In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, algebroid may refer to several distinct notions, which nevertheless all arise from generalising certain aspects of the theory of
algebras
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition ...
or
Lie algebras
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
.
*
Algebroid branch, a formal power series branch of an algebraic curve
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Algebroid cohomology
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Algebroid multifunction
*
Courant algebroid
In differential geometry, a field of mathematics, a Courant algebroid is a vector bundle together with an inner product and a compatible bracket more general than that of a Lie algebroid.
It is named after Theodore James Courant, Theodore Courant, ...
, an object generalising
Lie bialgebroids
*
Lie algebroid
In mathematics, a Lie algebroid is a vector bundle A \rightarrow M together with a Lie bracket on its space of sections \Gamma(A) and a vector bundle morphism \rho: A \rightarrow TM, satisfying a Leibniz rule. A Lie algebroid can thus be thought ...
, the infinitesimal counterpart of
Lie groupoids
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Atiyah algebroid
In mathematics, the Atiyah algebroid, or Atiyah sequence, of a principal '' G''-bundle '' P'' over a manifold '' M'', where '' G'' is a Lie group, is the Lie algebroid of the gauge groupoid of '' P''. Explicitly, it is given by the following shor ...
, a fundamental example of a Lie algebroid associated to a
principal bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equ ...
*
R-algebroid, a categorical construction associated to
groupoids
In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a:
* '' Group'' with a partial fun ...
{{disambig
Mathematics disambiguation pages