Lie algebra

A unitary -module over a commutative ring with a unit that is endowed with a bilinear mapping of into having the following two properties:

1) (hence the anti-commutative law );

2) (the Jacobi identity).

Thus, a Lie algebra is an algebra over (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 is said to be commutative if for all .

The most important case is that in which is a field (especially when or ) and is a vector space (of finite or infinite dimension) over .

Lie algebras appeared in mathematics at the end of the 19th century in connection with the study of Lie groups (cf. Lie group, see also Lie group, local; Lie transformation group; 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 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 [8], [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, which to a significant extent absorbs it and extremely outgrows it, or in the theory of finite groups, cf. 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 of all derivations (cf. Derivation in a ring) of any -algebra is a Lie algebra with the operation

The derivations of a Lie algebra of the form

are called inner derivations or adjoint transformations. In they form a subalgebra, , and the mapping is a homomorphism of Lie algebras (the adjoint representation of the Lie algebra ); its image is isomorphic to the quotient algebra of with respect to its centre

2) Another important source of Lie algebras is connected with the following simple observation. If is an associative algebra over (cf. Associative rings and algebras) with multiplication , then the multiplication in the -module specified by the rule

endows with the structure of a Lie algebra over . One says that is the Lie algebra associated with the associative algebra . Thus, the classical example of a Lie algebra is obtained if for one takes the (associative) algebra of all square matrices of order over .

The following four infinite series of subalgebras of a Lie algebra of this type are called classical ( is a field of characteristic zero):

One has , , , .

A remarkable result is that over an algebraically closed field of characteristic zero these Lie algebras, together with five exceptional Lie algebras , , , , , 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 , up to isomorphism (cf. Lie algebra, exceptional; Lie algebra, semi-simple).

3) One more source of Lie algebras is that of vector fields on a manifold (see [13], [14] and Vector field on a manifold). Let be the ring of -smooth functions on a -smooth manifold . The vector space of all -smooth vector fields on forms a Lie algebra with respect to the commutation operation (see Lie bracket), which plays an important role in the theory of manifolds; the Lie algebra coincides with the Lie algebra . Generally speaking, this algebra is infinite-dimensional. If is a Lie group, then the subspace of consisting of all left-invariant vector fields is a finite-dimensional subalgebra and is called the Lie algebra of the Lie group ; 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 analytic group.

If in the example above one replaces the ring by a commutative algebra of formal power series over a field , then instead of one obtains the Lie algebra of formal vector fields, which consists of the differential operators

The subalgebras , consisting of derivations that annihilate the exterior differential forms

respectively, and also the subalgebra of derivations that multiply the form

by elements of , constitute, together with the algebra , important classes of simple infinite-dimensional Lie algebras (Lie algebras of Cartan type). The algebra is called general, is called special, is called Hamiltonian, and is called a contact algebra. These algebras were encountered by S. Lie in the study of pseudo-groups of transformations ( or ), and were then investigated for various reasons by E. Cartan and others (see [15], [17], , [19]).

4) The following general construction associates a Lie -algebra to any group ; it is used in group theory (see Burnside problem, ). Let

be the lower central series of . Then is the direct sum of the additively written quotient groups and, by definition, the product of two elements and is the element of that is the class of the commutator of elements and that represent and , respectively. This operation can be extended by distributivity to arbitrary elements of . 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, which states that for any Lie algebra over a field there is an associative -algebra such that can be isomorphically imbedded in the Lie algebra associated with (see Universal enveloping algebra).

Let be a finite-dimensional Lie algebra over a field of characteristic zero. Then is linear, that is, is isomorphic to a subalgebra of a certain Lie algebra (Ado's theorem). In there is a unique largest solvable ideal , called the radical (see Lie algebra, solvable). Moreover, in there is a subalgebra (called a Levi subalgebra) such that is the direct sum of the vector spaces and , and any other subalgebra with this property can be transformed into by an automorphism of (the Levi–Mal'tsev theorem, cf. also Levi–Mal'tsev decomposition). Such a subalgebra is semi-simple (that is, its radical is equal to zero), and it can be characterized as a maximal semi-simple subalgebra of . Thus, 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 of the adjoint representation of in ). 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 such that is an ideal of and 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): 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 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 ).

Finite-dimensional Lie algebras over a field of characteristic 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 [23]). For any 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 [17], , [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 [20]. These algebras have a filtration for which the associated graded Lie algebra has the form 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, and also [17], , [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 [14]).

The abstract theory of infinite-dimensional Lie algebras (see [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; Lie algebras, variety of.

References

[1]	N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French) MR0682756 Zbl 0319.17002
[2]	N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979)) MR0148716 MR0143793 Zbl 0121.27504 Zbl 0109.26201
[3]	I. Kaplansky, "Lie algebras and locally compact groups" , Chicago Univ. Press (1971) MR0276398 Zbl 0223.17001
[4]	W. Magnus, A. Karrass, B. Solitar, "Combinatorial group theory: presentations in terms of generators and relations" , Wiley (Interscience) (1966) Zbl 0138.25604
[5]	L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian) MR0201557 Zbl 0022.17104
[6]	, Theórie des algèbres de Lie. Topologie des groupes de Lie , Sem. S. Lie , Ie année 1954–1955 , Ecole Norm. Sup. (1962) Zbl 0106.02701
[7]	J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French) MR0218496 Zbl 0132.27803
[8]	C. Chevalley, "Théorie des groupes de Lie" , 3 , Hermann (1955) MR0068552 Zbl 0186.33104 Zbl 0054.01303 Zbl 0063.00843
[9]	R.K. Amayo, I. Stewart, "Infinite-dimensional Lie algebras" , Noordhoff (1974) MR0396708 Zbl 0302.17006
[10]	J.E. Humphreys, "Introduction to Lie algebras and representation theory" , Springer (1972) MR0323842 Zbl 0254.17004
[11]	G.B. Seligman, "Modular Lie algebras" , Springer (1967) MR0245627 Zbl 0189.03201
[12]	V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) (Translated from Russian) Zbl 0692.70003 Zbl 0572.70001 Zbl 0647.70001
[13]	C. Godbillon, "Géométrie différentielle et mécanique analytique" , Hermann (1969) MR0242081 Zbl 0653.53001 Zbl 0284.53018
[14]	B.A. Dubrovin, A.T. Fomenko, S.P. Novikov, "Modern geometry" , Springer (1984) (Translated from Russian) MR0766739 MR0736837 Zbl 0582.55001 Zbl 0529.53002
[15]	, Internat. Congress Mathematicians (Nice, 1970) , Moscow (1972) pp. 111–117 (In Russian) (Lectures of Soviet Mathematicians)
[16a]	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) MR0094380 Zbl 0158.02605
[16b]	A.I. Kostrikin, "The Burnside problem" Izv. Akad. Nauk SSSR Ser. Mat. , 23 (1959) pp. 3–34 (In Russian) MR0132100 Zbl 0132.01702
[16c]	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 MR0274539 Zbl 0245.17008
[17]	V. Guillemin, S. Sternberg, "An algebraic model of transitive differential geometry" Bull. Amer. Math. Soc. , 70 (1964) pp. 16–47 MR0170295 Zbl 0121.38801
[18a]	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 Zbl 0222.17007
[18b]	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 MR0430000 Zbl 0317.17002
[19]	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 MR0252460 Zbl 0211.05304
[20]	E. Cartan, "Les groupes de transformations continues, infinis, simples" Ann. Sci. Ecole Norm. Sup. , 26 (1909) pp. 93–161
[21]	M. Lazard, "Sur les groupes nilpotents et les anneaux de Lie" Ann. Sci. Ecole Norm. Sup. , 71 (1954) pp. 101–190 MR0088496 Zbl 0055.25103
[22]	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
[23]	R.E. Block, "Determination of the differentiably simple rings with a minimal ideal" Ann. of Math. (2) , 90 (1969) pp. 433–459 MR0251088 Zbl 0216.07303

Comments

Let now be of characteristic . A simple Lie algebra of classical type in characteristic , , is one "like" the simple algebras in characteristic zero corresponding to the root systems , , , , , , , , . 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) defined over ; reduce the coefficients in the multiplication table of the basis elements modulo to obtain a Lie algebra over ; divide out, when necessary, the centre of this algebra over ; extend the scalars to .

Perhaps somewhat confusingly, the algebras of classical type in characteristic are taken to include the five exceptional types , , , , . The Lie algebras of classical type can also be characterized by the Mills–Seligman axioms, [11]: if is simple (and finite dimensional) and has a Cartan subalgebra which acts diagonally on each root space , if, moreover, for each root , and if whenever are roots, then not all , , are roots, then is classical. A simple Lie algebra with a projective representation with non-degenerate trace form is classical (and vice versa). If is algebraically closed, the simple Lie algebras of classical type are classified by their Dynkin diagrams, and over an arbitrary (perfect) field of characteristic the classification can then be attacked by means of the theory of forms (cf. Form of an (algebraic) structure). For instance, over the finite fields one finds for the number of (non-isomorphic) forms (for ) the following:

types , , , , , , : one;

types (), (), : two;

type : three.

Besides the simple Lie algebras of classical type over a field of characteristic there are many more. Some are described in 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 , , , , and they arise as (twisted) subalgebras of the algebras of derivations , which are defined as follows.

Define a co-algebra structure (cf. Co-algebra) on the algebra of polynomials by defining (making a Hopf algebra). The dual algebra is an infinite-dimensional associative and commutative algebra consisting of all formal sums , where ranges over all -tuples of non-negative integers. The multiplication is given by

For each -tuple of non-negative integers , let denote the span of the with for all . Then is a subalgebra of . Let be the derivation algebra of . The algebra is simple of dimension , where . The are the Jacobson–Witt algebras , cf. Witt algebra. For a description of the various (twisted) subalgebras of types , , of the cf. [a4]–[a7]. These are the Lie algebras of Cartan type. The generalized Kostrikin–Shafarevich conjecture states that every simple Lie algebra over is of classical type or of Cartan type.

A restricted Lie algebra over is such that for every there is a such that . If is simple, then 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 , [a5], [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 -root string through may include all roots , ; there may be infinitely non-conjugate Cartan subalgebras of a given dimension. Cf. [a8]–[a11].

References

[a1]	J.-P. Serre, "Algèbres de Lie semi-simples complexes" , Benjamin (1966) MR0215886 Zbl 0144.02105
[a2]	V.S. Varadarajan, "Lie groups, Lie algebras, and their representations" , Prentice-Hall (1974) MR0376938 Zbl 0371.22001
[a3]	D.J. Britten (ed.) F.W. Lemire (ed.) , Lie algebras and related topics , Amer. Math. Soc. (1986) MR0832191 Zbl 0574.00001
[a4]	R.E. Block, R.L. Wilson, "Classification of restricted simple Lie algebras" J. of Algebra , 114 (1988) pp. 115–259 MR931904 Zbl 0644.17008
[a5]	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 MR0758423 Zbl 0542.17003
[a6]	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 MR0832192 Zbl 0582.17009
[a7]	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 MR0412239 Zbl 0355.17012
[a8]	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 MR0832198 Zbl 0581.17006
[a9]	R.E. Block, "New simple Lie algebras of prime characteristic" Trans. Amer. Math. Soc. , 89 (1958) pp. 421–449 MR0100010 Zbl 0082.25104
[a10]	G. Brown, "Cartan subalgebras of Zassenhaus algebras" Canad. J. Math. , 27 (1975) pp. 1011–1021 MR0401851 Zbl 0329.17004
[a11]	H. Strade, "Cartanalgebren im modularen Liealgebren" Comm. in Algebra , 5 (1977) pp. 1335–1359