Non trivial massey triple product

Question

Suppose that $(A, d)$ is graded commutative algebra with three generators $a, b, c$, each of degree $1$ and $da = db =0$, $dc = ab$. Then, how can I construct a non-trivial Massey product? What changes if the degree of $a, b, c$ changes?

Answer 1

Typically to get a nontrivial Massey product $\langle u,v,w \rangle$, you need the products $uv$ and $vw$ to be boundaries. In your case you have just one nontrivial product being a boundary, namely $ab$. So it's conceivable that you could have a Massey product $\langle a, b, a \rangle$ or $\langle b, a, b \rangle$. I don't see other viable options. The first of these would be represented by $ac + ca$, which is zero in $A$, and similarly for the second. So there are no nontrivial Massey products here.

If you just want to mess around with things, you can try SageMath (a.k.a. Sage, "Software for Algebra and Geometry EXPERIMENTATION", emphasis on experimentation).

sage: # (everything after a hash mark "#" is a comment)
sage: A.<a,b,c> = GradedCommutativeAlgebra(QQ)  # define a graded commutative algebra A with three generators a, b, c. Coefficient ring is the rationals. By default, the generators are in degree 1.
sage: A.<a,b,c> = GradedCommutativeAlgebra(GF(3), degrees=(2,2,3))  # this would put the generators in degrees 2, 2, 3, respectively. Coefficient ring would be the field with 3 elements.
sage: A.<a,b,c> = GradedCommutativeAlgebra(QQ)  # back to the basic example
sage: d = A.differential({a: 0, b: 0, c: a*b})  # define a differential sending a -> 0, b -> 0, c -> a*b
sage: C = A.cdg_algebra(d)  # define a differential graded algebra C = (A, d)
sage: C.cohomology_generators(10)  # what are the algebra generators for the cohomology, up to degree 10?
{1: [a, b], 2: [a*c, b*c]}

There are no Massey products here.

We can produce a Massey product:

sage: B.<a,b,c,x,y> = GradedCommutativeAlgebra(QQ)
sage: d_B = B.differential({a: 0, b: 0, c: 0, x: a*b, y: b*c})
sage: D = B.cdg_algebra(d_B)

At this point, D has five generators and two nontrivial products are boundaries: $d(x) = ab$, $d(y) = bc$. So there will be a Massey product $\langle a, b, c \rangle$, represented by $xc + ay$. Let's find the algebra generators for the cohomology:

sage: D.cohomology_generators(10)
{1: [a, b, c], 2: [a*x, b*x, c*x - a*y, b*y, c*y], 3: [b*x*y]}

You can see a generator $cx-ay$ in degree 2, representing the Massey product. You can modify these examples: change the coefficient ring, change the number and/or degrees of the generators, change the differential, and see the effects.

Some of this code will be available, at least temporarily, at https://sagecell.sagemath.org/?q=rjfjdb, if you want to experiment.