How to multiply the elements of each site of tensor products to each other while using symbolic representation?

Question

If I have, say three sites in a symbolic operator which is of the form A = a⊗1⊗c and I want it to act on another operator B = a1⊗b⊗1 in order to find the commutator A.B – B.A, the result I expect would be A.B = a.a1⊗b⊗c, but I am unable to achieve such a result using the regular A.B command. A.B gives a⊗c.a1⊗b. I would also like to know how I can expand the dot product to multiple terms if they exist. Any help would be appreciated.

I tried the following:

A = TensorProduct[a, 1, c]

B = TensorProduct[a1, b, 1]

A.B

The output it gives is a⊗c.a1⊗b but the output I expect is a.a1⊗b⊗c

products symbolic tensors

I tried the following:

A = TensorProduct[a, 1, c]

B = TensorProduct[a1, b, 1]

A.B

The output it gives is a&otimes;c.a1&otimes;b but the output I expect is a.a1&otimes;b&otimes;c

Carl Woll · Accepted Answer

KroneckerProduct will work with symbolic tensors, although it needs some help from TensorExpand. First, set some assumptions on your tensors:
$Assumptions = (a | a1 | b | c) ∈ Matrices[{n, n}];

Then, your example is:
A = KroneckerProduct[a, IdentityMatrix[n], c];
B = KroneckerProduct[a1, b, IdentityMatrix[n]];

A.B

KroneckerProduct[a, IdentityMatrix[n], c].KroneckerProduct[a1, b,
IdentityMatrix[n]]

Using TensorExpand produces your desired result:
TensorExpand[A.B]

KroneckerProduct[a.a1, b, c]

How to multiply the elements of each site of tensor products to each other while using symbolic representation?

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