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Qiaochu Yuan
Qiaochu Yuan
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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, I believe that's all you needthe 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 thesome graded vector space $V = \bigoplus_{n \ge 1} V_n$ over $k$. ThisIf 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 $X = \prod_{n \ge 1} K(V_n, n)$,

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

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 in fact 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.

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, I believe that's all you need. (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 the graded vector space $V = \bigoplus_{n \ge 1} V_n$ over $k$. This is the cohomology of the product of Eilenberg-MacLane spaces $X = \prod_{n \ge 1} K(V_n, n)$, 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.

This can in fact 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.

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.

Source Link
Qiaochu Yuan
Qiaochu Yuan
  • 118.2k
  • 40
  • 447
  • 741

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, I believe that's all you need. (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 the graded vector space $V = \bigoplus_{n \ge 1} V_n$ over $k$. This is the cohomology of the product of Eilenberg-MacLane spaces $X = \prod_{n \ge 1} K(V_n, n)$, 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.

This can in fact 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.