Questions of the paper "PBW-pairs of varieties of linear algebras"

Question

I am reading this paper "PBW-pairs of varieties of linear algebras", the link is here:https://www.tandfonline.com/doi/abs/10.1080/00927872.2012.720867.

At page 672, there is a definition of PBW-pair.

A pair of varieties $(\mathcal{V}, \mathcal{W})$ with a multiplication changing functor $\mathcal{K}: \mathcal{V} \rightarrow \mathcal{W}$ is a PBW-pair, if it satisfies the following property:

(PBW) for every $A \in \mathcal{W}$, the epimorphism $\pi: \mathcal{U}(Ab A) \rightarrow gr \mathcal{U}(A)$ is an isomorphism.

Then there is a proposition giving the equivalent definition of a PBW-pair.

Proposition: A pair $(\mathcal{V}, \mathcal{W})$ is a PBW-pair if and only if it satisfies the conditions:

1. The canonical mapping $i: A \rightarrow \mathcal{U}(A)$ is injective for every $A \in \mathcal{W}$;
2. If $A \in \mathcal{W}$ has a base $\{e_j, j\in J \}$, then $\mathcal{U}(A)$ has a base formed by certain monomials (normal words) on $i(e_j)$, where the definition of normal words does not depend on a particular algebra $A$ (but depends on the pair $(\mathcal{V}, \mathcal{W})$).

I have seen the proof of this proposition. It could be seen as four steps:

• (1) For a PBW-pair $(\mathcal{V}, \mathcal{W})$, prove that (1) of the Proposition holds for every abelian $A \in \mathcal{W}$;

• (2) For a PBW-pair $(\mathcal{V}, \mathcal{W})$, prove that (2) of the Proposition holds for every abelian $A \in \mathcal{W}$;

• (3) For a PBW-pair $(\mathcal{V}, \mathcal{W})$, prove that (1), (2) of the Proposition holds for every $A \in \mathcal{W}$;

• (4) If a pair $(\mathcal{V} ,\mathcal{W})$ satisfies conditions 1,2 of the Proposition, prove that $(\mathcal{V}, \mathcal{W})$ is a PBW-pair.

I have some places confused in the proof of this proposition. The places are at step (2) and (4).

Question1: In step (2), they take an abelian $\mathcal{W}$-algebra $E$ of countable rank with a base $\{ e_n, n \in \mathbf{N} \}$. Then they choose in $\mathcal{U}(E)$ a base formed by monomials (normal words) $f_{k}(e_{i_1}, \cdots, e_{i_{n_k}})$, $i_1 < i_2 < \cdots< i_{n_{k}}, k \in K$.

I have not find the definition of $f_{k}(e_{i_1}, \cdots , e_{i_{n_k}}), k \in K$. Could anyone explain what they are?

Also I am confused that how to know the monomials $f_{k}(e_{i_1}, \cdots, e_{i_{n_k}})$, $i_1 < i_2 < \cdots< i_{n_{k}}, k \in K$ form a base of $\mathcal{U}(E)$? (I know that in Lie algebra and associative algebra case, if Lie algebra $L$ has a base $\{ x_1, x_2, \cdots \}$, then $\mathcal{U}(L)$ has a base of the form $x_{i_1} x_{i_2} \cdots x_{i_n}$ with $i_1 < i_2 < \cdots < i_n$, but getting this base is a proof. However there in the case of varieties, I do not understand what $f_{k}(e_{i_1}, \cdots, e_{i_{n_k}})$ are, and I do not know how to prove they are a base of $\mathcal{U}(E)$ ).

Question2: In step (4), if a pair $(\mathcal{V}, \mathcal{W})$ satisfies condition 1, 2 of the Proposition, then $\mathcal{U}(A)$ and $\mathcal{U}(Ab A)$ have bases formed by the normal words which include a base of Vect A. Therefore, the epimorphism $\pi$ maps the base of normal words of $\mathcal{U}(Ab A)$ to the base of normal words of $gr \mathcal{U}(A)$ and hence is an isomorphism.

I still do not understand the concrete forms of normal words of $\mathcal{U}(A)$ and $\mathcal{U}(Ab A)$. I am also confused at that why the epimorphism $\pi$ maps the base of normal words of $\mathcal{Ab A}$ to the base of normal words of $gr \mathcal{U}(A)$, could anyone explain it? (I know the process of Lie algebra and associative algebra case, but in varies of algebras, I do not understand it.)

Thank you very much for your help.