what is the basic difference between an algebra over a field and a vector space over a field?

Question

We know $\text{an algebra over a field is a vector space}$.

So what is the basic difference between an algebra over a field and a vector space over a field ?

what is the difference between an algebra over a field and a group algebra ?

Answer 1

An algebra over a field is, even if you ignore the field, a ring. Therefore, it has both addition and multiplication. Additionally, of course, multiplication with scalars is also defined. So, for $a,b\in A$ and $\alpha \in K$, where $A$ is an algebra over a field $K$, the terms $a+b, ab, \alpha a$ are all defined, and all elements of $A$.

A vector space, on the other hand, is (ignoring the field) merely an (Abelian) group - it has only one operation defined on it, addition. $a+b$ and $\alpha a$ are defined, but $ab$ is not.

A group algebra is a particular case of an algebra over a field. That is, for a field $K$ and group $G$, $K(G)$ is an algebra over $K$. Not all algebras over $K$ need to be group algebras, however.

Answer 2

It is not correct to say that an algebra over a field is a vector space. Rather, it is a vector space plus certain linear maps defined on this space that satisfy some requirements (depending on the definition, associativity, commutativity etc.).

So if you have an algebra, it gives you a well-defined vector space but a vector space does not give you a well-defined algebra, the data of the multiplication maps really matters. You can have two non-isomorphic algebras with isomorphic vector spaces, for example over $\mathbb{R}$ there is $\mathbb{R}[x]/(x^2+1)$ and $\mathbb{R}[x]/(x^2)$ (both are 2-dimensional as vector spaces but one has nilpotents and the other does not, so they are not isomorphic as rings). See here for some examples over an algebraically closed field.