The following is the answer due to @Nicolas-M-Thiéry

```
sage: Groups?
The category of (multiplicative) groups, i.e. monoids with
inverses.
```

Mind the *multiplicative*!

What you want is:

```
sage: V = FreeModule(CC,2)
sage: V in CommutativeAdditiveGroups()
True
```

or (better, but not imported by
default):

```
sage: from sage.categories.additive_groups import AdditiveGroups
sage: V in AdditiveGroups()
True
```

Now you can construct the group
algebra:

```
sage: C = V.algebra(QQ)
sage: C.category()
Category of commutative additive group algebras over Rational Field
sage: x = C.an_element()
sage: x
B[(1.00000000000000, 0.000000000000000)]
sage: 3 * x + 1
B[(0.000000000000000, 0.000000000000000)] + 3*B[(1.00000000000000, 0.000000000000000)]
```

Ah, but this is disappointing::

```
sage: (x+1)^2
TypeError: mutable vectors are unhashable
```

One would need to have a variant of
FreeModule that would guarantee that
vectors remain immutable upon
arithmetic.

In the mean time, you can use:

```
sage: V = CombinatorialFreeModule(CC, [0,1])
sage: C = V.algebra(QQ)
sage: x = C.an_element()
sage: x
B[2.00000000000000*B[0] + 2.00000000000000*B[1]]
sage: (x+1)^2
B[0] + 2*B[2.00000000000000*B[0] + 2.00000000000000*B[1]] + B[4.00000000000000*B[0] + 4.00000000000000*B[1]]
```