# Hopf algebra in derived category vector spaces

Let $$H$$ be a complex of vector spaces over some field $$k$$ which is endowed with the structure of a Hopf algebra object. I have heard several times that if $$H$$ is concentrated in positive or negative degrees, then the cohomology in degree zero of $$H$$ is a Hopf algebra in the usual sense. How does one prove such a statement and why is the condition on the degrees important?

• When restricted to these subcategories, the functor $H^0(-)$ is symmetric monoidal and so preserves the Hopf algebra structures. I'm really not sure that this is a research-level question though. – Denis Nardin May 6 at 11:30
• Could you actually explain your answer? – hopfology May 6 at 16:28
• (I've edited the previous comment in response to a flag.) Which part needs explaining? Do you know what a symmetric monoidal ($k$-linear) functor is? Do you understand that if a concept (such as the notion of Hopf algebra object) is definable in the language of symmetric monoidal $k$-linear categories, then a symmetric monoidal $k$-linear functor will map models of that concept in the domain category to ones in the codomain? – Todd Trimble Jun 8 at 11:26

If $$C$$ is a graded coalgebra (e.g. $$C$$= the homology of a d.g. Hopf algebra), then $$C_0$$ is not necesarily a subcoalgebra, because
$$\Delta(C_0)\subset (C\otimes C)_0=\oplus_{n\in\mathbb Z}C_n\otimes C_{-n}$$
For example, $$H=k\{x,y\}/(x^2=y^2=xy+yx)$$ is a graded Hopf algebra with $$|x|=1$$ and $$|y|=-1$$, both $$x$$ and $$y$$ primitives (in particular a d.g. Hopf algebra with $$d=0$$ and agree with its homology).
$$H_0=k\oplus kx\wedge y$$, and $$\Delta(x\wedge y)= (x\otimes 1+1\otimes x)(y\otimes 1+1\otimes y)$$ $$= x\wedge y\otimes 1+1\otimes x\wedge y+ x\otimes y-y\otimes x \notin H_0\otimes H_0$$
Of course if $$C=\oplus_{n\geq 0} C_n$$, then $$C_n=0$$ for $$n<0$$ and $$C_{-n}=0$$ for $$n>0$$, so, the only nonzero summand in $$\oplus_{n\in\mathbb Z}C_n\otimes C_{-n}$$ is $$C_0\otimes C_0$$.