6
$\begingroup$

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 $X$?

$\endgroup$
5
$\begingroup$

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.

$\endgroup$
6
  • $\begingroup$ 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 Oct 24 '15 at 22:23
  • 2
    $\begingroup$ @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 Oct 24 '15 at 22:34
  • $\begingroup$ One can get lots of noncommutative Hopf-algebras from loop spaces, but they'll always be cocommutative. $\endgroup$ – Jonathan Beardsley Oct 25 '15 at 1:23
  • 2
    $\begingroup$ 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 Oct 25 '15 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 Oct 25 '15 at 18:35

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.