# Checking the antipode for a dg-Hopf algebra on algebra generators

If $$H$$ is a bialgebra and $$S \colon H \to H$$ is an algebra antihomomorphism such that the identities $$$$m \circ (S \otimes {\operatorname{id}}) \circ \Delta(x) = \varepsilon(x) 1_H \quad \text{and} \quad m \circ ({\operatorname{id}} \otimes S) \circ \Delta(x) = \varepsilon(x) 1_H \tag{1}$$$$ hold on an algebra generating set of $$H$$ then it follows that $$S$$ is already an antipode for $$H$$, making it a Hopf algebra.

How does this criterion generalize to differential graded Hopf algebras?

(I want to use this generalization to show that for a dg-vector space $$V$$ the induced dg-bialgebra structure on $$\operatorname{T}(V)$$, for which $$V$$ consists of primitive elements, is a dg-Hopf algebra.)

My attempt: It tried to show that for a dg-bialgebra $$H$$ and a homomorphism of dg-algebras $$S \colon H \to H^{\mathrm{op}}$$ the identities $$(1)$$ only need to be checked on an algebra generating set of $$H$$. For this I tried to check that the sets \begin{align*} H' &= \{ x \in H \mid \Delta \circ (S \otimes {\operatorname{id}}) \circ m(x) = \varepsilon(x) 1_H \} \,, \\ H'' &= \{ x \in H \mid \Delta \circ ({\operatorname{id}} \otimes S) \circ m(x) = \varepsilon(x) 1_H \} \end{align*} are subalgebras of $$H$$ (since this is what is done in the non-dg case). But this doesn’t seem to work: We have for all $$x, y \in H$$ that \begin{align*} m \circ (S \otimes {\operatorname{id}}) \circ \Delta(xy) &= \sum_{(xy)} S( (xy)_{(1)} ) (xy)_{(2)} \\ &= \sum_{(x),(y)} (-1)^{|x_{(2)}| |y_{(1)}|} S( x_{(1)} y_{(1)} ) x_{(2)} y_{(2)} \\ &= \sum_{(x), (y)} (-1)^{|x_{(2)}| |y_{(1)}| + |x_{(1)}| |y_{(1)}|} S( y_{(1)} ) S( x_{(1)} ) x_{(2)} y_{(2)} \\ &= \sum_{(x), (y)} (-1)^{|x| |y_{(1)}|} S( y_{(1)} ) S( x_{(1)} ) x_{(2)} y_{(2)} \,. \end{align*} The first occuring sign comes from the identity $$\Delta(xy) = \sum_{(x),(y)} (-1)^{|x_{(2)}| |y_{(1)}|} x_{(1)} y_{(1)} \otimes x_{(2)} y_{(2)}$$ (which comes from $$\Delta \colon H \to H \otimes H$$ being a homomorphism of dg-algebras), and the second sign comes from $$S$$ being a homomorphism of dg-algebras into $$H^{\mathrm{op}}$$ (whose multiplication $$*$$ is given by $$x*y = (-1)^{|x| |y|} y x$$). I had hoped that the appearing signs cancel out, so that one can use for $$x, y \in H'$$ the equalities $$\sum_{(x)} S(x_{(1)}) x_{(2)} = \varepsilon(x) 1_H$$ and $$\sum_{(y)} S(y_{(1)}) y_{(2)} = \varepsilon(y) 1_H$$. But this does not seem to work.