Lie p-algebra

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An algebra $ L $ over a field $ k $ of characteristic $ p > 0 $ ( or, more generally, over a ring of prime characteristic $ p > 0 $ ), endowed with a $ p $ - mapping $ x \rightarrow x ^{[p]} $ such that the following relations are satisfied:$$ \mathop{\rm ad}\nolimits ( x ^{[p]} ) = ( \mathop{\rm ad}\nolimits \ x ) ^{p} , $$ $$ ( \lambda x ) ^{[p]} = \lambda ^{p} x ^{[p]} , $$ $$ ( x + y ) ^{[p]} = x ^{[p]} + y ^{[p]} + \Lambda _{p} ( x ,\ y ) . $$ Here $ \mathop{\rm ad}\nolimits \ x : \ y \rightarrow [ x ,\ y ] $ is the inner derivation of $ L $ defined by the element $ x \in L $ ( the adjoint transformation) and $ \Lambda _{p} ( x ,\ y ) $ is a certain element of $ L $ that is a linear combination of Lie monomials$$ ( \mathop{\rm ad}\nolimits \ x _{1} \dots \mathop{\rm ad}\nolimits \ x _{p-1} ) x $$ with $ x _{i} = x $ or $ y $ for all $ i = 1 \dots p - 1 $ .

A typical example of a Lie $ p $ - algebra is obtained if one considers an arbitrary associative algebra $ A $ over $ k $ ( cf. Associative rings and algebras) as a universal algebra, with the following two derivation operations:

i) $ ( x ,\ y ) \rightarrow [ x ,\ y ] = x y - y x $ ,

ii) $ x \rightarrow x ^{p} $ .

In particular, the property $ \mathop{\rm ad}\nolimits ( x ^{p} ) = ( \mathop{\rm ad}\nolimits \ x ) ^{p} $ is a direct consequence of the identity$$ ( \mathop{\rm ad}\nolimits \ x ) ^{n} y = \sum _{j=1} ^ n (-1) ^{j} \binom{n}{j} x ^{n-j} y x ^{j} $$ for $ n = p $ , in which case $ ( _{j} ^{n} ) = 0 $ for $ j = 1 \dots p - 1 $ . Since any Lie algebra can be imbedded in a suitably chosen associative algebra $ A $ with the operations i) and ii) (the Poincaré–Birkhoff–Witt theorem), one often replaces $ x ^{[p]} $ , with some risk of ambiguity, by $ x ^{p} $ .

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

For any Lie $ p $ - algebra $ L $ there is a $ p $ - universal (restricted universal) enveloping associative algebra $ U _{p} (L) $ . If $ \mathop{\rm dim}\nolimits _{k} \ L = n $ , then $ \mathop{\rm dim}\nolimits _{k} \ U _{p} (L) = p ^{n} $ . This remark shows that for an arbitrary Lie algebra it makes sense to talk about its smallest $ p $ - envelope, or about its $ p $ - closure.

An ordinary Lie subalgebra $ M $ ( Lie ideal) of $ L $ is called a $ p $ - subalgebra ($ p $ - ideal) if $ x ^{[p]} \in M $ for all $ x \in M $ . A homomorphism $ \phi : \ L \rightarrow K $ of Lie $ p $ - algebras is called a $ p $ - homomorphism if$$ \phi ( x ^{[p]} ) = ( \phi (x) ) ^{[p]} , x \in L . $$ If $ K $ is a linear Lie $ p $ - algebra over $ k $ , one also calls this a $ p $ - representation $ \phi $ of $ L $ .

The specification of a $ p $ - structure $ x \rightarrow x ^{[p]} $ on a Lie algebra $ L $ with basis $ \{ e _{1} ,\ e _{2} , . . . \} $ and zero centre $ Z (L) $ is uniquely and completely determined by specifying the images $ e _{i} ^{[p]} $ of the basis elements $ e _{i} $ . On the other hand, a commutative Lie algebra $ L $ , for which one always has $ \Lambda _{p} ( x ,\ y ) = 0 $ , is endowed with a $ p $ - structure by considering the pair $ ( L ,\ \pi ) $ , where $ \pi $ is an arbitrary $ p $ - semi-linear mapping,$$ \pi ( x + y ) = \pi (x) + \pi (y) , \pi ( \lambda x ) = \lambda ^{p} \pi (x) , \lambda \in k . $$ Over an algebraically closed field $ k $ , every finite-dimensional commutative Lie $ p $ - algebra splits into the direct sum $ L = L _{0} \oplus L _{1} $ of a torus$$ L _{0} = < e _{1} \dots e _{r} : e _{i} ^{[p]} = e _{i} > $$ and a nilpotent subalgebra (cf. Nilpotent algebra) $ L _{1} $ , where the identity$$ x ^ {[ p ^{m} ]} = ( x ^ {[ p ^{m-1} ]} ) ^{[p]} = 0 $$ holds for sufficiently large $ m $ ( see [1]).

Important sources of Lie $ p $ - algebras are the theory of algebraic groups, the theory of formal groups and the theory of inseparable fields (see [2]). The Lie algebra $ \mathop{\rm Der}\nolimits _{k} (A) $ of all derivations of an arbitrary algebra $ A $ is a $ p $ - subalgebra of $ \mathop{\rm End}\nolimits _{k} (A) $ .

The class of simple Lie $ p $ - algebras (restricted simple Lie algebras) is especially interesting for several reasons. For each finite-dimensional Lie algebra $ {\mathcal L} $ over the complex numbers $ \mathbf C $ , let $ {\mathcal L} _ {\mathbf Z} $ be the $ \mathbf Z $ - span of a Chevalley basis of $ {\mathcal L} $ , and extend scalars to $ k $ : $ {\mathcal L} _{k} = {\mathcal L} _ {\mathbf Z} \otimes k $ . The quotient algebra $ L = {\mathcal L} _{k} / Z ( {\mathcal L} _{k} ) $ is simple and restricted. The simple Lie algebras obtained in this way are known as algebras of classical type: $ A _{n} $ ( $ n \geq 1 $ ), $ B _{n} $ ( $ n \geq 3 $ ), $ C _{n} $ ( $ n \geq 2 $ ), $ D _{n} $ ( $ n \geq 4 $ ), $ G _{2} $ , $ F _{4} $ , $ E _{6} $ , $ E _{7} $ , $ E _{8} $ . Besides the classical algebras, there are four other classes of simple Lie $ p $ - algebras: general algebras $ W _{n} $ , $ n \geq 1 $ ( $ \mathop{\rm dim}\nolimits \ W _{n} = n p ^{m} $ ); special algebras $ S _{n} $ , $ n \geq 2 $ ( $ \mathop{\rm dim}\nolimits \ S _{n} = n ( p ^{n+1} - 1 ) $ ); Hamiltonian algebras $ H _{n} $ , $ n\geq 1 $ ( $ \mathop{\rm dim}\nolimits \ H _{n} = p ^{2n} - 2 $ ); contact algebras $ K _{n} $ , $ n \geq 2 $ ( $ \mathop{\rm dim}\nolimits \ K _{n} = p ^{2n-1} - \epsilon $ , where $ \epsilon = 0 $ for $ n + 1 \not\equiv 0 $ ( $ \mathop{\rm mod}\nolimits \ p $ ) and $ \epsilon = 1 $ for $ n + 1 \equiv 0 $ ( $ \mathop{\rm mod}\nolimits \ p $ )). The simple Lie $ p $ - algebras just described are called algebras of Cartan type. They are obtained by replacing the ring of power series $ \mathbf C [ [ X _{1} \dots X _{m} ] ] $ in the Lie–Cartan construction (see Lie algebra, 3)) by that of the $ p $ - truncated polynomials $ k [ X _{1} \dots X _{m} ;\ X _{1} ^{p} = 0 \dots X _{m} ^{p} = 0 ] $ , $ m = n ,\ n + 1 ,\ 2 n $ , or $ 2 n - 1 $ . In the symbols $ W _{n} ,\ S _{n} \dots $ the index $ n $ has an invariant meaning; namely, it is the dimension of a maximal toroidal subalgebra. The main Block–Wilson classification theorem [5]: Let $ L $ be a finite-dimensional simple Lie $ p $ - algebra over an algebraically closed field $ k $ of characteristic $ p > 7 $ ; then $ L $ is of classical or Cartan type. This result was conjectured by A.I. Kostrikin and I.R. Shafarevich (see [3]). It is not known whether the statement above will be true for $ p = 7 $ ( presumably so), but for $ p = 2 ,\ 3 ,\ 5 $ , however, the situation is necessarily more complicated. For example, for $ p = 3 $ the classical Lie algebra $ C _{2} $ is included in a parametric family of $ 10 $ - dimensional simple Lie $ p $ - algebras $ C _{2} ( \epsilon ) $ , $ \epsilon \in k $ .

The theory of modular Lie algebras, i.e. Lie algebras over fields of characteristic $ p > 0 $ , 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 $ W _{1} $ . Here it should be noted that there is a much more involved construction of the simple Lie algebras of Cartan type that are not $ p $ - 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. [6]).

References

[1]	N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979)) MR0148716 MR0143793 Zbl 0121.27504 Zbl 0109.26201
[2]	G.B. Seligman, "Modular Lie algebras" , Springer (1967) MR0245627 Zbl 0189.03201
[3]	A.I. Kostrikin, I.R. Shafarevich, "Cartan pseudogroups and Lie p-algebras" Soviet Math. Dokl. , 7 (1986) pp. 715–718 Dokl. Akad. Nauk SSSR , 168 (1966) pp. 740–742
[4]	H. Zassenhaus, "Ueber Liesche Ringe mit Primzahlcharacteristik" Abh. Math. Sem. Hansische Univ. , 13 (1939) pp. 1–100
[5]	R.E. Block, R.L. Wilson, "Classification of the restricted simple Lie algebras" J. of Algebra , 114 (1988) pp. 115–259 MR0931904
[6]	H. Strade, R. Farnsteiner, "Modular Lie algebras and their representations" , M. Dekker (1988) MR0929682 Zbl 0648.17003

Comments

In characteristic 2 and 3 there exist infinitely many simple Lie $ p $ - algebras, of dimension 31 and 10, respectively (cf. [a1]).

References

[a1]	V.G. Kac, B.Yu. Veisfeiler, "Exponentials in Lie algebras of characteristic " Math. USSR Izv. , 5 (1971) pp. 777–803 Izv. Akad. Nauk SSSR , 35 (1971) pp. 762–788 MR0306282 Zbl 0252.17003 Zbl 0245.17007
[a2]	A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201