An R-algebroid, , is constructed from a groupoid as follows. The object set of is the same as that of and is the freeR-module on the set , with composition given by the usual bilinear rule, extending the composition of .[1]
R-categoryedit
A groupoid can be regarded as a category with invertible morphisms. Then an R-category is defined as an extension of the R-algebroid concept by replacing the groupoid in this construction with a general category C that does not have all morphisms invertible.
R-algebroids via convolution productsedit
One can also define the R-algebroid, , to be the set of functionswith finite support, and with the convolution product defined as follows: .[2]
Only this second construction is natural for the topological case, when one needs to replace 'function' by 'continuous function with compact support', and in this case .
Brown, R.; Mosa, G. H. (1986). "Double algebroids and crossed modules of algebroids". Maths Preprint. University of Wales-Bangor.
Mosa, G.H. (1986). Higher dimensional algebroids and Crossed complexes (PhD). University of Wales. uk.bl.ethos.815719.
Mackenzie, Kirill C.H. (1987). Lie Groupoids and Lie Algebroids in Differential Geometry. London Mathematical Society Lecture Note Series. Vol. 124. Cambridge University Press. ISBN978-0-521-34882-9.
Mackenzie, Kirill C.H. (2005). General Theory of Lie Groupoids and Lie Algebroids. London Mathematical Society Lecture Note Series. Vol. 213. Cambridge University Press. ISBN978-0-521-49928-6.
Marle, Charles-Michel (2002). "Differential calculus on a Lie algebroid and Poisson manifolds". arXiv:0804.2451 [math.DG].
Weinstein, Alan (1996). "Groupoids: unifying internal and external symmetry". AMS Notices. 43: 744–752. arXiv:math/9602220. Bibcode:1996math......2220W. CiteSeerX10.1.1.29.5422.
January 01, 1970
algebroid, mathematics, constructed, starting, from, groupoids, these, more, abstract, concepts, than, algebroids, that, play, similar, role, theory, groupoids, that, algebras, theory, groups, thus, algebroid, thought, algebra, with, many, objects, contents, d. In mathematics R algebroids are constructed starting from groupoids These are more abstract concepts than the Lie algebroids that play a similar role in the theory of Lie groupoids to that of Lie algebras in the theory of Lie groups Thus a Lie algebroid can be thought of as a Lie algebra with many objects Contents 1 Definition 2 R category 3 R algebroids via convolution products 4 Examples 5 See also 6 ReferencesDefinition editAn R algebroid R G displaystyle R mathsf G nbsp is constructed from a groupoid G displaystyle mathsf G nbsp as follows The object set of R G displaystyle R mathsf G nbsp is the same as that of G displaystyle mathsf G nbsp and R G b c displaystyle R mathsf G b c nbsp is the free R module on the set G b c displaystyle mathsf G b c nbsp with composition given by the usual bilinear rule extending the composition of G displaystyle mathsf G nbsp 1 R category editA groupoid G displaystyle mathsf G nbsp can be regarded as a category with invertible morphisms Then an R category is defined as an extension of the R algebroid concept by replacing the groupoid G displaystyle mathsf G nbsp in this construction with a general category C that does not have all morphisms invertible R algebroids via convolution products editOne can also define the R algebroid R G R G b c displaystyle bar R mathsf G R mathsf G b c nbsp to be the set of functions G b c R displaystyle mathsf G b c longrightarrow R nbsp with finite support and with the convolution product defined as follows f g z f x g y z x y displaystyle displaystyle f g z sum fx gy mid z x circ y nbsp 2 Only this second construction is natural for the topological case when one needs to replace function by continuous function with compact support and in this case R C displaystyle R cong mathbb C nbsp Examples editEvery Lie algebra is a Lie algebroid over the one point manifold The Lie algebroid associated to a Lie groupoid See also editAlgebraic category Algebroid disambiguation Bialgebra Bicategory Convolution product Crossed module Double groupoid Higher dimensional algebra Hopf algebra Module mathematics Ring mathematics References edit Mosa 1986 Brown amp Mosa 1986 This article incorporates material from Algebroid Structures and Algebroid Extended Symmetries on PlanetMath which is licensed under the Creative Commons Attribution Share Alike License Sources Brown R Mosa G H 1986 Double algebroids and crossed modules of algebroids Maths Preprint University of Wales Bangor Mosa G H 1986 Higher dimensional algebroids and Crossed complexes PhD University of Wales uk bl ethos 815719 Mackenzie Kirill C H 1987 Lie Groupoids and Lie Algebroids in Differential Geometry London Mathematical Society Lecture Note Series Vol 124 Cambridge University Press ISBN 978 0 521 34882 9 Mackenzie Kirill C H 2005 General Theory of Lie Groupoids and Lie Algebroids London Mathematical Society Lecture Note Series Vol 213 Cambridge University Press ISBN 978 0 521 49928 6 Marle Charles Michel 2002 Differential calculus on a Lie algebroid and Poisson manifolds arXiv 0804 2451 math DG Weinstein Alan 1996 Groupoids unifying internal and external symmetry AMS Notices 43 744 752 arXiv math 9602220 Bibcode 1996math 2220W CiteSeerX 10 1 1 29 5422 Retrieved from https en wikipedia org w index php title R algebroid amp oldid 1073207189, wikipedia, wiki, book, books, library,
