In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, an algebra bundle is a fiber bundle

Fiber bundle

In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...

 whose fiber

Fiber

Fiber is a class of materials that are continuous filaments or are in discrete elongated pieces, similar to lengths of thread.They are very important in the biology of both plants and animals, for holding tissues together....

s are algebra

Algebra over a field

In mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it isan algebraic structure consisting of a vector space together with an operation, usually called multiplication, that combines any two vectors to form a third vector; to qualify as...

s and local trivializations respect the algebra structure. It follows that the transition function

Transition function

In mathematics, a transition function has several different meanings:* In topology, a transition function is a homeomorphism from one coordinate chart to another...

s are algebra isomorphisms. Since algebras are also vector space

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

s, every algebra bundle is a vector bundle

Vector bundle

In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...

.

Examples include the tensor bundle

Tensor bundle

In mathematics, the tensor bundle of a manifold is the direct sum of all tensor products of the tangent bundle and the cotangent bundle of that manifold. To do calculus on the tensor bundle a connection is needed....

, exterior bundle

Exterior bundle

In mathematics, the exterior bundle of a manifold M is the subbundle of the tensor bundle consisting of all antisymmetric covariant tensors. It has special significance, because one can define a connection-independent derivative on it, namely the exterior derivative.Sections of the exterior bundle...

, and symmetric bundle associated to a given vector bundle

Vector bundle

In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...

, as well as the Clifford bundle

Clifford bundle

In mathematics, a Clifford bundle is an algebra bundle whose fibers have the structure of a Clifford algebra and whose local trivializations respect the algebra structure...

associated to any Riemannian vector bundle.