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The following is the answer due to @nicolas-m-thiery

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]]

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No.2 Revision

updated 2016-07-01 21:32:48 +0200

kcrisman gravatar image

The following is the answer due to @nicolas-m-thiery

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

import AdditiveGroups sage: V in AdditiveGroups() True

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)]

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]]

+ 4.00000000000000*B[1]]

The following is the answer due to @nicolas-m-thiery

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]]

The following is the answer due to @nicolas-m-thiery@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]]

The following is the answer due to @Nicolas M Thiéry@462

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]]

The following is the answer due to @462@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]]

The following is the answer due to @Nicolas_M_Thiéry@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]]
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