Create FiniteDimensionalAlgebra from OperationTable - sage

Question

Let's assume we have an OperationTable -- T.
sage: T = OperationTable(Set, operation=operation, names='elements');
We would like to create a FiniteDimensionalAlgebra from T.
Question.
What is the fastest way to convert a table T to the form being expected by FiniteDimensionalAlgebra?
The list table must have the following form: there exists a finite-dimensional k-algebra of degree n with basis (e1,…,en)
(e1,… ,en) such that the i-th element of table is the matrix of right multiplication by ei with respect to the basis (e1,…,en).
Example.
An example of list of matrices accepted by FiniteDimensionalAlgebra over field QQ.
sage: B = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0,0], [0,1,0], [0,0,0]]), Matrix([[0,1,0], [0,0,0], [0,0,0]]), Matrix([[0,0,0], [0,0,0], [0,0,1]])])
An instance of T.
sage: T
. a b c d e f g h i j
+--------------------
a| a a a a a a a a a a
b| a a a a a a b b b b
c| a a a b b b b b c c
d| a a b b b c c c c d
e| a a b b c c c d d e
f| a a b c c c d e e e
g| a b b c c d e e f f
h| a b b c d e e f g g
i| a b c c d e f g g h
j| a b c d e e f g h i
sage: T.table()
[[0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 1, 1, 1, 1],
[0, 0, 0, 1, 1, 1, 1, 1, 2, 2],
[0, 0, 1, 1, 1, 2, 2, 2, 2, 3],
[0, 0, 1, 1, 2, 2, 2, 3, 3, 4],
[0, 0, 1, 2, 2, 2, 3, 4, 4, 4],
[0, 1, 1, 2, 2, 3, 4, 4, 5, 5],
[0, 1, 1, 2, 3, 4, 4, 5, 6, 6],
[0, 1, 2, 2, 3, 4, 5, 6, 6, 7],
[0, 1, 2, 3, 4, 4, 5, 6, 7, 8]]

Answer 1

I'm not sure if I have managed left/right correctly here, but something like this will work:
# elt_to_idx: looks like {'a': 0, 'b': 1, ...}
elt_to_idx = {b:a for (a,b) in enumerate(T.column_keys())}
mat_list = []
# Convert T to a list of matrices.
for a in T.column_keys():
# dictionary to define matrix for right multiplication by a:
dict_a = {}
for (idx_b, b) in enumerate(T.column_keys()):
# Each row (or should it be column?) should have a single entry of "1"
dict_a[idx_b, elt_to_idx[T[b, a]]] = 1
mat_list.append(matrix(QQ, 4, 4, dict_a))
FiniteDimensionalAlgebra(QQ, mat_list)