Mal'tsev algebra

Moufang–Lie algebra

An algebra over a field satisfying the identities

$$ x ^ {2} = 0 ,\ \ J ( x , y , x z ) = \ J ( x , y , z ) x , $$

where $ J ( x , y , z ) = ( x y ) z + ( z x ) y + ( y z ) x $ is the Jacobian of $ x , y , z $. Mal'tsev algebras are a natural generalization of Lie algebras. Any Mal'tsev algebra is a binary Lie algebra.

Mal'tsev algebras were introduced by A.I. Mal'tsev [1], who called them Moufang–Lie algebras because of the connection with analytic Moufang loops (cf. Moufang loop). The tangent algebra of a locally analytic Moufang loop is a Mal'tsev algebra. The converse is also true: Any finite-dimensional Mal'tsev algebra over a complete normed field of characteristic zero is the tangent algebra of some locally analytic Moufang loop.

There is a close connection between Mal'tsev algebras and alternative algebras (see Alternative rings and algebras). The commutator algebra of an arbitrary alternative algebra, that is, the algebra obtained by replacing the original multiplication by the commutator operation

$$ [ x , y ] = x y - y x , $$

is a Mal'tsev algebra.

Every simple Mal'tsev algebra (cf. Simple algebra) of characteristic $ p \neq 2 $ is either a Lie algebra or is a $ 7 $- dimensional algebra over its centroid. Every primary Mal'tsev algebra (for $ p \neq 2 $, cf. also Primary ring) is either a Lie algebra or can be imbedded as a subring in a suitable $ 7 $- dimensional simple algebra over some field. An arbitrary semi-primary Mal'tsev algebra (for $ p \neq 2 $) can be isomorphically imbedded as a subalgebra in the commutator algebra of an alternative algebra. The question of imbedding an arbitrary Mal'tsev algebra in the commutator algebra of an alternative algebra is open (1989).

Let $ Z ( A) $ be the Lie centre of a Mal'tsev algebra $ A $:

$$ Z ( A) = \ \{ {n \in A } : {J ( n , a , b ) = 0 \ \textrm{ for all } a , b \in A } \} . $$

For any ideal $ I $ of a semi-primary Mal'tsev algebra $ A $( for $ p \neq 2 $), $ Z ( I) = Z ( A) \cap I $.

The properties of an algebraic Mal'tsev algebra (cf. Algebraic algebra) are similar to the properties of an algebraic Lie algebra (cf. Lie algebra, algebraic). In any algebraic Mal'tsev algebra (for $ p \neq 2 $) there is a locally finite radical, that is, a maximal locally finite ideal such that the quotient algebra with respect to it does not contain locally finite ideals. A Mal'tsev algebra of characteristic $ p \geq n $ or $ p = 0 $ satisfying the $ n $- th Engel condition (see Engel algebra) is locally nilpotent (cf. Locally nilpotent algebra). The difference between Mal'tsev algebras and Lie algebras manifests itself in the passage from local nilpotency to global. There is, for example, a Mal'tsev algebra $ ( p = 0 ) $ satisfying the third Engel condition and which is solvable of index 2, but is not nilpotent (cf. Nilpotent algebra).

For a Mal'tsev algebra there is an analogue of Engel's theorem, which played a major role in the structure theory of Lie algebras: A Mal'tsev algebra satisfying the Engel condition and the maximum condition for subalgebras is nilpotent. This result also holds in the more general case of binary Lie algebras.

In every free Mal'tsev algebra (for $ p \neq 2 $) there is a non-zero Lie centre. A free Mal'tsev algebra (for $ p \neq 2 $) with three or more generators is not a primary algebra. A free Mal'tsev algebra (for $ p \neq 2 $) with nine or more generators contains trivial ideals.

If $ R _ {n} $ is the variety of Mal'tsev algebras generated by the free Mal'tsev algebra on $ n $ generators and $ p = 0 $, then the chain of varieties

$$ R _ {1} \subseteq R _ {2} \subseteq \dots $$

does not stabilize at any finite stage.

The theory of finite-dimensional Mal'tsev algebras and their representations is well-developed. The fundamental results are similar to the results in the theory of Lie algebras. There are analogues of Lie's classical theorems: if $ \rho $ is a split representation of a solvable Mal'tsev algebra of characteristic $ 0 $, then all matrices $ \rho ( x) $ can be simultaneously reduced to triangular form; if $ \rho $ is a split representation of a nilpotent Mal'tsev algebra on a space $ V $, then $ V $ decomposes into a direct sum of weight subspaces $ V _ \alpha $, and the matrices of the bounded operators $ \rho ( x) $ in $ V _ \alpha $ can be simultaneously reduced to triangular form with the numbers $ \alpha ( x) $ on the main diagonal (cf. Representation of a Lie algebra).

The following results are similar to Cartan's solvability criterion and semi-simplicity of Lie algebras: if $ \rho $ is a faithful representation of a Mal'tsev algebra $ A $( $ p = 0 $) and if the bilinear form on $ A $ associated with $ \rho $ is trivial, then $ A $ is solvable; if $ \rho $ is a representation of a semi-simple Mal'tsev algebra, then the trace form associated with $ \rho $ is non-degenerate. If the Killing form of $ A $ is non-degenerate, then $ A $ is semi-simple.

Any representation of a semi-simple Mal'tsev algebra with $ p = 0 $ is completely reducible (cf. Reducible representation). If $ S $ is the radical (maximal solvable ideal) of a Mal'tsev algebra $ A $, $ N $ the nil radical (maximal nilpotent ideal), then for any derivation (cf. Derivation in a ring) $ D $ on $ A $, $ S D \subseteq N $.

An arbitrary finite-dimensional Mal'tsev algebra $ A $ of characteristic zero is the direct sum (as linear spaces) of its radical $ S $ and a semi-simple subalgebra $ B $ isomorphic to the quotient algebra of $ A $ by $ S $. Two semi-simple factors are conjugate by inner automorphisms (an analogue of the Levi–Mal'tsev–Harish-Chandra theorem for Lie algebras).

References

[1]	A.I. Mal'tsev, "Analytic loops" Mat. Sb. , 36 : 3 (1955) pp. 569–576 (In Russian)
[2]	A. Sagle, "Malcev algebras" Trans. Amer. Math. Soc. , 101 : 3 (1961) pp. 426–458
[3]	E.N. Kuz'min, "Algebraic sets in Mal'tsev algebras" Algebra and Logic , 7 : 2 (1968) pp. 95–97 Algebra i Logika , 7 : 2 (1968) pp. 42–47
[4]	E.N. Kuz'min, "Mal'tsev algebras and their representations" Algebra and Logic , 7 : 4 (1968) pp. 233–244 Algebra i Logika , 7 : 4 (1968) pp. 48–69
[5]	E.N. Kuz'min, "On the relation between Mal'tsev algebras and analytic Moufang loops" Algebra and Logic , 10 : 1 (1971) pp. 1–14 Algebra i Logika , 10 : 1 (1971) pp. 3–22
[6]	E.N. Kuz'min, "Levi's theorem for Mal'tsev algebras" Algebra and Logic , 16 : 4 (1977) pp. 286–291 Algebra i Logika , 16 : 4 (1977) pp. 424–431
[7]	V.T. Filippov, "On Engelian Mal'tsev algebras" Algebra and Logic , 15 : 1 (1976) pp. 57–71 Algebra i Logika , 15 : 1 (1976) pp. 89–109
[8]	V.T. Filippov, "Mal'tsev algebras" Algebra and Logic , 16 : 1 (1977) pp. 70–74 Algebra i Logika , 16 : 1 (1977) pp. 101–108
[9]	A.N. Grishkov, "Analogues of Levi's theorem for Mal'tsev algebras" Algebra and Logic , 16 : 4 (1977) pp. 260–265 Algebra i Logika , 16 : 4 (1977) pp. 389–396
[10]	I.P. Shestakov, "A problem of Shirshov" Algebra and Logic , 16 : 2 (1977) pp. 153–166 Algebra i Logika , 16 : 2 (1977) pp. 227–246

Comments

For an arbitrary algebra $ A $ over a field $ F $, the centroid $ E $ of $ A $ is the set of elements of the module endomorphisms $ \mathop{\rm Hom} _ {F} ( A, A) $ of $ A $ over $ F $ which commute for all $ x, y \in A $ with the left and right multiplications $ L _ {x} , R _ {y} \in \mathop{\rm Hom} _ {F} ( A, A) $. If $ A $ is simple over $ F $, $ E $ is an extension field of $ F $, .

References

[a1]	A.A. Sagle, "Simple Malčev algebras over a field of characteristic zero" Pacific J. Math. , 12 (1962) pp. 1057–1078
[a2]	N. Jacobson, "Lie algebras" , Interscience (1962) pp. 291 ((also: Dover, reprint, 1979))