Define a vector space structure on $R^2$ as follows: $(a_1,b_1)\oplus(a_2,b_2)=(a_1+a_2,b_1+b_2+a_1 a_2)$, $k(a,b)=(ka,kb+\frac{k(k-1)}{2}a^2)$.
Can such a construction be realized from a familiar structure, by some "endomorphism" or "coordinate" change tricks? How to determine all such structures on a given vector space or module? Does this constructure arise naturally arise from other theories, say, for example, representation theory?
This is a "well-known" object, one of several things known as (part of?) a "Heisenberg group". It can be usefully modeled as the group of upper-triangular matrices of the form $\pmatrix{1&a&b\cr 0&1&a\cr 0&0&1}$ under matrix multiplication: $$ \pmatrix{1&a_1&b_1\cr 0&1&a_1\cr 0&0&1}\pmatrix{1&a_2&b_2\cr 0&1&a_2\cr 0&0&1} \;=\;\pmatrix{1&a_1+a_2&b_1+b_2+a_1a_2\cr 0&1&a_1+a_2\cr 0&0&1} $$ which accounts for the "mysterious group law".
Here is a general way to construct such "strange" structures.
Let $V$ be a vector space and $X$ a set and $f$ a bijective map from $X$ to $V$. Define a vector space structure on $X$ by $x + y = f^{-1}(f(x) + f(y))$ and $\alpha x = f^{-1}(\alpha f(x))$.
This turns $f$ into an isomorphism between $V$ and $X$ with this new structure as a vector space.
Given any such "strange" structure as a vector space, one can find a map like above to a vector space of the form $F^n$ for some field $F$ and some natural number $n$ (assuming the "strange" space is finite dimensional), simply because any finite dimensional space is isomorphic to some number of copies of the ground field.
(One can of course also do something similar for the infinite dimensional cases).
