View source for Lie p-algebra

''restricted Lie algebra''

An algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l0587202.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l0587203.png" /> of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l0587204.png" /> (or, more generally, over a ring of prime characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l0587205.png" />), endowed with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l0587206.png" />-mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l0587207.png" /> such that the following relations are satisfied:

<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/l058720/l0587208.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/l058720/l0587209.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/l058720/l05872010.png" /></td> </tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872011.png" /> is the inner derivation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872012.png" /> defined by the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872013.png" /> (the adjoint transformation) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872014.png" /> is a certain element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872015.png" /> that is a linear combination of Lie monomials

<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/l058720/l05872016.png" /></td> </tr></table>

with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872017.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872018.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872019.png" />.

A typical example of a Lie <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872020.png" />-algebra is obtained if one considers an arbitrary associative algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872021.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872022.png" /> (cf. [[Associative rings and algebras|Associative rings and algebras]]) as a [[Universal algebra|universal algebra]], with the following two derivation operations:

i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872023.png" />,

ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872024.png" />.

In particular, the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872025.png" /> is a direct consequence of the identity

<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/l058720/l05872026.png" /></td> </tr></table>

for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872027.png" />, in which case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872028.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872029.png" />. Since any [[Lie algebra|Lie algebra]] can be imbedded in a suitably chosen associative algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872030.png" /> with the operations i) and ii) (the Poincaré–Birkhoff–Witt theorem), one often replaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872031.png" />, with some risk of ambiguity, by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872032.png" />.

As in every structure theory, the structure-preserving mappings are of particular relevance.

For any Lie <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872033.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872034.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872036.png" />-universal (restricted universal) enveloping associative algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872038.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872039.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872040.png" />. This remark shows that for an arbitrary Lie algebra it makes sense to talk about its smallest <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872042.png" />-envelope, or about its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872044.png" />-closure.

An ordinary Lie subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872045.png" /> (Lie ideal) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872046.png" /> is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872048.png" />-subalgebra (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872050.png" />-ideal) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872051.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872052.png" />. A homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872053.png" /> of Lie <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872054.png" />-algebras is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872056.png" />-homomorphism if

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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872058.png" /> is a linear Lie <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872059.png" />-algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872060.png" />, one also calls this a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872062.png" />-representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872063.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872064.png" />.

The specification of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872065.png" />-structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872066.png" /> on a Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872067.png" /> with basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872068.png" /> and zero centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872069.png" /> is uniquely and completely determined by specifying the images <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872070.png" /> of the basis elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872071.png" />. On the other hand, a commutative Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872072.png" />, for which one always has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872073.png" />, is endowed with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872074.png" />-structure by considering the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872075.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872076.png" /> is an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872077.png" />-semi-linear mapping,

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Over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872079.png" />, every finite-dimensional commutative Lie <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872080.png" />-algebra splits into the direct sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872081.png" /> of a torus

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and a nilpotent subalgebra (cf. [[Nilpotent algebra|Nilpotent algebra]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872083.png" />, where the identity

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holds for sufficiently large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872085.png" /> (see [[#References|[1]]]).

Important sources of Lie <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872086.png" />-algebras are the theory of algebraic groups, the theory of formal groups and the theory of inseparable fields (see [[#References|[2]]]). The Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872087.png" /> of all derivations of an arbitrary algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872088.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872089.png" />-subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872090.png" />.

The class of simple Lie <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872091.png" />-algebras (restricted simple Lie algebras) is especially interesting for several reasons. For each finite-dimensional Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872092.png" /> over the complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872093.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872094.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872095.png" />-span of a Chevalley basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872096.png" />, and extend scalars to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872097.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872098.png" />. The quotient algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872099.png" /> is simple and restricted. The simple Lie algebras obtained in this way are known as algebras of classical type: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720100.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720101.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720102.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720103.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720104.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720105.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720106.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720107.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720108.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720110.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720111.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720112.png" />. Besides the classical algebras, there are four other classes of simple Lie <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720113.png" />-algebras: general algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720114.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720115.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720116.png" />); special algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720117.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720118.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720119.png" />); Hamiltonian algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720120.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720121.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720122.png" />); contact algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720123.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720124.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720125.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720126.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720127.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720128.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720129.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720130.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720131.png" />)). The simple Lie <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720132.png" />-algebras just described are called algebras of Cartan type. They are obtained by replacing the ring of power series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720133.png" /> in the Lie–Cartan construction (see [[Lie algebra|Lie algebra]], 3)) by that of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720134.png" />-truncated polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720135.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720136.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720137.png" />. In the symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720138.png" /> the index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720139.png" /> has an invariant meaning; namely, it is the dimension of a maximal toroidal subalgebra. The main Block–Wilson classification theorem [[#References|[5]]]: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720140.png" /> be a finite-dimensional simple Lie <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720141.png" />-algebra over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720142.png" /> of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720143.png" />; then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720144.png" /> is of classical or Cartan type. This result was conjectured by A.I. Kostrikin and I.R. Shafarevich (see [[#References|[3]]]). It is not known whether the statement above will be true for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720145.png" /> (presumably so), but for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720146.png" />, however, the situation is necessarily more complicated. For example, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720147.png" /> the classical Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720148.png" /> is included in a parametric family of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720149.png" />-dimensional simple Lie <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720150.png" />-algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720151.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720152.png" />.

The theory of modular Lie algebras, i.e. Lie algebras over fields of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720153.png" />, was created in the last half-century. It is symbolically said that its source is the discovery of E. Witt (1937) of the simple non-classical Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720154.png" />. Here it should be noted that there is a much more involved construction of the simple Lie algebras of Cartan type that are not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720155.png" />-algebras. By dropping the requirement of being restricted, additional difficulties arise also in the study of representations, cohomology, deformations, and other problems in the theory of modular Lie algebras. The study of interrelations between constructions taking these into account and not taking into account the restrictedness condition, forms an important part of the theory (cf. [[#References|[6]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Jacobson,   "Lie algebras" , Interscience  (1962)  ((also: Dover, reprint, 1979))</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.B. Seligman,   "Modular Lie algebras" , Springer  (1967)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.I. Kostrikin,   I.R. Shafarevich,   "Cartan pseudogroups and Lie <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720156.png" />-algebras"  ''Soviet Math. Dokl.'' , '''7'''  (1986)  pp. 715–718  ''Dokl. Akad. Nauk SSSR'' , '''168'''  (1966)  pp. 740–742</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  H. Zassenhaus,   "Ueber Liesche Ringe mit Primzahlcharacteristik"  ''Abh. Math. Sem. Hansische Univ.'' , '''13'''  (1939)  pp. 1–100</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  R.E. Block,   R.L. Wilson,   "Classification of the restricted simple Lie algebras"  ''J. of Algebra'' , '''114'''  (1988)  pp. 115–259</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  H. Strade,   R. Farnsteiner,   "Modular Lie algebras and their representations" , M. Dekker  (1988)</TD></TR></table>



====Comments====
In characteristic 2 and 3 there exist infinitely many simple Lie <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720157.png" />-algebras, of dimension 31 and 10, respectively (cf. [[#References|[a1]]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.G. Kac,   B.Yu. Veisfeiler,   "Exponentials in Lie algebras of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l058720158.png" />"  ''Math. USSR Izv.'' , '''5'''  (1971)  pp. 777–803  ''Izv. Akad. Nauk SSSR'' , '''35'''  (1971)  pp. 762–788</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Borel,   "Linear algebraic groups" , Benjamin  (1969)</TD></TR></table>