mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, an algebra bundle is a

fiber bundle In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...

whose

fiber Fiber or fibre (from la, fibra, links=no) is a natural or artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineering materials often incorporate ...

s are

algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...

s and

local trivialization In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...

s respect the algebra structure. It follows that the

transition functions In mathematics, a transition function may refer to: * a transition map between two charts of an atlas of a manifold or other topological space * the function that defines the transitions of a state transition system in computing, which may refer mo ...

are

algebra isomorphism In mathematics, an algebra homomorphism is a homomorphism between two associative algebras. More precisely, if and are algebras over a field (or commutative ring) , it is a function F\colon A\to B such that for all in and in , * F(kx) = kF ...

s. Since algebras are also

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...

s, every algebra bundle is a

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 (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...

. Examples include the

tensor-algebra bundle In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of being ...

exterior bundle In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction is analogous to the construction of the tensor produ ...

, and symmetric bundle associated to a given

, as well as the

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. There is a natural Clifford bundle associated to any (pseudo) Riemannian ...

associated to any Riemannian vector bundle.

References

*. *. *. *. Vector bundles {{algebra-stub

See also

References