View source for Lie algebra

A unitary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l0583701.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l0583702.png" /> over a commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l0583703.png" /> with a unit that is endowed with a bilinear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l0583704.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l0583705.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l0583706.png" /> having the following two properties:

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l0583707.png" /> (hence the anti-commutative law <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l0583708.png" />);

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l0583709.png" /> (the Jacobi identity).

Thus, a Lie algebra is an algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837010.png" /> (usually not associative); in the usual way one defines the concepts of a subalgebra, an ideal, a quotient algebra, and a homomorphism of Lie algebras. A Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837011.png" /> is said to be commutative if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837012.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837013.png" />.

The most important case is that in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837014.png" /> is a field (especially when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837015.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837016.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837017.png" /> is a vector space (of finite or infinite dimension) over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837018.png" />.

Lie algebras appeared in mathematics at the end of the 19th century in connection with the study of Lie groups (cf. [[Lie group|Lie group]], see also [[Lie group, local|Lie group, local]]; [[Lie transformation group|Lie transformation group]]; [[Lie theorem|Lie theorem]]), and in implicit form somewhat earlier in mechanics. The common prerequisite for such a concept to arise was the concept of an  "infinitesimal transformation" , which goes back at least to the time of the origin of infinitesimal calculus. The fact that integrals of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837019.png" /> of the Hamilton equation are closed with respect to the Poisson brackets, which satisfy the Jacobi identity, was one of the earliest observations to be expressed properly in the language of Lie algebras (see [[#References|[8]]], [[#References|[10]]]). The term  "Lie algebra"  itself was introduced by H. Weyl in 1934 (up to this time the terms  "infinitesimal transformations of the group in question"  or  "infinitesimal group"  had been used). In the course of time the role of Lie algebras increased in proportion to the place taken by Lie groups in mathematics (especially in geometry), and also in classical and quantum mechanics. In the first place this is explained by the special place of Lie algebras among many other varieties of universal algebras. Presently (1980-s) the apparatus of Lie algebras has been perceived not only as a useful and powerful tool in the linearization of group-theoretic problems (whether in the theory of Lie groups or in the theory of algebraic groups, cf. [[Algebraic group|Algebraic group]], which to a significant extent absorbs it and extremely outgrows it, or in the theory of finite groups, cf. [[Finite group|Finite group]], which rather stands by itself), but also as the source of beautiful and difficult problems in linear algebra.

There are several natural sources that provide important examples of Lie algebras.

1) In the framework of general algebra the significance of Lie algebras is determined first of all by the fact that the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837020.png" /> of all derivations (cf. [[Derivation in a ring|Derivation in a ring]]) of any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837021.png" />-algebra is a Lie algebra with the operation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837022.png" /></td> </tr></table>

The derivations of a Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837023.png" /> of the form

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are called inner derivations or adjoint transformations. In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837025.png" /> they form a subalgebra, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837026.png" />, and the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837027.png" /> is a homomorphism of Lie algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837028.png" /> (the adjoint representation of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837029.png" />); its image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837030.png" /> is isomorphic to the quotient algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837031.png" /> with respect to its centre

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837032.png" /></td> </tr></table>

2) Another important source of Lie algebras is connected with the following simple observation. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837033.png" /> is an associative algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837034.png" /> (cf. [[Associative rings and algebras|Associative rings and algebras]]) with multiplication <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837035.png" />, then the multiplication in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837036.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837037.png" /> specified by the rule

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837038.png" /></td> </tr></table>

endows <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837039.png" /> with the structure of a Lie algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837040.png" />. One says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837041.png" /> is the Lie algebra associated with the associative algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837042.png" />. Thus, the classical example of a Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837043.png" /> is obtained if for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837044.png" /> one takes the (associative) algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837045.png" /> of all square matrices of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837046.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837047.png" />.

The following four infinite series of subalgebras of a Lie algebra of this type are called classical (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837048.png" /> is a field of characteristic zero):

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837049.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837050.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837051.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837052.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837053.png" /></td> </tr></table>

One has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837057.png" />.

A remarkable result is that over an algebraically closed field of characteristic zero these Lie algebras, together with five exceptional Lie algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837062.png" />, of dimensions 14, 52, 78, 133, and 248, respectively, exhaust all simple (that is, non-commutative and not containing ideals other than 0 and the algebra itself) finite-dimensional Lie algebras over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837063.png" />, up to isomorphism (cf. [[Lie algebra, exceptional|Lie algebra, exceptional]]; [[Lie algebra, semi-simple|Lie algebra, semi-simple]]).

3) One more source of Lie algebras is that of vector fields on a manifold (see [[#References|[13]]], [[#References|[14]]] and [[Vector field on a manifold|Vector field on a manifold]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837064.png" /> be the ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837065.png" />-smooth functions on a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837066.png" />-smooth [[Manifold|manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837067.png" />. The vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837068.png" /> of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837069.png" />-smooth vector fields on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837070.png" /> forms a Lie algebra with respect to the commutation operation (see [[Lie bracket|Lie bracket]]), which plays an important role in the theory of manifolds; the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837071.png" /> coincides with the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837072.png" />. Generally speaking, this algebra is infinite-dimensional. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837073.png" /> is a Lie group, then the subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837074.png" /> consisting of all left-invariant vector fields is a finite-dimensional subalgebra and is called the Lie algebra of the Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837075.png" />; it plays an important role in the theory of Lie groups, making it possible to rephrase many properties of Lie groups in terms of Lie algebras. See also [[Lie algebra of an algebraic group|Lie algebra of an algebraic group]]; [[Lie algebra of an analytic group|Lie algebra of an analytic group]].

If in the example above one replaces the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837076.png" /> by a commutative algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837077.png" /> of formal power series over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837078.png" />, then instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837079.png" /> one obtains the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837080.png" /> of formal vector fields, which consists of the differential operators

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837081.png" /></td> </tr></table>

The subalgebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837082.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837083.png" /> consisting of derivations that annihilate the exterior differential forms

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837084.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837085.png" /></td> </tr></table>

respectively, and also the subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837086.png" /> of derivations that multiply the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837087.png" /></td> </tr></table>

by elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837088.png" />, constitute, together with the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837089.png" />, important classes of simple infinite-dimensional Lie algebras (Lie algebras of Cartan type). The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837090.png" /> is called general, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837091.png" /> is called special, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837092.png" /> is called Hamiltonian, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837093.png" /> is called a contact algebra. These algebras were encountered by S. Lie in the study of pseudo-groups of transformations (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837094.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837095.png" />), and were then investigated for various reasons by E. Cartan and others (see [[#References|[15]]], [[#References|[17]]], , [[#References|[19]]]).

4) The following general construction associates a Lie <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837096.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837097.png" /> to any group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837098.png" />; it is used in group theory (see [[Burnside problem|Burnside problem]], ). Let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l05837099.png" /></td> </tr></table>

be the lower central series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370100.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370101.png" /> is the direct sum of the additively written quotient groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370102.png" /> and, by definition, the product of two elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370103.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370104.png" /> is the element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370105.png" /> that is the class of the commutator of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370106.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370107.png" /> that represent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370108.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370109.png" />, respectively. This operation can be extended by distributivity to arbitrary elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370110.png" />. There are (see ) some generalizations of this construction.

==The structure of Lie algebras.==
One of the general results that show, in particular, that the construction 2) has, in a sense, a universal character is the [[Birkhoff–Witt theorem|Birkhoff–Witt theorem]], which states that for any Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370111.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370112.png" /> there is an associative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370113.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370114.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370115.png" /> can be isomorphically imbedded in the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370116.png" /> associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370117.png" /> (see [[Universal enveloping algebra|Universal enveloping algebra]]).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370118.png" /> be a finite-dimensional Lie algebra over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370119.png" /> of characteristic zero. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370120.png" /> is linear, that is, is isomorphic to a subalgebra of a certain Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370121.png" /> (Ado's theorem). In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370122.png" /> there is a unique largest solvable ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370123.png" />, called the radical (see [[Lie algebra, solvable|Lie algebra, solvable]]). Moreover, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370124.png" /> there is a subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370125.png" /> (called a Levi subalgebra) such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370126.png" /> is the direct sum of the vector spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370127.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370128.png" />, and any other subalgebra with this property can be transformed into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370129.png" /> by an automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370130.png" /> (the Levi–Mal'tsev theorem, cf. also [[Levi–Mal'tsev decomposition|Levi–Mal'tsev decomposition]]). Such a subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370131.png" /> is semi-simple (that is, its radical is equal to zero), and it can be characterized as a maximal semi-simple subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370132.png" />. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370133.png" /> is the semi-direct sum of a semi-simple and a solvable Lie algebra, which reduces the problem of classifying finite-dimensional Lie algebras over a field of characteristic zero to the description of the Lie algebras of these two types and of the action of a semi-simple Lie algebra on a solvable one (viz, the restriction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370134.png" /> of the adjoint representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370135.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370136.png" />). Although solvable Lie algebras are in a certain sense  "obtained"  from one-dimensional Lie algebras with trivial structure (namely, they have a chain of subalgebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370137.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370138.png" /> is an ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370139.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370140.png" /> is one-dimensional), their structure is so complicated that at present (1989) there is not even a proper formulation of the problem of classifying solvable Lie algebras. By contrast, the finite-dimensional semi-simple Lie algebras over a field of characteristic zero have been completely described (see [[Lie algebra, semi-simple|Lie algebra, semi-simple]]): Any such algebra splits into the direct sum of simple ideals (and conversely, the direct sum of simple Lie algebras is semi-simple). In the case of an algebraically closed field all simple Lie algebras have been explicitly listed (see 2) above); in the case of an arbitrary field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370141.png" /> there is a procedure for finding them, by means of which an explicit classification has been found in a number of cases (for example, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370142.png" />).

Finite-dimensional Lie algebras over a field of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370143.png" /> have not been investigated in nearly so much detail (even for algebraically closed fields). These Lie algebras have many specific properties. For example, even the description of semi-simple Lie algebras in terms of simple algebras has turned out to be by no means trivial (see [[#References|[23]]]). For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370144.png" /> there are parametric families of simple Lie algebras that are pairwise non-isomorphic to one another. The theory of Lie algebras for this case is in the process of being established, and in a curious way it reflects the features of two different classes of complex Lie algebras, finite-dimensional simple algebras and finite-dimensional transitive simple algebras corresponding to primitive Lie pseudo-groups (see [[#References|[17]]], , [[#References|[19]]]).

The study of infinite-dimensional Lie algebras was begun in the 19th century at the same time as the study of finite-dimensional Lie algebras. These Lie algebras appear naturally in the classification of primitive pseudo-groups of transformations, which was undertaken by Cartan in 1909 [[#References|[20]]]. These algebras have a filtration for which the associated graded Lie algebra has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370145.png" /> and is transitive. Infinite-dimensional graded Lie algebras are the subject of intensive research in which connections of these Lie algebras not only with classical geometrical questions but also with many other branches of mathematics have been discovered (see [[Lie algebra, graded|Lie algebra, graded]], and also [[#References|[17]]], , [[#References|[22]]]). Important examples of infinite-dimensional Lie algebras have appeared recently in the theory of certain equations in mathematical physics (for example, for the Korteweg–de Vries equation) and in formal variational calculus (see [[#References|[14]]]).

The abstract theory of infinite-dimensional Lie algebras (see [[#References|[9]]], for example) is now in the initial phase of development. The theory of representations of Lie algebras plays an important role both in the structure theory of Lie algebras and in the majority of applications to physics.

See also [[Superalgebra|Superalgebra]]; [[Lie algebras, variety of|Lie algebras, variety of]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley  (1975)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Jacobson,   "Lie algebras" , Interscience  (1962)  ((also: Dover, reprint, 1979))</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I. Kaplansky,   "Lie algebras and locally compact groups" , Chicago Univ. Press  (1971)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  W. Magnus,   A. Karrass,   B. Solitar,   "Combinatorial group theory: presentations in terms of generators and relations" , Wiley (Interscience)  (1966)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  L.S. Pontryagin,   "Topological groups" , Princeton Univ. Press  (1958)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> , ''Theórie des algèbres de Lie. Topologie des groupes de Lie'' , ''Sem. S. Lie'' , '''Ie année 1954–1955''' , Ecole Norm. Sup.  (1962)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  J.-P. Serre,   "Lie algebras and Lie groups" , Benjamin  (1965)  (Translated from French)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  C. Chevalley,   "Théorie des groupes de Lie" , '''3''' , Hermann  (1955)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  R.K. Amayo,   I. Stewart,   "Infinite-dimensional Lie algebras" , Noordhoff  (1974)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  J.E. Humphreys,   "Introduction to Lie algebras and representation theory" , Springer  (1972)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  G.B. Seligman,   "Modular Lie algebras" , Springer  (1967)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  V.I. Arnol'd,   "Mathematical methods of classical mechanics" , Springer  (1978)  (Translated from Russian)</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  C. Godbillon,   "Géométrie différentielle et mécanique analytique" , Hermann  (1969)</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top">  B.A. Dubrovin,   A.T. Fomenko,   S.P. Novikov,   "Modern geometry" , Springer  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top"> , ''Internat. Congress Mathematicians (Nice, 1970)'' , Moscow  (1972)  pp. 111–117  (In Russian)  (Lectures of Soviet Mathematicians)</TD></TR><TR><TD valign="top">[16a]</TD> <TD valign="top">  A.I. Kostrikin,   "On the connection between periodic groups and Lie rings"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''21'''  (1957)  pp. 289–310  (In Russian)</TD></TR><TR><TD valign="top">[16b]</TD> <TD valign="top">  A.I. Kostrikin,   "The Burnside problem"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''23'''  (1959)  pp. 3–34  (In Russian)</TD></TR><TR><TD valign="top">[16c]</TD> <TD valign="top">  A.I. Kostrikin,   "A parametric family of simple Lie algebras"  ''Math. USSR Izv.'' , '''4'''  (1970)  pp. 751–764  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''34'''  (1970)  pp. 744–756</TD></TR><TR><TD valign="top">[17]</TD> <TD valign="top">  V. Guillemin,   S. Sternberg,   "An algebraic model of transitive differential geometry"  ''Bull. Amer. Math. Soc.'' , '''70'''  (1964)  pp. 16–47</TD></TR><TR><TD valign="top">[18a]</TD> <TD valign="top">  V.G. [V.G. Kats] Kac,   "Simple irreducible Lie algebras of finite growth"  ''Math. USSR Izv.'' , '''2'''  (1968)  pp. 1271–1311  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''32'''  (1968)  pp. 1323–1367</TD></TR><TR><TD valign="top">[18b]</TD> <TD valign="top">  V.G. [V.G. Kats] Kac,   "Description of filtered Lie algebras with which graded Lie algebras of Cartan type are associated"  ''Math. USSR Izv.'' , '''8'''  (1974)  pp. 801–835  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''38'''  (1974)  pp. 800–834</TD></TR><TR><TD valign="top">[19]</TD> <TD valign="top">  A.I. Kostrikin,   I.R. Shafarevich,   "Graded Lie algebras of finite characteristic"  ''Math. USSR Izv.'' , '''3'''  (1969)  pp. 237–304  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''33'''  (1969)  pp. 251–322</TD></TR><TR><TD valign="top">[20]</TD> <TD valign="top">  E. Cartan,   "Les groupes de transformations continues, infinis, simples"  ''Ann. Sci. Ecole Norm. Sup.'' , '''26'''  (1909)  pp. 93–161</TD></TR><TR><TD valign="top">[21]</TD> <TD valign="top">  M. Lazard,   "Sur les groupes nilpotents et les anneaux de Lie"  ''Ann. Sci. Ecole Norm. Sup.'' , '''71'''  (1954)  pp. 101–190</TD></TR><TR><TD valign="top">[22]</TD> <TD valign="top">  I.M. Singer,   S. Sternberg,   "The infinite groups of Lie and Cartan, Part I (the transitive groups)"  ''J. d'Anal. Math.'' , '''15'''  (1965)  pp. 1–114</TD></TR><TR><TD valign="top">[23]</TD> <TD valign="top">  R.E. Block,   "Determination of the differentiably simple rings with a minimal ideal"  ''Ann. of Math. (2)'' , '''90'''  (1969)  pp. 433–459</TD></TR></table>



====Comments====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370146.png" /> now be of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370147.png" />. A simple Lie algebra of classical type in characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370149.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370150.png" />, is one  "like"  the simple algebras in characteristic zero corresponding to the root systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370151.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370152.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370153.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370154.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370155.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370156.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370157.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370158.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370159.png" />. Examples of these algebras are obtained by taking a Chevalley basis of one of these algebras; that gives a version (i.e. a form, see [[Form of an (algebraic) structure|Form of an (algebraic) structure]]) defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370160.png" />; reduce the coefficients in the multiplication table of the basis elements modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370161.png" /> to obtain a Lie algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370162.png" />; divide out, when necessary, the centre of this algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370163.png" />; extend the scalars to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370164.png" />.

Perhaps somewhat confusingly, the algebras of classical type in characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370165.png" /> are taken to include the five exceptional types <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370166.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370167.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370168.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370169.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370170.png" />. The Lie algebras of classical type can also be characterized by the Mills–Seligman axioms, [[#References|[11]]]: if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370171.png" /> is simple (and finite dimensional) and has a [[Cartan subalgebra|Cartan subalgebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370172.png" /> which acts diagonally on each root space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370173.png" />, if, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370174.png" /> for each root <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370175.png" />, and if whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370176.png" /> are roots, then not all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370177.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370178.png" />, are roots, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370179.png" /> is classical. A simple Lie algebra with a projective representation with non-degenerate trace form is classical (and vice versa). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370180.png" /> is algebraically closed, the simple Lie algebras of classical type are classified by their Dynkin diagrams, and over an arbitrary (perfect) field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370181.png" /> of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370182.png" /> the classification can then be attacked by means of the theory of forms (cf. [[Form of an (algebraic) structure|Form of an (algebraic) structure]]). For instance, over the finite fields one finds for the number of (non-isomorphic) forms (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370183.png" />) the following:

types <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370184.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370185.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370186.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370187.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370188.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370189.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370190.png" />: one;

types <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370191.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370192.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370193.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370194.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370195.png" />: two;

type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370196.png" />: three.

Besides the simple Lie algebras of classical type over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370197.png" /> of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370198.png" /> there are many more. Some are described in [[Witt algebra|Witt algebra]]. All the other known simple Lie algebras are simple Lie algebras of Cartan type. These are (twisted) analogues of the infinite-dimensional algebras of Lie and Cartan of types <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370199.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370200.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370201.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370202.png" />, and they arise as (twisted) subalgebras of the algebras of derivations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370203.png" />, which are defined as follows.

Define a co-algebra structure (cf. [[Co-algebra|Co-algebra]]) on the algebra of polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370204.png" /> by defining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370205.png" /> (making <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370206.png" /> a [[Hopf algebra|Hopf algebra]]). The dual algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370207.png" /> is an infinite-dimensional associative and commutative algebra consisting of all formal sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370208.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370209.png" /> ranges over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370210.png" />-tuples of non-negative integers. The multiplication is given by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370211.png" /></td> </tr></table>

For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370212.png" />-tuple of non-negative integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370213.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370214.png" /> denote the span of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370215.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370216.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370217.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370218.png" /> is a subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370219.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370220.png" /> be the derivation algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370221.png" />. The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370222.png" /> is simple of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370223.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370224.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370225.png" /> are the Jacobson–Witt algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370226.png" />, cf. [[Witt algebra|Witt algebra]]. For a description of the various (twisted) subalgebras of types <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370227.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370228.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370229.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370230.png" /> cf. [[#References|[a4]]]–[[#References|[a7]]]. These are the Lie algebras of Cartan type. The generalized Kostrikin–Shafarevich conjecture states that every simple Lie algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370231.png" /> is of classical type or of Cartan type.

A restricted Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370232.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370233.png" /> is such that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370234.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370235.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370236.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370237.png" /> is simple, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370238.png" /> is unique. The original Kostrikin–Shafarevich conjecture states that each simple finite-dimensional restricted Lie algebra is of classical type or of Cartan type. This has been proved by R.E. Block and R.L. Wilson for characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370239.png" />, [[#References|[a5]]], [[#References|[a4]]].

Simple Lie algebras of Cartan type are very different from those of classical type: they have zero Killing forms; their Cartan subalgebras may be nilpotent of any index and not all have the same dimension; the root spaces may be arbitrarily large; the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370240.png" />-root string through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370241.png" /> may include all roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370242.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058370/l058370243.png" />; there may be infinitely non-conjugate Cartan subalgebras of a given dimension. Cf. [[#References|[a8]]]–[[#References|[a11]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.-P. Serre,   "Algèbres de Lie semi-simples complexes" , Benjamin  (1966)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  V.S. Varadarajan,   "Lie groups, Lie algebras, and their representations" , Prentice-Hall  (1974)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D.J. Britten (ed.)  F.W. Lemire (ed.) , ''Lie algebras and related topics'' , Amer. Math. Soc.  (1986)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R.E. Block,   R.L. Wilson,   "Classification of restricted simple Lie algebras"  ''J. of Algebra'' , '''114'''  (1988)  pp. 115–259</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R.E. Block,   R.L. Wilson,   "The restricted simple Lie algebras are of classical or Cartan type"  ''Proc. Nat. Acad. Sci. USA'' , '''81'''  (1984)  pp. 5271–5274</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  R.E. Block,   R.L. Wilson,   "Restricted simple Lie algebras"  D.J. Britten (ed.)  F.W. Lemire (ed.) , ''Lie algebras and related topics'' , Amer. Math. Soc.  (1986)  pp. 3–18</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  R.L. Wilson,   "A structural characterization of the simple Lie algebras of generalized Cartan type over fields of prime characteristic"  ''J. of Algebra'' , '''40'''  (1976)  pp. 418–465</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  G. Benkart,   "Cartan subalgebras in Lie algebras of Cartan type"  D.J. Britten (ed.)  F.W. Lemire (ed.) , ''Lie algebras and related topics'' , Amer. Math. Soc.  (1986)  pp. 157–187</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  R.E. Block,   "New simple Lie algebras of prime characteristic"  ''Trans. Amer. Math. Soc.'' , '''89'''  (1958)  pp. 421–449</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  G. Brown,   "Cartan subalgebras of Zassenhaus algebras"  ''Canad. J. Math.'' , '''27'''  (1975)  pp. 1011–1021</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  H. Strade,   "Cartanalgebren im modularen Liealgebren"  ''Comm. in Algebra'' , '''5'''  (1977)  pp. 1335–1359</TD></TR></table>