Lie algebra

2020 Mathematics Subject Classification: Primary: 17B [MSN][ZBL]

A Lie algebra is a unitary $k$-module $L$ over a commutative ring $k$ with a unit that is endowed with a bilinear mapping $(x,y)\mapsto [x,y]$ of $L\times L$ into $L$ having the following two properties:

1) $[x,x] = 0$ (hence the anti-commutative law $[x,y]=-[y,x]$);

2) $[x,[y,z]]+[y,[z,x]]+[z,[x,y]] = 0$ (the Jacobi identity).

Thus, a Lie algebra is an algebra over $k$ (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 $L$ is said to be commutative if $[x,y] = 0$ for all $x,y\in L$.

The most important case is that in which $k$ is a field (especially when $k=\R$ or $\C$) and $L$ is a vector space (of finite or infinite dimension) over $k$.

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 $C^2$ 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 [Ch], [Hu]). 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 $\def\Der{\textrm{Der}}\Der(A)$ of all derivations (cf. Derivation in a ring) of any $k$-algebra is a Lie algebra with the operation

$$[D_1,D_2] = D_1\circ D_2 - D_2\circ D_1. $$ The derivations of a Lie algebra $L$ of the form

$$\def\ad{\textrm{ad}\;}\ad x \mapsto [x,y],\quad x,y \in L$$ are called inner derivations or adjoint transformations. In $\Der(L)$ they form a subalgebra, $\ad L$, and the mapping $x\mapsto \ad x$ is a homomorphism of Lie algebras $L\to \Der(L)$ (the adjoint representation of the Lie algebra $L$); its image $\ad L$ is isomorphic to the quotient algebra of $L$ with respect to its centre

$$Z(L) = \{ x\in L \;|\; [x,y] = 0 \textrm{ for all } y\in L\}.$$ 2) Another important source of Lie algebras is connected with the following simple observation. If $L$ is an associative algebra over $k$ (cf. Associative rings and algebras) with multiplication $(x,y)\mapsto xy$, then the multiplication in the $k$-module $L$ specified by the rule

$$(x,y)\mapsto [x,y] = xy-yx$$ endows $L$ with the structure of a Lie algebra over $k$. One says that $(L,[\;,\;])$ is the Lie algebra associated with the associative algebra $(L, \cdot)$. Thus, the classical example of a Lie algebra $(L,[\;,\;])$ is obtained if for $(L, \cdot)$ one takes the (associative) algebra $\def\M{\textrm{M}}\M_n(k)$ of all square matrices of order $n$ over $k$.

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

$$\def\tr{\textrm{tr}\;}A_n = \{x\in \M_{n+1}(k)\;|\; \tr x = 0 \},\quad n \ge 1; $$

$$B_n = \{x \in\M_{2n+1}(k)\;|\; xB+B^t x = 0\},\quad n\ge 2,$$

$$B=\begin{pmatrix} 1&0&0\\0&0&E_n\\0&E_n&0\end{pmatrix},\quad E_n = \begin{pmatrix}1&0&\cdots&0\\0&1&\cdots&0\\ \vdots &\ddots &\ddots & \vdots\\0&0&\cdots&1 \end{pmatrix}\in\M_n(k);$$

$$C_n = \{x\in\M_{2n}\;|\; xC+C^t x = 0\},\quad n\ge 3,\quad C = \begin{pmatrix}0&E_n\\-E_n&0\end{pmatrix};$$

$$D_n = \{x\in\M_{2n}\;|\; xD+D^t x = 0\},\quad n\ge 4,\quad C = \begin{pmatrix}0&E_n\\E_n&0\end{pmatrix}.$$

One has $\dim A_n = n(n+2)$, $\dim B_n = n(2n+1)$, $\dim C_n = n(2n+1)$, $\dim D_n = n(2n-1)$.

A remarkable result is that over an algebraically closed field of characteristic zero these Lie algebras, together with five exceptional Lie algebras $G_2$, $F_4$, $E_6$, $E_7$, $E_8$, 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 $k$, 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 [Go], [DuFoNo] and Vector field on a manifold). Let $F$ be the ring of $C^\infty$-smooth functions on a $C^\infty$-smooth manifold $M$. The vector space $\def\Vect{\textrm{Vect}}\Vect(M)$ of all $C^\infty$-smooth vector fields on $M$ 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 $\Vect(M)$ coincides with the Lie algebra $\Der(F)$. Generally speaking, this algebra is infinite-dimensional. If $M$ is a Lie group, then the subspace of $\Vect(M)$ consisting of all left-invariant vector fields is a finite-dimensional subalgebra and is called the Lie algebra of the Lie group $M$; 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 $F$ by a commutative algebra $\def\cO{\mathcal{O}}\cO_n(k) = k[[X_1,\dots,X_n]]$ of formal power series over a field $k$, then instead of $\Vect(M)$ one obtains the Lie algebra $W_n$ of formal vector fields, which consists of the differential operators

$$D=\sum_{i=1}^nf_i\frac{\partial}{\partial X_i},\quad f_i\in\cO_n(k).$$ The subalgebras $S_n\subset W_{n+1},$, $H_n\subset W_{2n}$ consisting of derivations that annihilate the exterior differential forms

$$\omega = dX_1\wedge\cdots\wedge dX_{n+1},$$

$$\omega = \sum_{i=1}^n dX_i\wedge dX_{i+n},$$ respectively, and also the subalgebra $K_n\subset W_{2n-1}$ of derivations that multiply the form

$$\omega = dX_{2n-1} + \sum_{i=1}^{n-1}(X_i dX_{i+n-1} - X_{i+n-1} dX_{i}$$ by elements of $\cO_n(k)$, constitute, together with the algebra $W_n$, important classes of simple infinite-dimensional Lie algebras (Lie algebras of Cartan type). The algebra $W_n$ is called general, $S_n$ is called special, $H_n$ is called Hamiltonian, and $K_n$ is called a contact algebra. These algebras were encountered by S. Lie in the study of pseudo-groups of transformations ($k=\R$ or $\C$), and were then investigated for various reasons by E. Cartan and others (see [GuSt], [KoSh]).

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

$$G=G_1\supseteq G_2\supseteq \cdots$$ be the lower central series of $G$. Then $L$ is the direct sum of the additively written quotient groups $G_i/G_{i+1}$ and, by definition, the product of two elements $\bar x\in G_i/G_{i+1}$ and $\bar y\in G_j/G_{j+1}$ is the element of $G_{i+j}/G_{i+j+1}$ that is the class of the commutator of elements $x\in G_i$ and $y\in G_j$ that represent $\bar x$ and $\bar y$, respectively. This operation can be extended by distributivity to arbitrary elements of $L$. 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 $L$ over a field $k$ there is an associative $k$-algebra $U$ such that $LL$ can be isomorphically imbedded in the Lie algebra $(U,[\;,\;])$ associated with $U$ (see Universal enveloping algebra).

Let $L$ be a finite-dimensional Lie algebra over a field $k$ of characteristic zero. Then $L$ is linear, that is, is isomorphic to a subalgebra of a certain Lie algebra $\M_n(k)$ (Ado's theorem). In $L$ there is a unique largest solvable ideal $R$, called the radical (see Lie algebra, solvable). Moreover, in $L$ there is a subalgebra $S$ (called a Levi subalgebra) such that $L$ is the direct sum of the vector spaces $S$ and $R$, and any other subalgebra with this property can be transformed into $S$ by an automorphism of $L$ (the Levi–Mal'tsev theorem, cf. also Levi–Mal'tsev decomposition). Such a subalgebra $S$ is semi-simple (that is, its radical is equal to zero), and it can be characterized as a maximal semi-simple subalgebra of $L$. Thus, $L$ 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 $\R$ of the adjoint representation of $S$ in $L$). 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 $L=L_0\supset L_1\supset \cdots \supset L_n = 0$ such that $L_i$ is an ideal of $L_{i-1}$ and $L_{i-1}/L_i$ 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 $k$ 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 $k=\R$).

Finite-dimensional Lie algebras over a field of characteristic $p>0$ 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 [Bl]). For any $p$ 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 [GuSt], , [KoSh]).

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 [Ca]. These algebras have a filtration for which the associated graded Lie algebra has the form $\bigoplus_{i=-1}^\infty G_i$ 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 [GuSt], , [SiSt]). 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 [DuFoNo]).

The abstract theory of infinite-dimensional Lie algebras (see [AmSt], 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.

Comments

Let $k$ now be of characteristic $p>0$. A simple Lie algebra of classical type in characteristic $p$, $p>0$, is one "like" the simple algebras in characteristic zero corresponding to the root systems $A_n, B_n, C_n, E_6$, $E_7$, $E_8$, $F_4$, $G_2$. 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 $\Z$; reduce the coefficients in the multiplication table of the basis elements modulo $p$ to obtain a Lie algebra over $Z/(p)$; divide out, when necessary, the centre of this algebra over $\Z/(p)$; extend the scalars to $k\supset \Z/(p)$.

Perhaps somewhat confusingly, the algebras of classical type in characteristic $p>0$ are taken to include the five exceptional types $E_6$, $E_7$, $E_8$, $F_4$, $G_2$. The Lie algebras of classical type can also be characterized by the Mills–Seligman axioms, [Se2]: if $\def\f#1{\mathfrak{#1}}\f G$ is simple (and finite dimensional) and has a Cartan subalgebra $\f h$ which acts diagonally on each root space $\def\a{\alpha}\def\b{\beta}\f g_\a$, if, moreover, $\dim [\f g_\a,\f g_{-\a}] = 1$ for each root $\a \ne 0$, and if whenever $\a,\b \ne 0$ are roots, then not all $\a+i\b $, $i\in\Z$, are roots, then $\f g$ is classical. A simple Lie algebra with a projective representation with non-degenerate trace form is classical (and vice versa). If $k$ is algebraically closed, the simple Lie algebras of classical type are classified by their Dynkin diagrams, and over an arbitrary (perfect) field $k$ of characteristic $p>0$ 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 $p\ne 2,3$) the following:

types $A_1$, $B$, $C$, $G_2$, $F_4$, $E_7$, $E_8$: one;

types $A_n$ ($n>1$), $D_n$ ($n>4$), $E_6$: two;

type $D_4$: three.

Besides the simple Lie algebras of classical type over a field $k$ of characteristic $p>0$ 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 $W$, $S$, $H$, $K$, and they arise as (twisted) subalgebras of the algebras of derivations $W(m;\bf n)$, which are defined as follows.

Define a co-algebra structure (cf. Co-algebra) on the algebra of polynomials $k[X_1,\dots,X_m]$ by defining $\mu(X_i)=1\otimes X_i+X_i\otimes 1$ (making $k[X_1,\dots,X_m]$ a Hopf algebra). The dual algebra $A(m)$ is an infinite-dimensional associative and commutative algebra consisting of all formal sums $\sum c_\a x^\a$, where $\a$ ranges over all $m$-tuples of non-negative integers. The multiplication is given by

$$x^\a x^\b = {\a+\b \choose \a} x^{\a+\b}.$$

For each $m$-tuple of non-negative integers ${\bf n}$, let $A(m;{\bf n})$ denote the span of the $x^\a$ with $\a_i < p^{n_i}$ for all $i$. Then $A(m;{\bf n})$ is a subalgebra of $A(m)$. Let $W(m;{\bf n})$ be the derivation algebra of $A(m;{\bf n})$. The algebra $W(m;{\bf n})$ is simple of dimension $mp^{|{\bf n}|}$, where $|{\bf n}| = n_1+\cdots + n_m$. The $W(m;(1,\dots,1))$ are the Jacobson–Witt algebras $W_n$, cf. Witt algebra. For a description of the various (twisted) subalgebras of types $S$, $H$, $K$ of the $W(m;{\bf n})$ cf. [BlWi]–[Wi]. These are the Lie algebras of Cartan type. The generalized Kostrikin–Shafarevich conjecture states that every simple Lie algebra over $k$ is of classical type or of Cartan type.

A restricted Lie algebra $\f g$ over $k$ is such that for every $x\in \f g$ there is a $y\in \f g$ such that $(\ad x)^p = \ad y$. If $\f g$ is simple, then $y$ 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 $p>7$, [BlWi2], [BlWi].

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 $\a$-root string through $\b$ may include all roots $\b+i\a$, $i=0,\dots,p-1$; there may be infinitely non-conjugate Cartan subalgebras of a given dimension. Cf. [Be]–[St].

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