Lie algebra, nilpotent

An algebra $ \mathfrak g $ over a field $ k $ that satisfies one of the following equivalent conditions:

1) there is a finite decreasing chain of ideals $ \{ \mathfrak g _ {i} \} _ {0 \leq i \leq n } $ of $ \mathfrak g $ such that $ \mathfrak g _ {0} = \mathfrak g $, $ \mathfrak g _ {n} = \{ 0 \} $ and $ [ \mathfrak g , \mathfrak g _ {i+1} ] \subset \mathfrak g _ {i+1} $ for $ 0 \leq i < n $;

2) $ C ^ {k} \mathfrak g = \{ 0 \} $( respectively, $ C _ {k} \mathfrak g = \mathfrak g $) for sufficiently large $ k $, where $ C ^ {k} \mathfrak g $ and $ C _ {k} \mathfrak g $ are the terms of the lower and upper central series, respectively;

3) there is a $ k $ such that $ \mathop{\rm ad} x _ {1} \dots \mathop{\rm ad} x _ {k} = 0 $ for any $ x _ {1} \dots x _ {k} \in \mathfrak g $.

An Abelian algebra is nilpotent. If $ V $ is a finite-dimensional vector space over $ k $ and $ F = \{ V _ {i} \} $ is a flag in it, then

$$ \mathfrak n ( F ) = \ \{ {x \in \mathop{\rm End} V } : {x V _ {i} \subset V _ {i-1} \ \textrm{ for all } i \geq 1 } \} $$

is a nilpotent subalgebra of the Lie algebra $ \mathfrak g \mathfrak l ( V) $ of all linear transformations of $ V $. If a basis is chosen in $ V $ that is compatible with the flag $ F $, then with respect to that basis the elements of the algebra $ \mathfrak n ( F ) $ are represented by upper-triangular matrices with zeros on the main diagonal. If $ F $ is a complete flag (that is, $ \mathop{\rm dim} V _ {k} = k $), then the corresponding nilpotent linear Lie algebra (cf. Lie algebra, linear) $ \mathfrak n ( m , k ) $ consists of all upper-triangular matrices of order $ m = \mathop{\rm dim} V $ with zeros on the main diagonal.

For any nilpotent Lie algebra the codimension of its commutator ideal is $ \mathop{\rm codim} [ \mathfrak g , \mathfrak g ] \geq 2 $ if $ \mathop{\rm dim} \mathfrak g > 1 $. In particular, if $ \mathop{\rm dim} \mathfrak g \leq 2 $, then $ \mathfrak g $ is Abelian. The unique non-Abelian three-dimensional nilpotent Lie algebra $ \mathfrak g $ is isomorphic to $ \mathfrak n ( 3 , K ) $. Nilpotent Lie algebras have been listed in a few small dimensions (for $ \mathop{\rm dim} \mathfrak g \leq 7 $ if $ k = \mathbf C $), but there is still (1989) no general approach to their classification.

Nilpotent Lie algebras (earlier they were called special Lie algebras or Lie algebras of rank 0) had already been encountered in the first stage of S. Lie's research on the integration of differential equations. The classification of solvable Lie algebras (cf. Lie algebra, solvable) reduces in a certain sense to the enumeration of nilpotent Lie algebras. In an arbitrary finite-dimensional Lie algebra there is a largest nilpotent ideal (the nil radical in the terminology of [2]). Another nilpotent ideal has also been considered — the intersection of the kernels of the irreducible finite-dimensional representations (the nilpotent radical, cf. also Representation of a Lie algebra) (see [1]), [4]). If $ \mathfrak r $ is the radical of the algebra $ \mathfrak g $, then the nilpotent radical $ \mathfrak n $ coincides with

$$ [ \mathfrak g , \mathfrak r ] = \ [ \mathfrak g , \mathfrak g ] \cap \mathfrak r . $$

The quotient algebra $ \mathfrak g / \mathfrak n $ is reductive (cf. Lie algebra, reductive) and $ \mathfrak n $ is the smallest ideal with this property. If $ \mathop{\rm char} k = 0 $, the nil radical consists of all $ x \in \mathfrak r $ such that $ \mathop{\rm ad} x $ is nilpotent.

In the study of reductive Lie algebras $ \mathfrak g $ over $ \mathbf C $, nilpotent subalgebras naturally arise, these are the nilpotent radicals of the parabolic subalgebras (cf. Parabolic subalgebra) of $ \mathfrak g $. In the case $ \mathfrak g = \mathfrak g \mathfrak l ( V) $ these nilpotent subalgebras coincide with the subalgebras $ \mathfrak n ( F ) $ considered above. The nilpotent radical of a Borel subalgebra (see Borel subgroup) of $ \mathfrak g $ is a maximal subalgebra of $ \mathfrak g $ that consists of nilpotent elements; it is unique up to conjugacy. A wider class of nilpotent Lie algebras is formed by arbitrary ideals of parabolic subalgebras of $ \mathfrak g $ consisting of nilpotent elements. In the case $ \mathfrak g = \mathfrak g \mathfrak l ( V) $ these nilpotent Lie algebras were classified in [6] (standard nil algebras), and in the general case in [7].

The centre of a nilpotent Lie algebra is non-trivial and any nilpotent Lie algebra can be obtained by a series of central extensions by means of nilpotent Lie algebras. The class of nilpotent Lie algebras is closed under transition to a subalgebra, a quotient algebra, a central extension, and a finite direct sum. In particular, any subalgebra of $ \mathfrak n ( n , K ) $ is nilpotent. Conversely, an arbitrary finite-dimensional nilpotent Lie algebra is isomorphic to a subalgebra of $ \mathfrak n ( m , K ) $ for some $ m $( if $ \mathop{\rm char} k = 0 $); this is a special case of Ado's theorem (see [1], [2]).

If $ \mathfrak g $ is an arbitrary finite-dimensional Lie algebra, then any nilpotent ideal of it is orthogonal to it with respect to the Killing form; in particular, for a nilpotent Lie algebra this form is trivial.

One of the main theorems in the theory of nilpotent Lie algebras is Engel's theorem: If $ \rho : \mathfrak g \rightarrow \mathfrak g \mathfrak l ( V) $ is a finite-dimensional representation of a nilpotent Lie algebra $ \mathfrak g $ and $ \rho ( x) $ is nilpotent for any $ x \in \mathfrak g $, then there is a complete flag $ F $ such that $ \rho ( \mathfrak g ) \subset \mathfrak n ( F ) $. Engel's theorem implies that a finite-dimensional Lie algebra $ \mathfrak g $ is nilpotent if and only if $ \mathop{\rm ad} ^ {n} x= 0 $ for some $ n $ and all $ x \in \mathfrak g $, that is, if any $ x \in \mathfrak g $ is nilpotent.

Engel's theorem contains a description of the nilpotent representations of nilpotent Lie algebras; the description of arbitrary finite-dimensional representations is due to H. Zassenhaus (see [2]): If the field $ k $ is algebraically closed and $ V $ is a finite-dimensional $ \mathfrak g $- module, then $ V = \oplus_{i=1}^ {n} V _ {i} $, where the submodules $ V _ {i} $ are such that the restriction of the action of any $ x \in \mathfrak g $ to them is the sum of a scalar operator and a nilpotent operator. If $ V $ is a finite-dimensional vector space over a field $ k $ of characteristic 0, then any algebraic nilpotent Lie algebra $ \mathfrak g \subset \mathfrak g \mathfrak l ( V) $ has the form $ \mathfrak g = \mathfrak a + \mathfrak n $, where $ \mathfrak a $ and $ \mathfrak n $ are the ideals consisting, respectively, of the semi-simple and the nilpotent linear transformations belonging to $ \mathfrak g $[5].

References

[1]	N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French)
[2]	N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979))
[3]	J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French)
[4]	, Theórie des algèbres de Lie. Topologie des groupes de Lie , Sem. S. Lie , Ie année 1954–1955 , Ecole Norm. Sup. (1955)
[5]	C. Chevalley, "Théorie des groupes de Lie" , 3 , Hermann (1955)
[6]	G.B. Gurevich, "Standard Lie algebras" Mat. Sb. , 35 (1954) pp. 437–460 (In Russian)
[7]	Yu.B. Khakimdzhanov, "Standard subalgebras of reductive Lie algebras" Vestn. Moskov. Univ. Mat. Mekh. : 6 (1974) pp. 49–55 (In Russian) (English abstract)

Comments

Let $ \mathfrak g $ be a Lie algebra. The lower central series of $ \mathfrak g $ consists of the ideals $ \mathfrak g ^ {0} = \mathfrak g $, $ \mathfrak g ^ {1} = [ \mathfrak g , \mathfrak g ] \dots \mathfrak g ^ {i+1} = [ \mathfrak g , \mathfrak g ^ {i} ] ,\dots $. It is also called the descending central series. The derived series is the series of ideals $ \mathfrak g ^ {(} 0) = \mathfrak g $, $ \mathfrak g ^ {(} 1) = [ \mathfrak g ^ {(} 0) , \mathfrak g ^ {(} 0) ] \dots \mathfrak g ^ {(} i+ 1) = [ \mathfrak g ^ {(} i) , \mathfrak g ^ {(} i) ] ,\dots $. The upper central series is defined by $ \mathfrak g _ {0} = $ centre of $ \mathfrak g = \{ {x \in \mathfrak g } : {[ x , y ] = 0 } \} $, and inductively $ \mathfrak g _ {i+1} $ is that ideal of $ \mathfrak g $ such that $ \mathfrak g _ {i+1} / \mathfrak g _ {i} $ is the centre of $ \mathfrak g / \mathfrak g _ {i} $.

References

[a1]	J.E. Humphreys, "Introduction to Lie algebras and representation theory" , Springer (1972)
[a2]	J.-P. Serre, "Algèbres de Lie semi-simples complexes" , Benjamin (1986)