symbols to denote mathematical structures

Question

I would like to collect a list of symbols used to denote some mathematical structures. For example, consider the following situation:

• A vector space is usually denoted by $V,V',W,W'$. Suppose in some set up all these $4$ letters has different meanings what are the other commonly used symbols for vector spaces?

Suppose I am working with some fields and the letters $\mathbb{K},\mathbb{k}$ and $\mathbb{F}$ have some other meanings. What are the other common notations for fields?

• for elements of vector space $v,w$ and what else?
• for differenial forms $\omega,\varphi$ and what else?
• for vector fields $X,Y$ and what else?

It would be useful if you can give some reference where the symbols you suggest are already used.

Answer 1

Concerning Galois theory, fields and field extensions are often denoted by $L/K$ with intermediate fields $E,M$ and $E,E',M,M',N,N'$ and so on. Specific fields have its own symbol, like $\Bbb F_p$ for the finite field with $p$ elements, or $\Bbb F_q$ with $q=p^n$. An algebraic closure of a field $K$ is often denoted by $\overline{K}$, e.g., the Galois extension $\overline{\Bbb Q}/\Bbb Q$ with absolute Galois group $\rm{Gal}(\overline{\Bbb Q}/\Bbb Q)$. The field of meromorphic functions for a Riemannian surface $X$ is often denoted by $\mathcal{M}(X)$.

Answer 2

I cannot see a problem. If V, W are already used for a different purposes then use $R, S, T, U$ or $ X, Y,Z$ for vector spaces.

Of course you can resort to a notation used in older German publications and use $\frak {V}$ , $\frak W$ for vector spaces, but I don't think that will delight your readers.

For vectors I have seen different notations: $$x,\bf{x},\vec{x},\frak x$$

but I think most of the time you do not have so many options.

I think it is easy for the reader if you use adjacent letters for the same kind of objects from different ranges of the alphabet that are distant, e.g. if $a,b,c$ are scalars and $x,y,z$ are vectors then $$(a+b)z=az+bz$$ $$c(x+y)=cx+cy$$ will be easy to read for a reader.

Often the start range of the alphabet is used for constant and the end range for variables.

Answer 3

It seems it is better to seek category by category. And in every category write the notation for related concepts and objects which live on a given category, something like:

I-Topological spaces category, TOP :

-the notation for the object: a topological space $(X, \tau$ ), which $X$ or $Y$ is a set and $\tau\subset P(X)$.

-for morphisms : ordinary continuous functions with symbols $f,g..$

-for open sets: $O, U, V..$

-other definition via closure operators : $(X,\it cl )$, $\it cl $ for closure operator..

II-Manifolds as a kind of topological spaces:

-the notation for a manifold: M, N

-for a chart on a manifold: $(U,\phi),(V,\psi)$..,

-for local tangent space on a manifold: $T_{P}M$.

III-Differential forms as geometric objects on manifolds:

-for differential forms: $\omega, \varphi, \eta,\xi$.

-for the class of smooth functions: $C^\infty$.

-for a tangent map and a cotangent map: $f_{*}$, $f^*$ respectively.

-for the family of alternative maps on the manifold $\Omega^p M$.

and so on.

Answer 4

As you note, many branches of mathematics have evolved conventions for which kinds of symbols represent different kinds of objects in that branch. Those conventions make it easier to read and write mathematics for local consumption.

I doubt that there is a single place listing those conventions area by area. You just learn them when you need them.