View source for Malcev algebra

{{for|the Lie algebras or groups |Malcev Lie algebra}}
In [[mathematics]], a '''Malcev algebra''' (or '''Maltsev algebra''' or '''[[Ruth Moufang|Moufang]]–[[Sophus Lie|Lie]] algebra''') over a [[field (mathematics)|field]] is a [[nonassociative algebra]] that is antisymmetric, so that

:<math>xy = -yx</math>

and satisfies the '''Malcev identity'''

:<math>(xy)(xz) = ((xy)z)x + ((yz)x)x + ((zx)x)y.</math>

They were first defined by [[Anatoly Maltsev]] (1955).

Malcev algebras play a role in the theory of [[Moufang loop]]s that generalizes the role of [[Lie algebras]] in the theory of [[group (mathematics)|group]]s.  Namely, just as the tangent space of the identity element of a [[Lie group]] forms a Lie algebra, the tangent space of the identity of a smooth Moufang loop forms a Malcev algebra.  Moreover, just as a Lie group can be recovered from its Lie algebra under certain supplementary conditions, a smooth Moufang loop can be recovered from its Malcev algebra if certain supplementary conditions hold.  For example, this is true for a connected, simply connected real-analytic Moufang loop.<ref>{{cite journal |url=http://www.heldermann-verlag.de/jlt/jlt03/NAGYLAT.PDF|title= Moufang loops and Malcev algebras|first1= Peter T. |last1=Nagy |journal=Seminar Sophus Lie|volume=3|year=1992|pages=65–68|citeseerx=10.1.1.231.8888}}</ref>

==Examples==
*Any [[Lie algebra]] is a Malcev algebra.
*Any [[alternative algebra]] may be made into a Malcev algebra by defining the Malcev product to be ''xy''&nbsp;&minus;&nbsp;''yx''.
*The 7-sphere may be given the structure of a smooth Moufang loop by identifying it with the unit [[octonions]].  The tangent space of the identity of this Moufang loop may be identified with the 7-dimensional space of imaginary octonions.  The imaginary octonions form a Malcev algebra with the Malcev product ''xy''&nbsp;&minus;&nbsp;''yx''.

== Kernel ==
{{unreferenced section|date=April 2025}}
In the case of Malcev algebras, this construction can be simplified. Every Malcev algebra has a special [[neutral element]] (the [[zero vector]] in the case of [[vector space]]s, the [[identity element]] in the case of [[commutative group]]s, and the [[zero element]] in the case of [[ring (mathematics)|ring]]s or modules). The characteristic feature of a Malcev algebra is that we can recover the entire equivalence relation ker ''f'' from the [[equivalence class]] of the neutral element.

To be specific, let ''A'' and ''B'' be Malcev algebraic structures of a given type and let ''f'' be a homomorphism of that type from ''A'' to ''B''. If ''e''<sub>''B''</sub> is the neutral element of ''B'', then the ''kernel'' of ''f'' is the [[preimage]] of the [[singleton set]] {''e''<sub>''B''</sub>}; that is, the [[subset]] of ''A'' consisting of all those elements of ''A'' that are mapped by ''f'' to the element ''e''<sub>''B''</sub>.
The kernel is usually denoted {{nowrap|ker ''f''}} (or a variation). In symbols:
: <math> \operatorname{ker} f = \{a \in A : f(a) = e_{B}\} .</math>

Since a Malcev algebra homomorphism preserves neutral elements, the identity element ''e''<sub>''A''</sub> of ''A'' must belong to the kernel. The homomorphism ''f'' is injective if and only if its kernel is only the singleton set {{mset|''e''<sub>''A''</sub>}}.

The notion of [[ideal (ring theory)|ideal]] generalises to any Malcev algebra (as [[linear subspace]] in the case of vector spaces, [[normal subgroup]] in the case of groups, two-sided ideals in the case of rings, and [[submodule]] in the case of [[module (algebra)|module]]s). 
It turns out that ker ''f'' is not a [[subalgebra]] of ''A'', but it is an ideal.
Then it makes sense to speak of the [[quotient (universal algebra)|quotient algebra]] {{nowrap|''G'' / (ker ''f'')}}.
The first isomorphism theorem for Malcev algebras states that this quotient algebra is naturally isomorphic to the image of ''f'' (which is a subalgebra of ''B'').

The connection between this and the congruence relation for more general types of algebras is as follows.
First, the kernel-as-an-ideal is the equivalence class of the neutral element ''e''<sub>''A''</sub> under the kernel-as-a-congruence. For the converse direction, we need the notion of [[quotient]] in the Mal'cev algebra (which is [[division (mathematics)|division]] on either side for groups and [[subtraction]] for vector spaces, modules, and rings).
Using this, elements ''a'' and ''b'' of ''A'' are equivalent under the kernel-as-a-congruence if and only if their quotient ''a''/''b'' is an element of the kernel-as-an-ideal.

==See also==

*[[Malcev-admissible algebra]]

==Notes==
{{reflist}}

==References==
{{refbegin}}
*{{citation |first1=Alberto |last1=Elduque |first2=Hyo C. |last2=Myung |title=Mutations of alternative algebras |publisher=Kluwer |year=1994 |isbn=0-7923-2735-7}}
*{{springer|id=M/m062170|first1=V.T. |last1=Filippov|title=Mal'tsev algebra}}
*{{Citation | last1=Mal'cev | first1=A. I. | title=Analytic loops |mr=0069190 | year=1955 | journal=Mat. Sb. |series=New Series | volume=36 | pages=569–576|language=Russian | issue=78}}
{{refend}}

[[Category:Non-associative algebras]]
[[Category:Lie algebras]]


{{algebra-stub}}