Lie algebra, graded

A Lie algebra $\mathfrak{g}$ over a field $K$ that is graded by means of an Abelian group $A$, that is, which splits into a direct sum of subspaces ${\mathfrak{g}}_{\alpha }$, $\alpha \in A$, in such a way that $[{\mathfrak{g}}_{\alpha },{\mathfrak{g}}_{\beta }]\subseteq {\mathfrak{g}}_{\alpha +\beta }$. If $A$ is an ordered group, then for every filtered Lie algebra (cf. Filtered algebra) the graded algebra associated with it is a graded Lie algebra.

Graded Lie algebras play an important role in the classification of simple finite-dimensional Lie algebras, Jordan algebras and their generalizations, and primitive pseudo-groups of transformations (see [3], [4]). For any semi-simple real Lie algebra its Cartan decomposition can be regarded as a ${\mathbf{Z}}_2$- grading. The local classification of symmetric Riemannian spaces reduces to the classification of ${\mathbf{Z}}_2$- graded simple complex Lie algebras [6].

Some constructions of graded Lie algebras.[edit]

1) Let $U$ be an associative algebra (cf. Associative rings and algebras) endowed with an increasing filtration $(U_k:k\in \mathbf{Z})$, suppose that $[U_k,U_l]\subset U_{k+l-d}$, where $d$ is a fixed natural number, and let ${\mathfrak{u}}_k=U_{k+d}/U_{k+d-1}$. Then the commutation operation in $U$ induces in the space $\mathfrak{u}=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{\mathfrak{u}}_k$ the structure of a $\mathbf{Z}$- graded Lie algebra. In this way one can obtain some Lie algebras of functions with the Poisson brackets as commutator. In the next two examples, $U_k=U_1^k$ for $k>0$ and $U_k=0$ for $k<0$.

a) Let $U$ be the algebra of linear differential operators with polynomial coefficients and let $U_1$ be the subspace spanned by its generators $p_i=\partial /\partial x_i$, $q_i=x_i$, $i=1\dots m$. Then $[U_k,U_l]\subset U_{k+l-2}$ and $\mathfrak{u}$ is the Lie algebra of polynomials in $p_i$ and $q_i$ with the usual Poisson brackets.

b) Let $U$ be the universal enveloping algebra of a finite-dimensional Lie algebra $\mathfrak{g}$ and let $U_1=\mathfrak{g}$. Then $[U_k,U_l]\subset U_{k+l-1}$ and $\mathfrak{u}$ is canonically isomorphic (as a vector space) to the symmetric algebra over $\mathfrak{g}$, that is, to the algebra of polynomials on the dual space ${\mathfrak{g}}^{\ast }$( the Poincaré–Birkhoff–Witt theorem). If $\mathfrak{g}$ is the Lie algebra of a connected Lie group $G$, then the commutator of elements of $\mathfrak{u}$ can be interpreted either as the Poisson brackets for the corresponding left-invariant functions on the cotangent bundle $T^{\ast }G$, or as the Poisson brackets on each orbit of the co-adjoint representation, defined by means of the standard symplectic structure on these orbits.

2) Suppose that $\mathrm{char}k\ne 2$ and that $E$ is an $n$- dimensional vector space over $k$ endowed with a non-singular quadratic form $Q$; let $e_1\dots e_n$ be an orthogonal basis of $E$. The decomposition of the Clifford algebra $C(Q)$ into the sum of one-dimensional subspaces $⟨e_{i_1}\dots e_{i_k}⟩$, $i_1<\cdots <i_k$, is a ${\mathbf{Z}}_2^n$- grading of it. For $n=2m$ the elements of the algebra $C(Q)$ with zero trace form a simple graded Lie algebra of type $A_N$, $N=2^m-1$; its grading has a high degree of symmetry; in particular, all graded subspaces are equivalent. Similar gradings (by means of various finite groups) exist for other simple Lie algebras [1].

3) To every Lie pseudo-group of transformations corresponds a Lie algebra of vector fields. The germ $l$ of this Lie algebra at any point has a natural $\mathbf{Z}$- filtration

$$l=l_{-1}\supset l_0\supset l_1\supset \dots ,$$

where $l_k$ contains the germs of those vector fields whose coordinates can be expanded in power series without terms of degree less than $k+1$. The associated graded Lie algebra can be interpreted as a Lie algebra of polynomial vector fields.

The classification of simple graded Lie algebras.[edit]

To simple primitive Lie pseudo-groups correspond the following four series of simple infinite-dimensional graded Lie algebras (see [5]):

$W_n$, the Lie algebra of all polynomial vector fields in the $n$- dimensional affine space;

$S_n$, its subalgebra consisting of vector fields with zero divergence;

$H_n$, where $n=2m$, the subalgebra consisting of vector fields that annihilate the differential form

$$\sum _{i=1}^{m}d{x}_i\wedge d{x}_{m+i}$$

(Hamiltonian vector fields);

$K_n$, where $n=2m+1$, the subalgebra consisting of vector fields that multiply the differential form

$$\sum _{i=1}^m(x_{m+i}d{x}_i-x_{i}d{x}_{m+i})+d{x}_n$$

by a function.

Over fields of characteristic $p>0$ one can define simple finite-dimensional graded Lie algebras analogous to $W_n$, $S_n$, $H_n$, and $K_n$( see [5]).

Simple graded Lie algebras of another type are obtained in the following way [4]. Let $\mathfrak{g}=\mathfrak{g}(A)$ be the Lie algebra defined by means of an indecomposable Cartan matrix $A=\Vert a_{ij}\Vert$, $i,j=1\dots n$( from now on the notation of the article Cartan matrix is used). The algebra $\mathfrak{g}$ is endowed with a ${\mathbf{Z}}^k$- grading so that $h_i\in {\mathfrak{g}}_0$, $e_i\in {\mathfrak{g}}_{{\alpha }_i}$, $f_i\in {\mathfrak{g}}_{-{\alpha }_i}$, where ${\alpha }_i$ is the row $(0\dots 1\dots 0)$ with $1$ in the $i$- th place. Elements $\alpha \in {\mathbf{Z}}^n$ for which ${\mathfrak{g}}_{\alpha }\ne 0$ are called roots, and the ${\alpha }_i$ are called simple roots. Any root is a linear combination of simple roots with integer coefficients of the same sign and $\dim {\mathfrak{g}}_{\alpha }<\mathrm{\infty }$ for any $\alpha \in {\mathbf{Z}}^n$. The quotient algebra ${\mathfrak{g}}^{\prime }(A)$ of $\mathfrak{g}$ with respect to its centre, which lies in ${\mathfrak{g}}_0$, is simple as a graded algebra, that is, it does not have non-trivial graded ideals.

Let $R$ be the totality of linear combinations of rows of the matrix $A$ with positive coefficients. Then one of the following cases holds:

(P) $R$ contains a row all elements of which are positive;

(Z) $R$ contains a zero row;

(N) $R$ contains a row all elements of which are negative.

In the case (P), $\mathfrak{g}(A)={\mathfrak{g}}^{\prime }(A)$ is a simple finite-dimensional Lie algebra. In the case (N), $\mathfrak{g}(A)$ is a simple infinite-dimensional Lie algebra. In the case (Z), the algebra ${\mathfrak{g}}^{\prime }={\mathfrak{g}}^{\prime }(A)$ is simple only as a graded algebra. It can be converted in a $K[u,u^{-1}]$- algebra so that: a) $u{\mathfrak{g}}_{\alpha }^{\prime }={\mathfrak{g}}_{\alpha +\nu }^{\prime }$, where $\nu$ is a row of positive numbers; and b) the quotient algebra ${\mathfrak{g}}^{\prime }/(1-u){\mathfrak{g}}^{\prime }=\bar{\mathfrak{g}}$ is a simple finite-dimensional Lie algebra. The greatest common divisor of all components ${\nu }_i$ of the row $\nu$, which is equal to 1, 2 or 3, is called the index of the algebra ${\mathfrak{g}}^{\prime }$.

The following table is a list of all simple graded Lie algebras with Cartan matrix of type (Z). Here the algebra ${\mathfrak{g}}^{\prime }$ is denoted by the same symbol as the associated simple finite-dimensional Lie algebra $\bar{\mathfrak{g}}$, but with the addition of its index, given in brackets.

The diagram of simple roots describes the matrix $A$. Its vertices correspond to the simple roots; the $i$- th and $j$- th vertices are joined by an $(a_{ij}{a}_{ji})$- multiple edge, directed from the $i$- th vertex to the $j$- th if $|a_{ij}|>|a_{ji}|$, and undirected if $|a_{ij}|=|a_{ji}|$. Above the vertices stand the numbers ${\nu }_i$.

By means of graded Lie algebras with Cartan matrix of type (Z) one can classify ${\mathbf{Z}}_m$- graded simple finite-dimensional Lie algebras (see [4], [2]). Namely, let ${\mathfrak{g}}^{\prime }={\mathfrak{g}}^{\prime }(A)$, where $A$ satisfies condition (Z), and let $p:{\mathbf{Z}}^n\to \mathbf{Z}$ be a homomorphism such that $p({\alpha }_i)\ge 0$ and $p(\nu )=m$. Then ${\mathfrak{g}}_k^{\prime }=\sum _{p(\alpha )=k}{\mathfrak{g}}_{\alpha }^{\prime }$ for any $k\in \mathbf{Z}$ is mapped isomorphically onto the subspace $\bar{\mathfrak{g}}{}_k\subset \bar{\mathfrak{g}}$, which depends only on the residue of $k$ modulo $m$, and the decomposition $\bar{\mathfrak{g}}=\sum _{k=0}^{m-1}\bar{\mathfrak{g}}{}_k$ is a ${\mathbf{Z}}_m$- grading of $\bar{\mathfrak{g}}$. If the field $K$ is algebraically closed, then by the method described one obtains, without repetition, all ${\mathbf{Z}}_m$- graded simple finite-dimensional Lie algebras over $K$. The index of ${\mathfrak{g}}^{\prime }$ is equal to the order of the automorphism $\theta :x\mapsto (\exp (2\pi ik/m))x$, $x\in {\mathfrak{g}}_k$, of the algebra $\bar{\mathfrak{g}}$ modulo the group of inner automorphisms.

There is a classification of simple $\mathbf{Z}$- graded Lie algebras $\mathfrak{g}=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{\mathfrak{g}}_k$ satisfying the conditions: a) $\dim {\mathfrak{g}}_k\le C|k{|}^N$ for some $C$ and $N$; b) $\mathfrak{g}$ is generated by the subspace ${\mathfrak{g}}_{-1}+{\mathfrak{g}}_0+{\mathfrak{g}}_1$; and c) the representation of ${\mathfrak{g}}_0$ on ${\mathfrak{g}}_{-1}$ is irreducible. In this case either $\mathfrak{g}$ is finite-dimensional or it is one of the algebras $W_n$, $S_n$, $H_n$, $K_n$, or it is the algebra ${\mathfrak{g}}^{\prime }(A)$ defined by a Cartan matrix of type (Z), endowed with a suitable $\mathbf{Z}$- grading [4].

A Lie superalgebra is sometimes called a ${\mathbf{Z}}_2$- graded Lie algebra.

References[edit]

Comments[edit]

The Lie algebras $\mathfrak{g}(A)$ are called Kac–Moody algebras; they have close connections with many areas of mathematics and mathematical physics (cf. [a2]).

There is a second notion which also sometimes goes by the name of graded Lie algebra. This is a $\mathbf{Z}$- or $\mathbf{Z}/(2)$- graded vector space $V=\oplus V_i$ with a multiplication

$$[,]:V\times V\to V$$

such that

$$[V_i,V_j]\subset V_{i+j},$$

$$[x,y]=(-1{)}^{ij+1}[y,x],$$

for all $x\in V_i$, $y\in V_j$, and

$$(-1{)}^{ik}[[x,y],z]+(-1{)}^{ji}[[y,z],x]+(-1{)}^{kj}[[z,x],y]=0,$$

for all $x\in V_i$, $y\in V_j$, $z\in V_k$. One also says that $V=\oplus V_i$ has been equipped with a graded Lie product or graded Lie bracket.

Graded Lie brackets naturally arise, for instance, in cohomology theory in the context of deformations of algebras and complex structures, [a4]. A graded vector space $V$ with a graded Lie bracket is not a Lie algebra in the usual sense of the word. A Lie superalgebra is a $\mathbf{Z}/(2)$- graded vector space with a $\mathbf{Z}/(2)$- graded Lie bracket.

Of fundamental importance in recent progress in quantum field theory is the Virasoro algebra. This is a ( $\mathbf{Z}$- graded) Lie algebra with a basis $L_k$( $k\in \mathbf{Z}$) and $c$, and the following commutation relations:

$$[L_m,L_n]=(m-n)L_{m+n}+\frac{m^3-m}{12}{\delta }_{m,-n}c.$$

See [a1].

References[edit]