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$\mathbb{Z}$-graded algebras and tensor products
Let $A = \bigoplus_{k \in \mathbb{Z}} A_k$ be a not necessarily commutative $\mathbb{Z}$-graded unital algebra over a field $\mathbb{K}$, and assume that it is strongly graded:
$$
A_kA_l = A_{k+l}.
$$
...