algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...

, the Milnor–Moore theorem, introduced by classifies an important class of Hopf algebras, of the sort that often show up as

cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...

rings in

algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

. The theorem states: given a connected, graded,

cocommutative In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams ...

Hopf algebra Hopf is a German surname. Notable people with the surname include: *Eberhard Hopf (1902–1983), Austrian mathematician *Hans Hopf (1916–1993), German tenor *Heinz Hopf (1894–1971), German mathematician *Heinz Hopf (actor) (1934–2001), Swedis ...

''A'' over a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

of characteristic zero with

\dim A_n < \infty

for all ''n'', the natural Hopf algebra homomorphism :

U(P(A)) \to A

from the

universal enveloping algebra In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal enveloping algebras are used in the representati ...

of the graded

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...

P(A)

of primitive elements of ''A'' to ''A'' is an isomorphism. Here we say ''A'' is connected if

A_0

is the field and

A_n = 0

for negative ''n''. The universal enveloping algebra of a graded Lie algebra ''L'' is the quotient of the tensor algebra of ''L'' by the two-sided ideal generated by all elements of the form

xy- (-1)^yx -,y /math>.

In

, the term usually refers to the corollary of the aforementioned result, that for a pointed,

simply connected space In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...

''X'', the following isomorphism holds: :

U(\pi_(\Omega X) \otimes \Q) \cong H_(\Omega X;\Q),

where

\Omega X

denotes the

loop space In topology, a branch of mathematics, the loop space Ω''X'' of a pointed topological space ''X'' is the space of (based) loops in ''X'', i.e. continuous pointed maps from the pointed circle ''S''1 to ''X'', equipped with the compact-open topology ...

of ''X'', compare with Theorem 21.5 from . This work may also be compared with that of .

References

* * * * * *

External links

* Theorems about algebras Hopf algebras {{abstract-algebra-stub