$. This involves introducing the appropriate analogue of the category of analytic contravariant functors, namely a certain category of towers of polynomial functors on $\mathbf{gr}$. This category is abelian and has a natural symmetric monoidal structure induced by the usual tensor product of functors. Moreover, the projective generators of this category are described in terms of the Mal'cev functors that are introduced here. It follows that this category is equivalent to the category of right modules over the PROP associated to the Lie operad. As a fundamental example, the Passi functors arising from the group ring functors are described explicitly. The theory is applied to consider bifunctors on $\mathbf{gr}$. This allows the $\mathbb{Q}$-linearization of the category of free groups to be described, up to polynomial filtration. As a stronger application of the theory, this is generalized to the Casimir PROP associated to the Lie operad, as studied by Hinich and Vaintrob. Up to polynomial filtration, this recovers the category $\mathbf{A}$ introduced by Habiro and Massuyeau in their study of bottom tangles in handlebodies.Comment: 43 pages. Comments welcom

Axiomatic framework for the BGG Category O

We introduce a general axiomatic framework for algebras with triangular decomposition, which allows for a systematic study of the Bernstein-Gelfand-Gelfand Category $\mathcal{O}$. The framework is stated via three relatively simple axioms; algebras satisfying them are termed "regular triangular algebras (RTAs)". These encompass a large class of algebras in the literature, including (a) generalized Weyl algebras, (b) symmetrizable Kac-Moody Lie algebras $\mathfrak{g}$, (c) quantum groups $U_q(\mathfrak{g})$ over "lattices with possible torsion", (d) infinitesimal Hecke algebras, (e) higher rank Virasoro algebras, and others. In order to incorporate these special cases under a common setting, our theory distinguishes between roots and weights, and does not require the Cartan subalgebra to be a Hopf algebra. We also allow RTAs to have roots in arbitrary monoids rather than root lattices, and the roots of the Borel subalgebras to lie in cones with respect to a strict subalgebra of the Cartan subalgebra. These relaxations of the triangular structure have not been explored in the literature. We then study the BGG Category $\mathcal{O}$ over an RTA. In order to work with general RTAs - and also bypass the use of central characters - we introduce conditions termed the "Conditions (S)", under which distinguished subcategories of Category $\mathcal{O}$ possess desirable homological properties, including: (a) being a finite length, abelian, self-dual category; (b) having enough projectives/injectives; or (c) being a highest weight category satisfying BGG Reciprocity. We discuss whether the above examples satisfy the various Conditions (S). We also discuss two new examples of RTAs that cannot be studied using previous theories of Category $\mathcal{O}$, but require the full scope of our framework. These include the first construction of algebras for which the "root lattice" is non-abelian.Comment: 59 pages, LaTeX. This paper supersedes (and goes far beyond) the older preprint arXiv:0811.2080 (31 pages), which has been completely rewritte

Open-string BRST cohomology for generalized complex branes

It has been shown recently that the geometry of D-branes in general topologically twisted (2,2) sigma-models can be described in the language of generalized complex (GC) structures. On general grounds, such D-branes (called GC branes) must form a category. We compute the BRST cohomology of open strings with both ends on the same GC brane. In mathematical terms, we determine spaces of endomorphisms in the category of GC branes. We find that the BRST cohomology can be expressed as the cohomology of a Lie algebroid canonically associated to any GC brane. In the special case of B-branes, this leads to an apparently new way to compute Ext groups of holomorphic line bundles supported on complex submanifolds: while the usual method leads to a spectral sequence converging to the Ext, our approach expresses the Ext group as the cohomology of a certain differential acting on the space of smooth sections of a graded vector bundle on the submanifold. In the case of coisotropic A-branes, our computation confirms a proposal of Orlov and one of the authors (A.K.)