Symmetry, Integrability and Geometry: Methods and Applications On Free Pseudo-Product Fundamental Graded Lie Algebras

In this paper we first state the classification of the prolongations of complex free fundamental graded Lie algebras. Next we introduce the notion of free pseudo-product fundamental graded Lie algebras and study the prolongations of complex free pseudo-product fundamental graded Lie algebras. Furthermore we investigate the automorphism group of the prolongation of complex free pseudo-product fundamental graded Lie algebras.


Introduction
Let m = p<0 g p be a graded Lie algebra over the field R of real numbers or the field C of complex numbers, and let µ be a positive integer. The graded Lie algebra m = p<0 g p is called a fundamental graded Lie algebra if the following conditions hold: (i) m is finite-dimensional; (ii) g −1 = {0}, and m is generated by g −1 . Moreover a fundamental graded Lie algebra m = p<0 g p is said to be of the µ-th kind if g −µ = {0}, and g p = {0} for all p < −µ. It is shown that every fundamental graded algebra m = p<0 g p is prolonged to a graded Lie algebra g(m) = p∈Z g(m) p satisfying the following conditions: (i) g(m) p = g p for all p < 0; (ii) for X ∈ g(m) p (p 0), [X, m] = {0} implies X = 0; (iii) g(m) is maximum among graded Lie algebras satisfying conditions (i) and (ii) above. The graded Lie algebra g(m) is called the prolongation of m. Note that g(m) 0 is the Lie algebra of all the derivations of m as a graded Lie algebra.
Let m = p<0 g p be a fundamental graded Lie algebra of the µ-th kind, where µ 2. The fundamental graded Lie algebra m is called a free fundamental graded Lie algebra of type (n, µ) if the following universal properties hold: (i) dim g −1 = n; (ii) Let m = p<0 g p be a fundamental graded Lie algebra of the µ-th kind and let ϕ be a surjective linear mapping of g −1 onto g −1 . Then ϕ can be extended uniquely to a graded Lie algebra epimorphism of m onto m .
In Section 3 we see that a universal fundamental graded Lie algebra b(V, µ) of the µ-th kind introduced by N. Tanaka [11] becomes a free fundamental graded Lie algebra of type (n, µ), where µ 2, and V is a vector space such that dim V = n 2.
In [13], B. Warhurst gave the complete list of the prolongations of real free fundamental graded Lie algebras by using a Hall basis of a free Lie algebra. The complex version of his theorem has the completely same form except for the ground number field as follows: Theorem I. Let m = p<0 g p be a free fundamental graded Lie algebra of type (n, µ) over C.
The first purpose of this paper is to give a proof of Theorem I by using the classification of complex irreducible transitive graded Lie algebras of finite depth (cf. [6]). Note that Warhurst's methods in [13] are available to the proof of Theorem I.
Next we introduce the notion of free pseudo-product fundamental graded Lie algebras. Let m = p<0 g p be a fundamental graded Lie algebra, and let e and f be nonzero subspaces of g −1 .
Let m = p<0 g p be a pseudo-product fundamental graded Lie algebra with a pseudo-product structure (e, f), and let g(m) = p∈Z g(m) p be the prolongation of m. Moreover let g 0 be the Lie algebra of all the derivations of m as a graded Lie algebra preserving e and f. Also for p 1 we set g p = {X ∈ g(m) p : [X, g k ] ⊂ g p+k for all k < 0} inductively. Then the direct sum g = p∈Z g p becomes a graded subalgebra of g(m), which is called the prolongation of (m; e, f). Let m = p<0 g p be a pseudo-product fundamental graded Lie algebra of the µ-th kind with pseudo-product structure (e, f), where µ 2. The pseudo-product fundamental graded Lie algebra m = p<0 g p is called a free pseudo-product fundamental graded Lie algebra of type (m, n, µ) if the following conditions hold: (i) dim e = m and dim f = n; (ii) Let m = p<0 g p be a pseudo-product fundamental graded Lie algebra of the µ-th kind with pseudo-product structure (e , f ) and let ϕ be a surjective linear mapping of g −1 onto g −1 such that ϕ(e) ⊂ e and ϕ(f) ⊂ f . Then ϕ can be extended uniquely to a graded Lie algebra epimorphism of m onto m .
The main purpose of this paper is to prove the following theorem.
Theorem II. Let m = p<0 g p be a free pseudo-product fundamental graded Lie algebra of type (m, n, µ) with pseudo-product structure (e, f) over C, and let g = p∈Z g p be the prolongation of (m; e, f). If g 1 = {0}, then g = p∈Z g p is a finite-dimensional simple graded Lie algebra of type Let g = p∈Z g p be the prolongation of a free pseudo-product fundamental graded Lie algebra m = p<0 g p with pseudo-product structure (e, f) over C. We denote by Aut(g; e, f) 0 the group of all the automorphisms as a graded Lie algebra preserving e and f, which is called the automorphism group of the pseudo-product graded Lie algebra g = p∈Z g p . In Section 9, we show that Aut(g; e, f) 0 is isomorphic to GL(e) × GL(f).

Notation and conventions
(1) From Section 2 to the last section, all vector spaces are considered over the field C of complex numbers.
(2) Let V be a vector space and let W 1 and W 2 be subspaces of V . We denote by W 1 ∧ W 2 the subspace of Λ 2 V spanned by all the elements of the form w 1 ∧ w 2 (w 1 ∈ W 1 , w 2 ∈ W 2 ).

Free fundamental graded Lie algebras
First of all we give several definitions about graded Lie algebras. Let g be a Lie algebra. Assume that there is given a family of subspaces (g p ) p∈Z of g satisfying the following conditions: (ii) dim g p < ∞ for all p ∈ Z; (iii) [g p , g q ] ⊂ g p+q for all p, q ∈ Z.
Under these conditions, we say that g = p∈Z g p is a graded Lie algebra (GLA). Moreover we define the notion of homomorphism, isomorphism, monomorphism, epimorphism, subalgebra and ideal for GLAs in an obvious manner.
Next we define fundamental GLAs. A GLA m = p<0 g p is called a fundamental graded Lie algebra (FGLA) if the following conditions hold: , and m is generated by g −1 , or more precisely g p−1 = [g p , g −1 ] for all p < 0.
If an FGLA m = p<0 g p is of depth µ, then m is also said to be of the µ-th kind. Moreover an Let m = p<0 g p be an FGLA of the µ-th kind, where µ 2. m is called a free fundamental graded Lie algebra of type (n, µ) if the following conditions hold: (i) dim g −1 = n; (ii) Let m = p<0 g p be an FGLA of the µ-th kind and let ϕ be a surjective linear mapping of Then ϕ can be extended uniquely to a GLA epimorphism of m onto m .
Proposition 2.1. Let n and µ be positive integers such that n, µ 2.
(1) There exists a unique free FGLA of type (n, µ) up to isomorphism.
(2) Let m = p<0 g p be a free FGLA of type (n, µ). We denote by Der(m) 0 the Lie algebra of all the derivations of m preserving the gradation of m. Then the mapping Φ : is a Lie algebra isomorphism.
(1) The uniqueness of a free FGLA of type (n, µ) follows from the definition. We set X = {1, . . . , n}. Let L(X) be the free Lie algebra on X (see [1, Chapter II, § 2]) and let i : X → L(X) be the canonical injection. We define a mapping φ of X into Z by φ(k) = −1 (k ∈ X). The mapping φ defines the natural gradation (L(X) p ) p<0 on L(X) such that: . Note that if n > 1, then L(X) p = 0 for all p < 0. We set a = p<−µ L(X) p ; then a is a graded ideal of L(X) and the factor GLA m = L(X)/a becomes an FGLA of the µ-th kind. We put a p = a ∩ L(X) p and g p = L(X) p /a p . Now we prove that m = p<0 g p is a free FGLA of type (n, µ). Let m = p<0 g p be an FGLA of the µ-th kind and let ϕ be a surjective linear mapping of g −1 onto g −1 . Let h be a mapping of X into m defined by h(k) = ϕ(i(k)) (k ∈ X). Then there exists a Lie algebra homomorphismh of L(X) into m such thath•i = h. Since L(X) (resp. m ) is generated by L(X) −1 (resp. g −1 ),h is surjective. Since m = p<0 g p is of the µ-th kind,h(a) = 0, soh induces a GLA epimorphism L(ϕ) of m onto m such that L(ϕ)|g −1 = ϕ. The homomorphism L(ϕ) is unique, because m = p<0 g p is generated by g −1 . Thus m is a free FGLA of type (n, µ).
(2) Assume that m is a free FGLA constructed in (1). Let φ be an endomorphism of g −1 . By Corollary to Proposition 8 of [1, Chapter II, § 2, no. 8], φ can be extended uniquely to a unique derivation D of L(X). Since D(L(X) −1 ) = φ(L(X) −1 ) = φ(g −1 ) ⊂ L(X) −1 , and since L(X) is generated by L(X) −1 , we see that D(L(X) p ) ⊂ L(X) p and D(a) ⊂ a. Thus there is a derivation of D φ of m such that π • D = D φ • π, where π is the natural projection of L(X) onto m. The correspondence gl(g −1 ) φ → D φ ∈ Der(m) 0 is an injective linear mapping. Hence dim gl(g −1 ) dim Der(m) 0 . On the other hand, since m is generated by g −1 , the mapping Φ is a Lie algebra monomorphism. Therefore Φ is a Lie algebra isomorphism. (1) From the proof of Proposition 2.1, there exists a unique GLA homomorphism L(ϕ) of m into m such that L(ϕ)|g −1 = ϕ.
(2) Let m = p<0 g p be an FGLA of the µ-th kind, and let ϕ be a linear mapping of g −1 into g −1 . Assume that m = p<0 g p is a free FGLA. By the uniqueness of L(ϕ • ϕ), we see (3) Assume that m = p<0 g p is a free FGLA and ϕ is injective. By the result of (2), L(ϕ) is a monomorphism.
(4) Let W be an m-dimensional subspace of g −1 with m 2. By the result of (3), the subalgebra of m generated by W is a free FGLA of type (m, µ).

Universal fundamental graded Lie algebras
Following N. Tanaka [11], we introduce universal FGLAs of the µ-th kind.
to be the identity mapping. For k −3, we define b(V ) k and B k inductively as follows: We stands for the cyclic sum with respect to X, Y , Z, and X u denotes the b(V ) u - Let µ be a positive integer. Assume that µ 2 and dim V = n 2. Since becomes an FGLA of µ-th kind, which is called a universal fundamental graded Lie algebra of the µ-th kind. By [11,Proposition 3.2], b(V, µ) is a free FGLA of type (n, µ).

The prolongations of fundamental graded Lie algebras
Following N. Tanaka [11], we introduce the prolongations of FGLAs. Let m = p<0 g p be an FGLA. A GLA g(m) = p∈Z g(m) p is called the prolongation of m if the following conditions hold: We construct the prolongation g(m) = p∈Z g(m) p of m. We set g(m) p = g p (p < 0). We define subspaces g(m) k (k 0) of Hom(m, Next for k > 0 we define g(m) k (k 1) inductively as follows: It follows easily that [X, Y ] ∈ g(m) k+l . With this bracket operation, g(m) = p∈Z g(m) p becomes a graded Lie algebra satisfying conditions (i), (ii) and (iii) above. Let m and g(m) be as above. Assume that we are given a subalgebra g 0 of g(m) 0 . We define subspaces g k (k 1) of g(m) k inductively as follows: If we put g = p∈Z g p , then g = p∈Z g p becomes a transitive graded Lie subalgebra of g(m), which is called the prolongation of (m, g 0 ).
By Proposition 2.1 (2) we get the following proposition.
is an isomorphism.
Conversely we obtain the following proposition. Proof . We put n = dim g −1 . We consider a universal FGLA b( is an isomorphism, ϕ can be extended to be a homomorphismφ of p 0b Let a be the kernel ofφ; then a is a graded ideal of p 0b Also each a p is ab( From µ 3 it follows that ϕ is an isomorphism.

Finite-dimensional simple graded Lie algebras
Following [15], we first state the classification of finite-dimensional simple GLAs.
Let g = p∈Z g p be a finite-dimensional simple GLA of the µ-th kind over C such that the negative part g − is an FGLA. Let h be a Cartan subalgebra of g 0 ; then h is a Cartan subalgebra of g such that E ∈ h, where E is the element of g 0 such that [E, x] = px for all x ∈ g p and p. Let ∆ be a root system of (g, h). For α ∈ ∆, we denote by g α the root space corresponding to α. We set h R = {h ∈ h : α(h) ∈ R for all α ∈ ∆} and let (h 1 , . . . , h l ) be a basis of h R such that h 1 = E. We define the set of positive roots ∆ + as the set of roots which are positive with respect to the lexicographical ordering in h * R determined by the basis (h 1 , . . . , h l ) of h R . Let Π ⊂ ∆ + be the corresponding simple root system. We denote by {m 1 , . . . , m l } the coordinate functions corresponding to Π, i.e., for α ∈ ∆, we can write α = We set α i (E) = s i and s = (s 1 , . . . , s l ); then each s i is a non-negative integer. For α ∈ ∆, we call the integer s (α) = l i=1 m i (α)s i the s-length of α. We put ∆ p = {α ∈ ∆ : s (α) = p}, Π p = ∆ p ∩ Π and I = {i ∈ {1, . . . , l} : s i = 1}. Let θ be the highest root of g; then s (θ) = µ. Also since the g 0 -module g −µ is irreducible, dim g −µ = 1 if and only if θ, α ∨ i = 0 for all i ∈ {1, . . . , l} \ I, where {α ∨ i } is the simple root system of the dual root system ∆ ∨ of ∆ corresponding to {α i }. In our situation, since g − is generated by g −1 , we have s i = 0 or 1 for all i. The l-tuple s = (s 1 , . . . , s l ) of non-negative integers is determined only by the ordering of (α 1 , . . . , α l ). In what follows, we assume that the ordering of (α 1 , . . . , α l ) is as in the table of [2]. If g has the Dynkin diagram of type X l (X = A, . . . , G), then the simple GLA g = p∈Z g p is said to be of type (X l , Π 1 ). Here we remark that for an automorphismμ of the Dynkin diagram, a simple GLA of type (X l , Π 1 ) is isomorphic to that of type (X l ,μ(Π 1 )). We will identify a simple GLA of type (X l , Π 1 ) with that of type (X l ,μ(Π 1 )).
For i ∈ I, we put ∆ For i ∈ I, we denote by g (i) the subalgebra of g generated by g 1 ; then g (i) is a simple GLA whose Dynkin diagram is the connected component containing the vertex i of the subdiagram of X l corresponding to vertices ({1, . . . , l} \ I) ∪ {i}. We denote by θ (i) the highest root of g (i) . Then [g if and only if m i (θ (i) ) = 1. From Theorem 5.2 of [15], we obtain the following theorem: Theorem 5.1. Let g = p∈Z g p be a finite-dimensional simple GLA over C such that g − is an FGLA and the g 0 -module g −1 is irreducible. Then g = p∈Z g p is the prolongation of g − except for the following cases: (a) g − is of the first kind; (b) g − is of the second kind and dim g −2 = 1.
Let g = p∈Z g p be a finite-dimensional simple GLA. Now we assume that g 0 is isomorphic to gl(g −1 ); then the g 0 -module g −1 is irreducible. The derived subalgebra [g 0 , g 0 ] of g 0 is a semisimple Lie algebra whose Dynkin diagram is the subdiagram of X l consisting of the vertices {1, . . . , l} \ I. Since [g 0 , g 0 ] is of type A l−1 and since the g 0 -module g −1 is elementary, (X l , ∆ 1 ) is one of the following cases: From this result and Propositions 4.1 and 4.2, we get the following theorem: Theorem 5.2. Let g = p∈Z g p be a finite-dimensional simple GLA of type (X l , Π 1 ) over C satisfying the following conditions: (i) g − is an FGLA of the µ-th kind; (ii) The g 0 -module g −1 is irreducible; (iii) g 0 is isomorphic to gl(g −1 ); (iv) g is the prolongation of g − .
Then g − is a free FGLA of type (l, µ), and g = p∈Z g p is one of the following types: Let W (m) be the Lie algebra consisting of all the polynomial vector fields For an m-tuple s = (s 1 , . . . , s m ) of positive integers, we denote by W (m; s) p the subspaces of W (m) consisting of those polynomial vector fields (6.1) such that the polynomials P i are contained in A(m; s) p+s i ; then W (m; s) = p∈Z W (m; s) p is a transitive GLA. In particular, We now consider the following differential form (Here the action of D on the differential forms is extended from its action A(2n + 1) by requiring that D be derivation of the exterior algebra satisfying [3,5]).
From Proposition 2.2 of [6], we get Theorem 6.1. Let g = p∈Z g p be a transitive GLA over C satisfying the following conditions: (i) g − is an FGLA of the µ-th kind; (ii) g is infinite-dimensional; (iii) The g 0 -module g −1 is irreducible; (iv) g is the prolongation of g − .
Then µ 2 and g = p∈Z g p is isomorphic to W (m; 1 m ) or K(n).

Classif ication of the prolongations of free fundamental graded Lie algebras
Let m = p<0 g p be a free FGLA of type (n, µ) over C, and let g(m) = p∈Z g(m) p be the prolongation of m. First of all, we assume that dim g(m) = ∞. By Theorem 6.1, g(m) is isomorphic to K(m) as a GLA, where n = 2m. Since K(m) 0 is isomorphic to csp(m, C) and since g(m) 0 is isomorphic to gl(n, C), we see that m = 1. Therefore g(m) is isomorphic to K(1) as a GLA. Next we assume that dim g(m) < ∞ and g(m) 1 = 0. Since the g(m) 0 -module g(m) −1 is irreducible, g(m) is a finite-dimensional simple GLA (see [4,7]). By Theorem 5.2, g(m) is isomorphic to one of the following types: Thus we get a proof of the following theorem: Theorem 7.1. Let m = p<0 g p be a free FGLA of type (n, µ) over C, and let g(m) = p∈Z g(m) p be the prolongation of m. Then one of the following cases occurs: (a) (n, µ) = (n, 2) (n 2), (2,3). In this case, g(m) 1 = {0}.

Free pseudo-product fundamental graded Lie algebras
An FGLA m = p<0 g p equipped with nonzero subspaces e, f of g −1 is called a pseudo-product FGLA if the following conditions hold: The pair (e, f) is called the pseudo-product structure of the pseudo-product FGLA m = p<0 g p .
We will also denote by the triplet (m; e, f) the pseudo-product FGLA m = On the other hand, since m is a free FGLA, . From this fact it follows that m = l = 1.
(2) Assume that m = n = 1 and ϕ is injective. ThenL(ϕ)(m) is a graded subalgebra of m isomorphic to a free FGLA of type (2, µ). From this result, the subalgebra of m generated by a nonzero element X of e and a nonzero element Y of f is a free FGLA of type (2, µ).
Let m = p<0 g p be a pseudo-product FGLA of the µ-th kind with pseudo-product structure (e, f). We denote by g 0 the Lie algebra of all the derivations of m preserving the gradation of m, e and f: The prolongation g = p∈Z g p of (m, g 0 ) is called the prolongation of (m; e, f).
A transitive GLA g = p∈Z g p is called a pseudo-product GLA if there are given nonzero subspaces e and f of g −1 satisfying the following conditions: (i) The negative part g − is a pseudo-product FGLA with pseudo-product structure (e, f); The pair (e, f) is called the pseudo-product structure of the pseudo-product GLA g = p∈Z g p . If the g 0 -modules e and f are irreducible, then the pseudo-product GLA g = p∈Z g p is said to be of irreducible type.
The following lemma is due to N. Tanaka (cf. [9]). Here we give a proof for the convenience of the readers. (1) If g − is non-degenerate, then g is finite-dimensional.
(2) If g = p∈Z g p is of irreducible type and µ 2, then g is finite-dimensional. (2) We may assume that g 1 = {0}. By [16, Lemma 2.4], the g 0 -modules e, f are not isomorphic to each other. We put d = {X ∈ g −1 : [X, The prolongation of a pseudo-product FGLA becomes a pseudo-product GLA. By Proposition 8.2 (2), the prolongation of a free pseudo-product FGLA is a pseudo-product GLA of irreducible type. By Lemma 8.1 (2), the prolongation of a free pseudo-product FGLA is finitedimensional. Proposition 8.3. Let m = p<0 g p be a free pseudo-product FGLA of type (m, n, µ) with pseudoproduct structure (e, f) and let g = p∈Z g p be the prolongation of (m; e, f).
(3) g −3 is isomorphic to S 2 (e) ⊗ f ⊕ S 2 (f) ⊗ e as a g 0 -module. In particular, dim g −3 = Moreover, a −3 is isomorphic to This completes the proof. be the prolongation of (m; e, f). Assume that g 0 is isomorphic to gl(e) ⊕ gl(f) as a Lie algebra.
(1) If µ = 2, then m = p<0 g p be a free pseudo-product FGLA. p . We set a p = a ∩ǧ p ; then a = p 0 a p . Since the restriction of Also each a p is aǧ 0 -submodule ofǧ p . Since theǧ 0 -moduleǧ −2 is irreducible (Proposition 8.3 (2)), ϕ|g −2 is injective. If µ = 2, then ϕ is an isomorphism. This proves the assertion (1). Now we assume that µ 3. Theň  Example 8.1. Let V and W be finite-dimensional vector spaces and k 1. We set The bracket operation of C k (V, W ) is defined as follows: Equipped with this bracket operation, C k (V, W ) becomes a pseudo-product FGLA of the (k +1)th kind with pseudo-product structure (V, W ⊗ S k (V * )), which is called the contact algebra of order k of bidegree (n, m), where n = dim V and m = dim W (cf. [14, p. 133]). We assume that C k (V, W ) is a free pseudo-product FGLA. Since Thus we obtain that C k (V, W ) is a free pseudo-product FGLA if and only if k = 1, n = 1. We set e = g (m) . Then (g − ; e, f) is a pseudo-product FGLA. Since dim e = m, dim f = n and dim g −2 = mn, the pseudo-product FGLA (g − ; e, f) is a free pseudo-product FGLA of type (m, n, 2) (Proposition 8.3 (2)). Also g = p∈Z g p is the prolongation of g − except for the following cases (see [15]): (1) m = n = 1. In this case, the prolongation of g − is isomorphic to K(1).
(2) m = 1 or n = 1 and l = max{m, n} 2. In this case, the prolongation of g − is isomorphic to W (l + 1; s), where s = (1, 2, . . . , 2). Example 8.3. Let V and W be finite-dimensional vector spaces such that dim V = m 1 and dim W = n 1. We set The bracket operation of m is defined as follows: where v, v ∈ V and w, w ∈ W . Equipped with this bracket operation, m becomes a free pseudoproduct FGLA of type (m, n, 3) with pseudo-product structure (V, W ) (Proposition 8.3).
Proposition 9.1. Let m = p<0 g p be an FGLA and let g(m) = p∈Z g(m) p be the prolongation of m. The mapping Φ : Aut(g(m)) 0 φ → φ|m ∈ Aut(m) 0 is an isomorphism.
Proof . It is clear that Φ is a group homomorphism. We prove that Φ is injective. Let φ be an element of Ker Φ. Assume that φ(X) = X for all X ∈ g(m) p (p < k). For X ∈ g(m) k , Y ∈ g −1 , Since [X, Y ] ∈ g(m) k−1 , we have [φ(X) − X, Y ] = 0. By transitivity, φ(X) = X. By induction, we have proved φ to be the identity mapping. Hence Φ is a monomorphism. We prove that Φ is surjective. Let ϕ ∈ Aut(m) 0 . We construct the mapping ϕ p : g(m) p → g(m) p inductively as follows: First for X ∈ g(m) 0 , we set ϕ 0 (X) = ϕXϕ −1 . Then for Y, Z ∈ m so ϕ 0 (X) ∈ g(m) 0 . Furthermore we can prove easily that [ϕ 0 (X), ϕ p (Y )] = ϕ p ([X, Y ]) for X ∈ g 0 and Y ∈ g p (p 0). Here for p < 0 we set ϕ p = ϕ|g(m) p . Assume that we have defined linear isomorphisms ϕ p of g(m) p onto itself (0 p < k) such that for X ∈ g(m) r , Y ∈ g(m) s (r + s < k, r < k, s < k). For X ∈ g(m) k we define ϕ k (X) ∈ Hom(m, p k−1 g(m) p ) k as follows: Y ∈ g s , s < 0.
For a pseudo-product GLA g = p∈Z g p with pseudo-product structure (e, f), we denote by Aut(g; e, f) 0 the group of all the automorphisms of g preserving the gradation of g, e and f: Aut(g; e, f) 0 = {ϕ ∈ Aut(g) 0 : ϕ(e) = e, ϕ(f) = f}.