scholarly journals Equivariant K-theory, groupoids and proper actions

2011 ◽  
Vol 9 (3) ◽  
pp. 475-501
Author(s):  
Jose Cantarero
Keyword(s):  
Vector Bundles ◽  
Cohomology Theory ◽  
Dimensional Vector ◽  
Lie Groupoids ◽  
Proper Actions ◽  
Lie Groupoid ◽  
Finite Dimensional ◽  
Cw Complexes ◽  
K Theory ◽  
Completion Theorem

AbstractIn this paper we define complex equivariant K-theory for actions of Lie groupoids using finite-dimensional vector bundles. For a Bredon-compatible Lie groupoid , this defines a periodic cohomology theory on the category of finite -CW-complexes. We also establish an analogue of the completion theorem of Atiyah and Segal. Some examples are discussed.

scholarly journals On local integration of Lie brackets

2020 ◽  
Vol 2020 (760) ◽  
pp. 267-293 ◽  
Author(s):  
Alejandro Cabrera ◽  
Ioan Mărcuţ ◽  
María Amelia Salazar
Keyword(s):  
Vector Field ◽  
Lie Algebroids ◽  
Lie Algebroid ◽  
Lie Theory ◽  
Lie Groupoids ◽  
Lie Groupoid ◽  
Finite Dimensional ◽  
Local Integration ◽  
Lie Brackets ◽  
Complete Account

AbstractWe give a direct, explicit and self-contained construction of a local Lie groupoid integrating a given Lie algebroid which only depends on the choice of a spray vector field lifting the underlying anchor map. This construction leads to a complete account of local Lie theory and, in particular, to a finite-dimensional proof of the fact that the category of germs of local Lie groupoids is equivalent to that of Lie algebroids.

scholarly journals Moduli of endomorphisms of semistable vector bundles over a compact Riemann surface

1990 ◽  
Vol 32 (1) ◽  
pp. 1-12 ◽  
Author(s):  
L. Brambila Paz
Keyword(s):  
Riemann Surface ◽  
Vector Bundles ◽  
Compact Riemann Surface ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Finite Dimensional ◽  
Semistable Vector Bundles

Mumford and Suominen in [8] and Newstead in [11] have considered the moduli problem of classifying the endomorphisms of finite-dimensional vector spaces. Using similar ideas we consider the moduli problem for endomorphisms of indecomposable semistable vector bundles over a compact connected Riemann surface of genus g ≥ 2.

scholarly journals The completion theorem in K-theory for proper actions of a discrete group

Topology ◽  
2001 ◽  
Vol 40 (3) ◽  
pp. 585-616 ◽  
Author(s):  
Wolfgang Lück ◽  
Bob Oliver
Keyword(s):  
Discrete Group ◽  
Proper Actions ◽  
K Theory ◽  
Completion Theorem

scholarly journals Universal connections on Lie groupoids

2003 ◽  
Vol 2003 (23) ◽  
pp. 1465-1480
Author(s):  
Efstathios Vassiliou ◽  
Apostolos Nikolopoulos
Keyword(s):  
Vector Bundles ◽  
General Construction ◽  
Lie Groupoids ◽  
Principal Bundles ◽  
Lie Groupoid ◽  
Linear Connections ◽  
Connections On Principal Bundles

Given a Lie groupoidΩ, we construct a groupoidJ1Ωequipped with a universal connection from which all the connections ofΩare obtained by certain pullbacks. We show that this general construction leads to universal connections on principal bundles (considered by García (1972)) and universal linear connections on vector bundles (ultimately related with those of Cordero et al. (1989)).

A closer look at the stable completion

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Inverse Limit ◽  
Unit Vector ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Compact Sets ◽  
Linear Topology ◽  
Topological Embedding ◽  
Definable Set ◽  
Finite Dimensional ◽  
The One

This chapter introduces the concept of stable completion and provides a concrete representation of unit vector Mathematical Double-Struck Capital A superscript n in terms of spaces of semi-lattices, with particular emphasis on the frontier between the definable and the topological categories. It begins by constructing a topological embedding of unit vector Mathematical Double-Struck Capital A superscript n into the inverse limit of a system of spaces of semi-lattices L(Hsubscript d) endowed with the linear topology, where Hsubscript d are finite-dimensional vector spaces. The description is extended to the projective setting. The linear topology is then related to the one induced by the finite level morphism L(Hsubscript d). The chapter also considers the condition that if a definable set in L(Hsubscript d) is an intersection of relatively compact sets, then it is itself relatively compact.

scholarly journals A note on extensions of multilinear maps defined on multilinear varieties

2021 ◽  
pp. 1-26
Author(s):  
W. T. Gowers ◽  
L. Milićević
Keyword(s):  
Finite Fields ◽  
Vector Spaces ◽  
Restriction Map ◽  
Zero Set ◽  
Dimensional Vector ◽  
Prime Field ◽  
Finite Dimensional ◽  
Multilinear Maps ◽  
Multilinear Map

Abstract Let $G_1, \ldots , G_k$ be finite-dimensional vector spaces over a prime field $\mathbb {F}_p$ . A multilinear variety of codimension at most $d$ is a subset of $G_1 \times \cdots \times G_k$ defined as the zero set of $d$ forms, each of which is multilinear on some subset of the coordinates. A map $\phi$ defined on a multilinear variety $B$ is multilinear if for each coordinate $c$ and all choices of $x_i \in G_i$ , $i\not =c$ , the restriction map $y \mapsto \phi (x_1, \ldots , x_{c-1}, y, x_{c+1}, \ldots , x_k)$ is linear where defined. In this note, we show that a multilinear map defined on a multilinear variety of codimension at most $d$ coincides on a multilinear variety of codimension $O_{k}(d^{O_{k}(1)})$ with a multilinear map defined on the whole of $G_1\times \cdots \times G_k$ . Additionally, in the case of general finite fields, we deduce similar (but slightly weaker) results.

scholarly journals Vector bundles associated to Lie algebras

2016 ◽  
Vol 2016 (716) ◽  
Author(s):  
Jon F. Carlson ◽  
Eric M. Friedlander ◽  
Julia Pevtsova
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Vector Bundles ◽  
Coherent Sheaves ◽  
Restricted Lie Algebra ◽  
Finite Dimensional

AbstractWe introduce and investigate a functorial construction which associates coherent sheaves to finite dimensional (restricted) representations of a restricted Lie algebra

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