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Let $k$ denote an algebraically closed field of characteristic $0$. Suppose $K=\bigoplus_{i\geq 0}K(i)$ is a Hopf $k$-algebra which admits a connected Hopf-grading (that is, a grading which is both an algebra and coalgebra grading, with $K(0)=k$). Call such a Hopf algebra a connected Hopf-graded Hopf algebra.

Connected Hopf-graded Hopf $k$-algebras arise naturally in algebraic toplogy when studying the cohomology rings (with coeffecients in $k$) of $H$-spaces. I assume (although I'm not 100% sure), that not all such Hopf algebras arise as cohomology algebras in this way. My question is therefore the following:

Let $K$ be a connected Hopf-graded Hopf algebra. Which additional properties on $K$ guarantee that it can be viewed as the cohomology ring $K^{*}(X;k)$ of some $H$-space $X$?

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Any Hopf algebra of the form $H^{\bullet}(X, k)$ is necessarily graded commutative, and in addition to the conditions you've given so far, the only remaining condition is a mild cardinality condition. (You do not need to assume that $k$ is algebraically closed, only that it has characteristic zero.)

Any such Hopf algebra $K^{\bullet}$ is isomorphic, as an algebra, to the symmetric algebra (in the graded sense) of some graded vector space $V = \bigoplus_{n \ge 1} V_n$ over $k$. If this graded vector space has a graded predual $W = \bigoplus_{n \ge 1} W_n$ (so that $V_n \cong W_n^{\ast}$), which in particular is the case if each $V_n$ is finite-dimensional, then this is the cohomology of the product of Eilenberg-MacLane spaces

$$X = \prod_{n \ge 1} K(W_n, n)$$

and now the comultiplication $\psi : K^{\bullet} \to K^{\bullet} \otimes K^{\bullet}$ induces an H-space structure $X \times X \to X$ in the obvious way. Otherwise, I think some fiddling with universal coefficients shows that no candidate $X$ exists.

This correspondence can be upgraded to an equivalence of categories. In the case $k = \mathbb{Q}$ see, for example, May and Ponto's More Concise Algebraic Topology, Theorem 9.1.4.

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  • $\begingroup$ Thanks! I must admit, I don't know much algebraic topology, my background is in ring theory/ Hopf algebras. Do you know of any situations in algebraic toplogy where noncommutatve graded Hopf algebras may arise? $\endgroup$
    – Paul Gilmartin
    Commented Oct 24, 2015 at 22:23
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    $\begingroup$ @Paul: noncommutative but (graded) cocommutative Hopf algebras arise as the homology, as opposed to the cohomology, of H-spaces. I don't know of any source in algebraic topology of Hopf algebras which are neither (graded) commutative nor cocommutative. $\endgroup$
    – Qiaochu Yuan
    Commented Oct 24, 2015 at 22:34
  • $\begingroup$ One can get lots of noncommutative Hopf-algebras from loop spaces, but they'll always be cocommutative. $\endgroup$
    – Jonathan Beardsley
    Commented Oct 25, 2015 at 1:23
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    $\begingroup$ Each cohomology group of a space is either finitely generated or uncountable. Wouldn't this give counterexamples to the claim that every commutative connected Hopf algebra arises as the cohomology ring of a space? $\endgroup$
    – Sebastian Goette
    Commented Oct 25, 2015 at 18:33
  • $\begingroup$ @Sebastian: oh, hmm. I'd somehow convinced myself that this wasn't a problem, but you're right. I need to modify the answer a bit. $\endgroup$
    – Qiaochu Yuan
    Commented Oct 25, 2015 at 18:35

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