Does phrase «Space over vector field» make any sense?

[SVN]

I met in several sources (textbooks) phrase «Space can be constructed over any field». But it is always illustrated with linear space over scalar field (or sometimes over ring). Does it make any sense to talk about spaces over vector fields? What kinds of spaces are they? What about tensor fields? Can they spawn spaces as well?

By the way Google search by «space over vector field» (quotes included) returns single hit, scientific paper which is too specialised for me to understand.

And the second question related to the first one (I am not a mathematician, so I am sorry if my questions appear to be naïve). What is a vector field from the viewpoint of abstact algebra? In abstract algebra chapters, they mention several classes of set-based structures, namely, groups, rings and fields (scalar ones!), linear spaces, different algebras, but not vector fields.

I am under impression that in terms of operations a vector field does have all characteristics of an algebra: addition, multiplication by scalar, vector product (although, dot product seems to be some additional constraint, does it?), but I am puzzled that I never saw it stated clearly in any text I checked.

 

[cpsinkule]

a vector field just means the assignment of a vector at each point on a manifold. the only additional algebraic structure one can impose over a vector space (that I am aware of, at least) is an algebra. from the viewpoint of abstract algebra, a vector field is the assignment of a linear vector space to every point on the manifold called the tangent space (and cotangent space, its dual). a tensor is a multi linear map from the vector spaces into other vector spaces (or R or C). a tensor field associates to each point a tensor that maps n-fold products of the tangent\cotangent spaces into m-fold products of the tangent\cotangent space. The most common algebra associated to higher-dimensional vector spaces is the Exterior algebra, or grassman algebra. There is no simple explanations for these algebras, it is a generalization of the cross product.

 

[lavinia]

And the second question related to the first one (I am not a mathematician, so I am sorry if my questions appear to be naïve). What is a vector field from the viewpoint of abstact algebra? In abstract algebra chapters, they mention several classes of set-based structures, namely, groups, rings and fields (scalar ones!), linear spaces, different algebras, but not vector fields.

I am under impression that in terms of operations a vector field does have all characteristics of an algebra: addition, multiplication by scalar, vector product (although, dot product seems to be some additional constraint, does it?), but I am puzzled that I never saw it stated clearly in any text I checked.

- If R is any ring (not necessarily commutative) , then an R-module is an Abelian group with a scalar multiplication by elements of R. If R is field then the R-module is called a vector space.

An Algebra is an R-module with a multiplication of is own elements. The multiplication must be bi-linear. I am not sure but maybe R must be commutative to have an algebra.

For instance the polynomials in a single variable with coefficients in the rational numbers is an algebra,

 

[Fredrik]

I met in several sources (textbooks) phrase «Space can be constructed over any field». But it is always illustrated with linear space over scalar field (or sometimes over ring). Does it make any sense to talk about spaces over vector fields? What kinds of spaces are they? What about tensor fields? Can they spawn spaces as well?

This concept of "field" doesn't have anything to do with scalar, vector or tensor fields. Those other types of field are functions. This type of field is a mathematical structure, like a group or a vector space. The field associated with a vector space is the set whose elements are referred to as "scalars".

A field is a set equipped with two binary operations (called addition and multiplication) as well as two unary operations ($x\mapsto -x$ and $x\mapsto x^{-1}$). The set contains two elements 0 and 1, which are usually assumed to be different. These operations and the elements 0 and 1 are assumed to satisfy the field axioms.

1. $\forall x\forall y\forall z~~(x+y)+z=x+(y+z)$ (Addition is associative).
2. $\forall x~~ x+0=0+x=x$ (0 is an identity element with respect to the addition operation).
3. $\forall x~~ x+(-x)=-x+x=0$ (For all x, -x is an inverse of x, with respect to the addition operation).
...
and so on. There are ten axioms. You can find the rest at Wikipedia.

The two most useful fields by far are $\mathbb R$ and $\mathbb C$.

And the second question related to the first one (I am not a mathematician, so I am sorry if my questions appear to be naïve). What is a vector field from the viewpoint of abstact algebra? In abstract algebra chapters, they mention several classes of set-based structures, namely, groups, rings and fields (scalar ones!), linear spaces, different algebras, but not vector fields.

It's a term from differential geometry, not abstract algebra. A vector field is a function X defined on a subset U of a smooth manifold M such that for each p in U, X(p) is an element of the tangent space of M at p.

 

[micromass]

A vector field on a set $U$ can be naturally identified with an $\mathcal{C}^\infty(U)$-module. In fact, the Serre-Swan theorem gives a kind of converse to this association and basically says that vector fields are equivalent to certain types of modules.

https://en.wikipedia.org/wiki/Serre–Swan_theorem

 

[SVN]

Thanks everyone for replies. The blurred picture seems to get good deal of clarity now. Just to be sure I got your ideas I try express them in less rigorous terms, please, let me know if I err.

The term «field» may refer to two (at least two) different concepts: set-based structure from abstact algebra similar to the notion of ring and functional fields (scalar, vector, tensor) defined on a manifold. So as long as topology, i. e. differentiable or smooth manifolds, are not involved, the concepts of functional fields lack any meaning.

For instance, elements of linear space (we forget about manifolds at the moment) can by definition be multiplied by elements of underlying scalar field. But this understanding of scalar field differs from what is meant when talking about field of scalar function defined on a manifold (or ℝⁿ for that matter)?

A field can be considered as a module of a set but again it necessarily involves manifolds.