Here is a function to produce the desired quotient algebra.
Given generator names, a maximum degree, and a base ring,
it builds the free algebra on the generators with the
requested names over the requested ring, and outputs the
quotient modulo terms of degree higher than the specified
maximum degree.
The following function works with one-letter generators.
def my_algebra(gen_names, max_degree=2, base_ring=QQ):
r"""
Return the quotient of the free algebra on these generators
by terms of degree ``max_degree + 1``.
"""
n = max_degree
R = base_ring
words = [w for W in [Words(gen_names, j) for j in range(n + 1)] for w in W]
names = [str(w) for w in words[1:]]
F = FreeAlgebra(R, names=names)
FM = F.monoid()
gens = FM.gens()
mons = (FM.one(),) + gens
r = len(mons)
M = MatrixSpace(R, r)
def row(m, mm):
if len(m) + len(mm) > n:
return [0] * r
res = [0] * r
res[words.index(mm * m)] = 1
return res
mats = [M([row(m, mm) for mm in words]) for m in words[1:]]
return FreeAlgebraQuotient(F, mons, mats, names=names)
This variant accepts multicharacter generator names:
def my_algebra(gen_names, max_degree=2, base_ring=QQ):
r"""
Return the quotient of the free algebra on these generators
by terms of degree ``max_degree + 1``.
"""
n = max_degree
R = base_ring
word_options = dict(WordOptions())
WordOptions(letter_separator='')
words = [w for W in [Words(gen_names, j) for j in range(n + 1)] for w in W]
names = [str(w) for w in words[1:]]
F = FreeAlgebra(R, names=names)
FM = F.monoid()
gens = FM.gens()
mons = (FM.one(),) + gens
r = len(mons)
M = MatrixSpace(R, r)
def row(m, mm):
if len(m) + len(mm) > n:
return [0] * r
res = [0] * r
res[words.index(mm * m)] = 1
return res
mats = [M([row(m, mm) for mm in words]) for m in words[1:]]
WordOptions(**word_options)
return FreeAlgebraQuotient(F, mons, mats, names=names)
This function gives the multiplication table:
def algebra_table(G):
gens = (G.one(),) + G.gens()
prods = [[x * y for y in gens] for x in gens]
hr = gens
hc = ('*',) + gens
return table(prods, header_row=hr, header_column=hc)
Some trials in the Sage REPL:
sage: G = my_algebra(['a', 'b'], max_degree=2, base_ring=QQ)
sage: a, b = G.gen(0), G.gen(1)
sage: (a * b + a) * b
ab
sage: algebra_table(G)
* | 1 a b aa ab ba bb
+----+----+----+----+----+----+----+----+
1 | 1 a b aa ab ba bb
a | a aa ab 0 0 0 0
b | b ba bb 0 0 0 0
aa | aa 0 0 0 0 0 0
ab | ab 0 0 0 0 0 0
ba | ba 0 0 0 0 0 0
bb | bb 0 0 0 0 0 0
sage: G = my_algebra(['x1', 'x2'], max_degree=2, base_ring=QQ)
sage: a, b = G.gen(0), G.gen(1)
sage: (a * b + a) * b
x1x2
sage: algebra_table(G)
* | 1 x1 x2 x1x1 x1x2 x2x1 x2x2
+------+------+------+------+------+------+------+------+
1 | 1 x1 x2 x1x1 x1x2 x2x1 x2x2
x1 | x1 x1x1 x1x2 0 0 0 0
x2 | x2 x2x1 x2x2 0 0 0 0
x1x1 | x1x1 0 0 0 0 0 0
x1x2 | x1x2 0 0 0 0 0 0
x2x1 | x2x1 0 0 0 0 0 0
x2x2 | x2x2 0 0 0 0 0 0
sage: G = my_algebra(['x_1', 'x_2'], max_degree=2, base_ring=QQ)
sage: a, b = G.gen(0), G.gen(1)
sage: (a * b + a) * b
x_1x_2
sage: algebra_table(G)
* | 1 x_1 x_2 x_1x_1 x_1x_2 x_2x_1 x_2x_2
+--------+--------+--------+--------+--------+--------+--------+--------+
1 | 1 x_1 x_2 x_1x_1 x_1x_2 x_2x_1 x_2x_2
x_1 | x_1 x_1x_1 x_1x_2 0 0 0 0
x_2 | x_2 x_2x_1 x_2x_2 0 0 0 0
x_1x_1 | x_1x_1 0 0 0 0 0 0
x_1x_2 | x_1x_2 0 0 0 0 0 0
x_2x_1 | x_2x_1 0 0 0 0 0 0
x_2x_2 | x_2x_2 0 0 0 0 0 0
In Jupyter, latexing is off for names such as x1x2 or x_1x_2.
Not sure how to fix that.
