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If $H$ is a bialgebra and $S \colon H \to H$ is an algebra antihomomorphism such that the identities \begin{equation} 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} \end{equation} 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.

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