In mathematics, a Malcev Lie algebra, or Mal'tsev Lie algebra, is a generalization of a rational nilpotent
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...
, and Malcev groups are similar. Both were introduced by , based on the work of .
Definition
According to a Malcev Lie algebra is a rational Lie algebra
together with a complete, descending
-vector space filtration
, such that:
*
*
* the associated graded Lie algebra
is generated by elements of degree one.
Applications
Relation to Hopf algebras
showed that Malcev Lie algebras and Malcev groups are both equivalent to complete
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), Sw ...
s, i.e., Hopf algebras ''H'' endowed with a
filtration
Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filte ...
so that ''H'' is isomorphic to
. The functors involved in these equivalences are as follows: a Malcev group ''G'' is mapped to the completion (with respect to the
augmentation ideal In algebra, an augmentation ideal is an ideal that can be defined in any group ring.
If ''G'' is a group and ''R'' a commutative ring, there is a ring homomorphism \varepsilon, called the augmentation map, from the group ring R /math> to R, defin ...
) of its
group ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the gi ...
Q''G'', with inverse given by the group of ''grouplike elements'' of a Hopf algebra ''H'', essentially those elements 1 + ''x'' such that
. From complete Hopf algebras to Malcev Lie algebras one gets by taking the (completion of)
primitive elements, with inverse functor given by the completion of 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 representa ...
.
This equivalence of categories was used by to prove that, after tensoring with Q, relative ''K''-theory K(''A'', ''I''), for a nilpotent ideal ''I'', is isomorphic to relative
cyclic homology In noncommutative geometry and related branches of mathematics, cyclic homology and cyclic cohomology are certain (co)homology theories for associative algebras which generalize the de Rham (co)homology of manifolds. These notions were independen ...
HC(''A'', ''I''). This theorem was a pioneering result in the area of
trace methods.
Hodge theory
Malcev Lie algebras also arise in the theory of
mixed Hodge structure
In algebraic geometry, a mixed Hodge structure is an algebraic structure containing information about the Cohomology#Cohomology of varieties, cohomology of general algebraic variety, algebraic varieties. It is a generalization of a Hodge structure, ...
s.
References
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*
*
*{{Citation , last1=Quillen , first1=Daniel , author1-link=Daniel Quillen , title=Rational homotopy theory , jstor=1970725 , mr=0258031 , year=1969 , journal=
Annals of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study.
History
The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as t ...
, series=Second Series , issn=0003-486X , volume=90 , issue=2 , pages=205–295 , doi=10.2307/1970725
Hodge theory
Lie algebras