Milnor–Moore theorem

In <a href="/facts/Algebra/nMdqczjo">algebra</a>, the Milnor–Moore theorem, introduced by <a href="/facts/John_Milnor/LTf24paN">John W. Milnor</a> and <a href="/facts/John_Coleman_Moore/aqw1FFvY">John C. Moore</a> (1965) classifies an important class of <a href="/facts/Hopf_algebras/3QF6g0tT">Hopf algebras</a>, of the sort that often show up as <a href="/facts/Cohomology/J7FJzTNh">cohomology</a> rings in <a href="/facts/Algebraic_topology/2wA8yrjT">algebraic topology</a>.
The theorem states: given a connected, graded, <a href="/facts/Cocommutative/pvx52xS6">cocommutative</a> <a href="/facts/Hopf_algebra/3QF6g0tT">Hopf algebra</a> A over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> of <a href="/facts/Characteristic_(algebra)/D2VzcQaG">characteristic</a> zero with 
 
 
 
 dim
 ⁡
 
 A
 
 n
 
 
 <
 ∞
 
 
 {\displaystyle \dim A_{n}<\infty }
 
 for all n, the natural Hopf algebra homomorphism

U
        (
        P
        (
        A
        )
        )
        →
        A
      
    
    {\displaystyle U(P(A))\to A}

from the <a href="/facts/Universal_enveloping_algebra/VJ3Syz4W">universal enveloping algebra</a> of the graded <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a> 
 
 
 
 P
 (
 A
 )
 
 
 {\displaystyle P(A)}
 
 of <a href="/facts/Primitive_elements_(Hopf_algebra)/2QKxzuWk">primitive elements</a> of A to A is an isomorphism. Here we say A is connected if 
 
 
 
 
 A
 
 0
 
 
 
 
 {\displaystyle A_{0}}
 
 is the field and 
 
 
 
 
 A
 
 n
 
 
 =
 0
 
 
 {\displaystyle A_{n}=0}
 
 for negative n. The universal enveloping algebra of a graded Lie algebra L is the quotient of the <a href="/facts/Tensor_algebra/H7CKKryf">tensor algebra</a> of L by the two-sided ideal generated by all elements of the form 
 
 
 
 x
 y
 −
 (
 −
 1
 
 )
 
 
 |
 
 x
 
 |
 
 
 |
 
 y
 
 |
 
 
 
 y
 x
 −
 [
 x
 ,
 y
 ]
 
 
 {\displaystyle xy-(-1)^{|x||y|}yx-[x,y]}
 
.
In <a href="/facts/Algebraic_topology/2wA8yrjT">algebraic topology</a>, the term usually refers to the corollary of the aforementioned result, that for a <a href="/facts/Pointed_space/V4DTgRCd">pointed</a>, <a href="/facts/Simply_connected_space/xFlta63J">simply connected space</a> X, the following isomorphism holds:

U
        (
        
          π
          
            ∗
          
        
        (
        Ω
        X
        )
        ⊗
        
          Q
        
        )
        ≅
        
          H
          
            ∗
          
        
        (
        Ω
        X
        ;
        
          Q
        
        )
        ,
      
    
    {\displaystyle U(\pi _{\ast }(\Omega X)\otimes \mathbb {Q} )\cong H_{\ast }(\Omega X;\mathbb {Q} ),}

where 
 
 
 
 Ω
 X
 
 
 {\displaystyle \Omega X}
 
 denotes the <a href="/facts/Loop_space/S5xkdCkx">loop space</a> of X, 
compare with Theorem 21.5 from Félix, Halperin & Thomas (2001). This work may also be compared with that of (Halpern 1958a, 1958b). Here the multiplication on the right hand side induced by the product 
 
 
 
 Ω
 X
 ×
 Ω
 X
 →
 Ω
 X
 
 
 {\displaystyle \Omega X\times \Omega X\rightarrow \Omega X}
 
, and then by the Eilenberg-Zilber multiplication 
 
 
 
 
 C
 
 ∗
 
 
 (
 Ω
 X
 )
 ×
 
 C
 
 ∗
 
 
 (
 Ω
 X
 )
 →
 
 C
 
 ∗
 
 
 (
 Ω
 X
 )
 
 
 {\displaystyle C_{*}(\Omega X)\times C_{*}(\Omega X)\rightarrow C_{*}(\Omega X)}
 
.
On the left hand side, since 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is simply connected, 
 
 
 
 
 π
 
 ∗
 
 
 (
 Ω
 X
 )
 ⊗
 
 Q
 
 
 
 {\displaystyle \pi _{\ast }(\Omega X)\otimes \mathbb {Q} }
 
 is a 
 
 
 
 
 Q
 
 
 
 {\displaystyle \mathbb {Q} }
 
-vector space; the notation 
 
 
 
 U
 (
 V
 )
 
 
 {\displaystyle U(V)}
 
 stands for the universal enveloping algebra.

<ul><li><a href="/facts/John_Milnor/LTf24paN">Milnor, John W.</a>; Moore, John C. (1965). <a href="https://doi.org/10.2307/1970615">"On the structure of Hopf algebras"</a>. <a href="/facts/Annals_of_Mathematics/CUYktUqD">Annals of Mathematics</a>. 81 (2): 211–264. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1970615">10.2307/1970615</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1970615">1970615</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0174052">0174052</a>.</li>
<li><a href="/facts/Spencer_Bloch/xywM6bIw">Bloch, Spencer</a>. <a href="https://web.archive.org/web/20100610114451/http://math.uchicago.edu/~mitya/bloch-hopf/hopf3.pdf">"Lecture 3 on Hopf algebras"</a> (PDF). Archived from <a href="http://www.math.uchicago.edu/~mitya/bloch-hopf/hopf3.pdf">the original</a> (PDF) on 2010-06-10. Retrieved 2014-07-18.</li>
<li><a href="/facts/Spencer_Bloch/xywM6bIw">Spencer Bloch</a>, "Three Lectures on Hopf algebras and Milnor–Moore theorem". Notes by Mitya Boyarchenko.</li>
<li>Félix, Yves; <a href="/facts/Stephen_Halperin/2VySVfGV">Halperin, Stephen</a>; <a href="/facts/Jean-Claude_Thomas/2fDrobXP">Thomas, Jean-Claude</a> (2001). Rational homotopy theory. Graduate Texts in Mathematics. Vol. 205. New York: Springer-Verlag. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-1-4613-0105-9">10.1007/978-1-4613-0105-9</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-95068-0. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1802847">1802847</a>. (<a href="https://www.amazon.com/Rational-Homotopy-Theory-Graduate-Mathematics/dp/0387950680">Book description and contents at the Amazon web page</a>)</li>
<li>Halpern, Edward (1958a), "Twisted polynomial hyperalgebras", <a href="/facts/Memoirs_of_the_American_Mathematical_Society/L6BHqgsE">Memoirs of the American Mathematical Society</a>, 29: 61 pp, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0104225">0104225</a></li>
<li>Halpern, Edward (1958b), "On the structure of hyperalgebras. Class 1 Hopf algebras", <a href="/facts/Portugaliae_Mathematica/qFvMakRN">Portugaliae Mathematica</a>, 17 (4): 127–147, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0111023">0111023</a></li>
<li><a href="/facts/J._Peter_May/GPbRcPTW">May, J. Peter</a> (1969). <a href="http://www.math.uchicago.edu/~may/PAPERS/9.pdf">"Some remarks on the structure of Hopf algebras"</a> (PDF). <a href="/facts/Proceedings_of_the_American_Mathematical_Society/W3wRF3MV">Proceedings of the American Mathematical Society</a>. 23 (3): 708–713. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2036615">10.2307/2036615</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2036615">2036615</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0246938">0246938</a>.(Broken link)</li></ul>

Milnor–Moore theorem open-in-new

Milnor–Moore theorem