I'm working my way through Dae-Woong Lee's 2020 paper, Algabraic Inverses on Lie Algebra Comultiplications, and I'm coming to a case where it looks like a perturbation must be zero (which makes it an awfully poor perturbation). This is because it looks like the perturbation must be the sum of a set of lie brackets, each of which are zero.
To quote everything I think is important (in case the link goes bad), in this paper, Lee starts by defining two graded vector spaces, $A$ and $B$ and their associated Lie algebras $L(A)$ and $L(B)$. From there, they define the coproduct of these lie algebras, $L(A)\sqcup L(B)$ to be $L(A\oplus B)$. The exact meaning of this notation is not specified, but based on the way it is is used in the examples that follow, I presume it to be the direct sum of $L(A)$ and $L(B)$. After all, they use the phrase "the coproduct," so it seems reasonable to assume it is the obvious one.
Lee then focuses on a special case, $L(A)\sqcup L(A)$ where A has generators $a_1, a_2, \ldots, a_n$. To disambiguate between the two algebras, he puts a prime on the generators from the right side, so the generators of $L(A)\sqcup L(A)$ are the set $\{a_1, a_2, \ldots, a_n, a^\prime_1, a^\prime_2, \ldots, a^\prime_n\}$. Good so far, its just notation. They also mention "For notational convenience, we will make use of polynomials to indicate the Lie brackets $[ , ]$; that is $[a, b]=ab$ in the Lie algebras." So I am primed to recognize polynomials as Lie brackets.
I run into trouble at equation (1) when he defines a map, $\phi: L(A)\to L(A)\sqcup L(A)$ which he defines to be a perturbation something he calls a comultiplication of $L(A)$ He defines $$\phi(a_s)=a_s + a^\prime_s + P_s$$
The $a_s + a^\prime_s$ bit makes sense from his definition of a comultiplication (omitted here). What is giving me pause is his definition of $P_s$:
$P_s$ is the $s$th perturbation of $\phi$ consisting of a[sic] polynomials as Lie brackets in the set $\{a_1, a_2, \ldots, a_n, a^\prime_1, a^\prime_2, \ldots, a^\prime_n\}$, each term of which contains at least one generator in $\{a_1, a_2, \ldots, a_n\}$ and at least one generator in $\{a^\prime_1, a^\prime_2, \ldots, a^\prime_n\}$.
This is where I have trouble, terms in such a polynomial would be of the form $a_i a^\prime_j$, also written $[a_i, a^\prime_j]$ But these are part of a direct sum, so such a bracket must be zero (implicitly they are $[(a_i, 0), (0, a^\prime_j)] = ([a_i, 0], [(0, a^\prime_j]) = (0, 0)$), and the sum of zeros would be zero, so as best as I can read it, the perturbation itself must be zero. This makes for a poor perturbation, as the whole point is to make them vanishingly small but not zero. So I must have missed something.
I feel like this construction would make sense if $P_s$ was constructed in the underlying vector space of $L(A)\sqcup L(A)$, but the wording draws attention to the use of Lie brackets here. Is my understanding of the direct sum flawed?