Gradings on Lie Algebras of Cartan and Melikian Type

Gradings on Lie Algebras of Cartan and Melikian Type Jason McGraw Department of Mathematics and Statistics Memorial University of Newfoundland St. John's, Canada jason.mcgraw@mun.ca July 22, 2010

Jason McGraw (MUN) Innite Dimensional Lie Algebras Workshop July 2010 2 / 36 Outline 1 Brief Overview of Gradings on Simple Lie Algebras 2 Brief Review of the Cartan Lie Algebras and Melikian Algebras 3 Canonical Z-Gradings 4 Automorphisms 5 Normalizers of Tori 6 Conclusion

Brief Overview of Gradings on Simple Lie Algebras Jason McGraw (MUN) Innite Dimensional Lie Algebras Workshop July 2010 3 / 36 Denition of a Grading Denition A grading Γ on an algebra A by a group G, also called a G -grading, is the decomposition of A as the direct sum of subspaces A g, Γ : A = g G A g, such that A g A g A g g for all g, g G. For g G, the subspace A g is called the homogeneous space of degree g, and any nonzero element y A g is called homogeneous of degree g.

Brief Overview of Gradings on Simple Lie Algebras Jason McGraw (MUN) Innite Dimensional Lie Algebras Workshop July 2010 4 / 36 Group-Equivalent and Isomorphic Gradings Denition The set Supp Γ = {g G A g 0} is called the support of the grading Γ : A = g G A g. By Supp Γ we denote the subgroup of G generated by Supp Γ. Note: We usually require that the grading group is generated by the support. Denition Two gradings A = g G A g and A = h G A h by a group G on an algebra A are called group-equivalent if there exist Ψ Aut A and θ Aut G such that Ψ(A g ) = A θ(g) for all g G. If θ is the identity, we call the gradings isomorphic. Our goal is to classify all gradings on an algebra up to group-equivalence or, if possible, up to isomorphism. The dierence between isomorphic gradings is a matter of "relabelling" a basis of homogeneous elements.

Brief Overview of Gradings on Simple Lie Algebras Jason McGraw (MUN) Innite Dimensional Lie Algebras Workshop July 2010 5 / 36 General Properties Lemma Let A = g G A g be a grading by a group G on an algebra A and φ a homomorphism of G. The associated φ(g)-grading on A can be dened if one sets A = h φ(g) A h, where A h = g G, φ(g)=h A g. Lemma Let L be a simple Lie algebra. If L = g G L g is a G -grading of L such that the support generates G, then G is abelian. For the rest of the talk we assume that the grading group is nitely generated and abelian.

Brief Overview of Gradings on Simple Lie Algebras Jason McGraw (MUN) Innite Dimensional Lie Algebras Workshop July 2010 6 / 36 General Properties Lemma Let A be an algebra over a eld of characteristic p, G a nitely generated abelian group (without p-torsion if p > 0) and Γ : A = g G A g a G -grading on A. Then there is a homomorphism of Ĝ into Aut A dened by the following action: χ y = χ(g)y, for all y A g, g G, χ Ĝ. The image of Ĝ in Aut A is a semisimple abelian algebraic subgroup (quasi-torus). Conversely, given a quasi-torus Q in Aut A, we obtain the eigenspace decomposition of A with respect to Q, which is a grading by the group of regular characters of the algebraic group Q, G = X(Q). For example, let L = L 0 L 1 be a Z 2 -grading. Then Q is the group of order 2 generated by ψ Aut L such that ψ(y) = y for all y L 0 and ψ(y) = y for all y L 1.

Brief Overview of Gradings on Simple Lie Algebras General Properties Lemma Let A be an algebra over a eld of positive characteristic p, G = Z p and Γ : A = g G A g a G -grading on A. Then there is a homomorphism of G = 1 into Der A dened by the following. For all i Z p and y A i set 1 y = iy. Conversely, given a semisimple derivation with eigenvalues in Z p F we obtain the eigenspace decomposition of A with respect to this derivation, which is a grading by Z p. Jason McGraw (MUN) Innite Dimensional Lie Algebras Workshop July 2010 7 / 36

Brief Overview of Gradings on Simple Lie Algebras Jason McGraw (MUN) Innite Dimensional Lie Algebras Workshop July 2010 8 / 36 General Properties Lemma Let L be a nite-dimensional simple Lie algebra over an algebraically closed eld of characteristic p and G a group. If Γ : L = g G L g is a G -grading and Supp Γ = G then the following are true: G is a nitely generated abelian group; G = G 1 G 2 where G 1 is a group that has no p-torsion and G 2 is a p-group generated by l elements where l is minimal; there exists a quasi-torus Q of Aut A isomorphic to Ĝ1 and the subspaces L g 1 = g 2 G 2 L (g1,g 2) are the eigenspaces of Q; there exists an epimorphism φ : G 2 Z l p and a set of l semisimple derivations {D i } l i=1 of L such that the subspaces L h = g 1 G 1, φ(g L 2)=h (g 1,g 2), h Z l p are the eigenspaces with respect to {D i } l i=1 ; Q and the derivations D i, 1 i l, commute.

Brief Overview of Gradings on Simple Lie Algebras Jason McGraw (MUN) Innite Dimensional Lie Algebras Workshop July 2010 9 / 36 General Properties Let L be a nite dimensional simple Lie algebra L and Γ : L = g G L g be a grading by a group G without p-torsion. Then there is a quasi-torus Q of Aut L isomorphic to Ĝ such that L g are the eigenspaces of L with respect to Q. If Γ : L = g G L g is a G -grading isomorphic to Γ then by denition there is a Ψ Aut L such that L g = Ψ(L g ). The quasi-torus of Aut L associated to the Γ grading is ΨQΨ 1. It follows that if we can classify all quasi-tori of the automorphism group of a simple nite dimensional Lie algebra L, up to conjugation by an automorphism of L, then we classify all gradings by groups with no p-torsion up to isomorphism.

Brief Overview of Gradings on Simple Lie Algebras Jason McGraw (MUN) Innite Dimensional Lie Algebras Workshop July 2010 10 / 36 General Properties The following theorem is a result of Platonov. Theorem Any quasi-torus of an algebraic group is contained in the normalizer of a maximal torus. We will show that, up to conjugation, a quasi-tori of the automorphism group of a simple graded Cartan Lie algebra or Melikian algebra is contained in a maximal torus of automorphism group.

Brief Review of the Cartan Lie Algebras and Melikian Algebras Jason McGraw (MUN) Innite Dimensional Lie Algebras Workshop July 2010 11 / 36 Most of the following lemmas and theorems can be found in Simple Lie Algebras over Fields of Positive Characteristic, Volume I by Helmut Strade.

Brief Review of the Cartan Lie Algebras and Melikian Algebras Jason McGraw (MUN) Innite Dimensional Lie Algebras Workshop July 2010 12 / 36 O(m;n) Denition Let O(m; n) to be the commutative algebra O(m; n) := 0 a τ(n) α(a)x (a) over a eld of characteristic p, where τ(n) = (p n1 1,..., p nm 1), with multiplication ( ) x (a) x (b) a + b = x (a+b), a ( ) a + b m ( ) ai + b where = i m (a i + b i )! =. a a i a i!b i! i=1 i=1

Brief Review of the Cartan Lie Algebras and Melikian Algebras Jason McGraw (MUN) Innite Dimensional Lie Algebras Workshop July 2010 13 / 36 O(m;n) For convenience, let x i := x (ɛ i ), ɛ i := (0,..., 0, 1, 0..., 0) where the 1 is at the i-th position. The sequence of maps γ 0, γ 1,... on O(m; n) (1) := α(a)x (a) to 0<a τ(n) O(m; n), where γ r (x i ) = x (r) i for all 1 i m, r 0, denes a system of divided powers on O(m; n) (1). The set {x i } m i=1 and the divided power maps generate O(m; n). Let i Der O(m; n) such that i (x (r) i ) = x (r 1) i, i (x j ) = 0 for 0 r, i j. The derivation i is a special derivation of O(m; n).

Brief Review of the Cartan Lie Algebras and Melikian Algebras Jason McGraw (MUN) Innite Dimensional Lie Algebras Workshop July 2010 14 / 36 W(m;n) Denition Let W (m; n) be the Lie algebra W (m; n) := with the commutator dened by 1 i m f i i f i O(m; n) [f i, g j ] = f ( i g) j g( j f ) i, f, g O(m; n). The Lie algebras W (m; n) can be viewed as Lie algebras of special derivations of O(m; n) endowed with the system of divided powers mentioned previously. These algebras are called the Witt algebras.

Brief Review of the Cartan Lie Algebras and Melikian Algebras Jason McGraw (MUN) Innite Dimensional Lie Algebras Workshop July 2010 15 / 36 The remaining simple graded Cartan type Lie algebras are subalgebras of W (m; n). When dealing with the Hamiltonian and contact algebras in m variables (types H(m; n) and K(m; n) on the next slide), it is useful to introduce the following notation: { i i + r, if 1 i r, = i r, if r + 1 i 2r; { 1, if 1 i r, σ(i) = 1, if r + 1 i 2r; where m = 2r in the case of H(m; n) and 2r + 1 in the case of K(m; n). We will also need the following dierential forms: ω S := dx 1 dx m, m 3; r ω H := dx i dx i, m = 2r; i=1 2r ω K := dx m + σ(i)x i dx i, m = 2r + 1. i=1

Brief Review of the Cartan Lie Algebras and Melikian Algebras Jason McGraw (MUN) Innite Dimensional Lie Algebras Workshop July 2010 16 / 36 Simple Graded Cartan Lie Algebras Denition We dene the special, Hamiltonian and contact algebras as follows: S(m; n) := {D W (m; n) D(ω S ) = 0}, m 3, H(m; n) := {D W (m; n) D(ω H ) = 0}, m = 2r, K(m; n) := {D W (m; n) D(ω K ) O(m; n)ω K }, m = 2r + 1, respectively. The above algebras are not simple in general. Lemma The Lie algebras S(m; n) (1), H(m; n) (2) and K (m; n) (1) are simple.

Brief Review of the Cartan Lie Algebras and Melikian Algebras Jason McGraw (MUN) Innite Dimensional Lie Algebras Workshop July 2010 17 / 36 Melikian Algebras The next type of simple Lie algebras we will consider are the Melikian algebras which are dened over elds of characteristic 5. Set W (2; n) = O(2; n) 1 + O(2; n) 2, f 1 1 + f 2 2 := f 1 1 + f 2 2 for all f 1, f 2 O(2; n) and div : W (m; n) O(m; n) the linear map dened by div(x (a) i ) = i (x (a) ) = x (a ε i ).

Brief Review of the Cartan Lie Algebras and Melikian Algebras Jason McGraw (MUN) Innite Dimensional Lie Algebras Workshop July 2010 18 / 36 Melikian Algebras Denition Let M(2; n) := O(2; n) W (2; n) W (2; n) be the algebra over a eld F of characteristic 5 whose multiplication is dened by the following equations. For all D, E W (2; n), f, f i, g i O(2; n) we set [D, Ẽ] := [D, E] + 2 div(d)ẽ, [D, f ] := D(f ) 2 div(d)f, [f, Ẽ] := fe [f 1, f 2 ] := 2(f 1 1 (f 2 ) f 2 1 (f 1 )) 2 + 2(f 2 2 (f 1 ) f 1 2 (f 2 )) 1. [f 1 1 + f 2 2, g 1 1 + g 2 2 ] := f 1 g 2 f 2 g 1. We refer to the Lie algebras M(2; n) as the Lie algebras of Melikian type.

Jason McGraw (MUN) Innite Dimensional Lie Algebras Workshop July 2010 19 / 36 Canonical Z-Gradings Canonical Gradings on O(m; n), W (m; n) Denition The Z-gradings on O(m; n) and W (m; n), O(m; n) = k Z O(m; n) k, W (m; n) = k Z W (m; n) k, where W (m; n) k = O(m; n) k = Span{x (a) a 1 + + a m = k}, m O(m; n) k+1 i = Span{x (a) i a 1 + +a m = k +1, 1 i m}, i=1 are called the canonical Z-gradings on O(m; n) and W (m; n), respectively. We denote the canonical Z-gradings on O(m; n) and W (m; n) by Γ O and Γ W, respectively and their degrees by deg O and deg W, respectively.

Canonical Z-Gradings Jason McGraw (MUN) Innite Dimensional Lie Algebras Workshop July 2010 20 / 36 Canonical Grading on M(2; n) Denition Set deg M (x (a) i ) := 3 deg W (x (a) i ), deg M (x (a) i ) := 3 deg W (x (a) i ) + 2, deg M (x (a) ) := 3 deg O (x (a) ) 2. The subspaces M k = Span{y M(2; n) deg M (y) = k} for k Z dene the canonical Z-grading on M(2; n). Note: W (2; n) = i Z M 3i. Hence W (2; n) is a graded subalgebra of M(2; n) with respect to the canonical Z-grading on M(2; n).

Canonical Z-Gradings Jason McGraw (MUN) Innite Dimensional Lie Algebras Workshop July 2010 21 / 36 For example, deg O (x 1 x (2) 2 ) = 3, deg W (x 1 x (2) 2 2) = 3 1 = 2, deg M (x 1 x (2) 2 2) = 3(2) = 6, deg M (x 1 x (2) 2 2 ) = 3(2) + 2 = 8, deg M (x 1 x (2) 2 ) = 3(3) 2 = 7,

Canonical Z-Gradings Jason McGraw (MUN) Innite Dimensional Lie Algebras Workshop July 2010 22 / 36 Canonical Filtrations Denition For any Z-grading Γ : A = k Z A k on an algebra A with a nite support Supp Γ there is an induced ltration A s = A (s) A (s 1) A (q) = A, where A (k) = l k A l for k Z, q = min(supp Γ) and s = max(supp Γ). We call the induced ltrations of the canonical Z-gradings on O(m; n), W (m; n) and M(2; n) the canonical ltrations. For example, O(2; (1, 1)) (2) = Span{x (a) a 1 + a 2 2}, W (2; (1, 1)) (1) = Span{x (a) 1, x (a) 2 a 1 + a 2 2}.

Automorphisms Continuous Automorphisms of O(m; n) Denition Let A(m, n) be the set of all m-tuples (y 1,..., y m ) O(m; n) m where each y i = 0<a τ(n) for which det(α j (ɛ i )) 1 i,j m F. Theorem α i (a)x (a), and α i (p l ɛ j ) = 0, if n i + l > n j, (Theorem 6.3.2 in SLAOFPC by Strade) Let Aut c O(m; n) be the set of maps ρ from O(m; n) to O(m; n), dened by the tuples (y 1,..., y m ) A(m, n) such that ρ α(a)x (a) = m α(a) (y i ) (a i ). 0 a τ(n) 0 a τ(n) Then Aut c O(m; n) is a subgroup of Aut O(m; n). Jason McGraw (MUN) Innite Dimensional Lie Algebras Workshop July 2010 23 / 36 i=1

Automorphisms Jason McGraw (MUN) Innite Dimensional Lie Algebras Workshop July 2010 24 / 36 Divided Power Automorphisms Note: Each automorphism ψ Aut c (O(m; n)) is a divided power automorphism. That is, ψ(x (a i ) i ) = (ψ(x i )) (a i ). It follows that ψ is dened by its action on V := Span{x i 1 i m}. Also, Aut c O(m; n) respects the canonical ltration on O(m; n). That is ψ(o(m; n) (k) ) = O(m; n) (k).

Automorphisms Jason McGraw (MUN) Innite Dimensional Lie Algebras Workshop July 2010 25 / 36 Automorphism of the Cartan Lie Algebras Let Φ be the map from Aut c O(m; n) to Aut W (m; n) dened on ψ Aut c O(m; n) by Φ(ψ)(D) = ψ D ψ 1, where D W (m; n) and the elements of W (m; n) are viewed as derivations of O(m; n). Theorem (Theorem 7.3.2 in SLAOFPC by Strade) The map Φ : Aut c O(m; n) W (m; n) is an isomorphism of groups provided that (m; n) (1; 1) if p = 3. Also, except for the case of H(m, (n 1 ; n 2 )) (2) with m = 2 and min{n 1, n 2 } = 1 if p = 3, Aut S(m; n) (1) = Φ({ψ Aut c O(m; n) ψ(ω S ) F ω S }), Aut H(m; n) (2) = Φ({ψ Aut c O(m; n) ψ(ω H ) F ω H }), Aut K(m; n) (1) = Φ({ψ Aut c O(m; n) ψ(ω K ) O(m; n) ω K }).

Automorphisms Jason McGraw (MUN) Innite Dimensional Lie Algebras Workshop July 2010 26 / 36 Automorphism of the Cartan Lie Algebras Note: Aut W (m; n) respects the canonical ltration on W (m; n). For convenience, set W(m; n) := Φ 1 (Aut W (m; n)) = Aut c O(m; n), S(m; n) := Φ 1 (Aut S(m; n) (1) ) = {ψ Aut c O(m; n) ψ(ω S ) F ω S }, H(m; n) := Φ 1 (Aut H(m; n) (2) ) = {ψ Aut c O(m; n) ψ(ω H ) F ω H }, K(m; n) := Φ 1 (Aut K(m; n) (1) ) = {ψ Aut c O(m; n) ψ(ω K ) O(m; n) ω K }.

Automorphisms Jason McGraw (MUN) Innite Dimensional Lie Algebras Workshop July 2010 27 / 36 Automorphisms of Melikian Algebras Lemma The following are true. Any automorphism Ψ of M(2; n) respects the canonical ltration of M (2; n). For every automorphism ψ of W (2; n) there exists an automorphism ψ M of M(2; n) which respects W (2; n) and whose restriction to W (2; n) is ψ. If Θ Aut W M(2; n) is such that π(θ) = Id W then for y M k, k Z, there exists β F such that Θ(y) = β i y and β 3 = 1. Denition Let Aut W M(2; n) = {Ψ Aut M(2; n); Ψ(W (2; n)) = W (2; n)} be the subgroup of automorphisms that leave W (2; n) invariant and let π : Aut W M(2; n) Aut W (2; n) be the restriction map.

Jason McGraw (MUN) Innite Dimensional Lie Algebras Workshop July 2010 28 / 36 Tori of Aut X (2) (m; n) Normalizers of Tori Lemma The following groups are maximal tori of Aut W (m; n), Aut S(m; n) (1), Aut H(m; n) (2) and Aut K(m; n) (1), respectively: T W = T S = {Ψ Aut W (m; n) Ψ(x (a) i ) = t a t 1 i x (a) i, t (F ) m }, T H = {Ψ Aut W (m; n) Ψ(x (a) i ) = t a ti 1 x (a) i, t (F ) m, t i t i = t j t j, 1 i, j r}, T K = {Ψ Aut W (m; n) t a t 1 i x (a) i, t (F ) m, t i t i = t j t j = t m, 1 i, j r}, T M = {Ψ Aut M(2; n) Ψ(x (a) i ) = t 3a1 1 t 3a2 2 ti 3, Ψ(x (a) i ) = t 3a1 1 t 3a2 2 ti 3 t 1 t 2, Ψ(x (a) ) = t 3a1 1 t 3a2 2 t 1 1 t 1 2, (t 1, t 2 ) (F ) 2 }.

Jason McGraw (MUN) Innite Dimensional Lie Algebras Workshop July 2010 29 / 36 Tori of X (m; n) Normalizers of Tori Corollary Let X = W, S, H or K. The maximal tori T X in Aut X (2) (m; n) described in previous lemma correspond, under the algebraic group isomorphism Φ, to the following maximal tori in X (m; n): T W = T S = {ψ W(m; n) ψ(x i ) = t i x i, t i F }, T H = {ψ W(m; n) ψ(x i ) = t i x i, t i F, t i t i = t j t j, }, T K = {ψ W(m; n) ψ(x i ) = t i x i, t i F, t i t i = t j t j = t m, 1 i, j r}.

Normalizers of Tori Jason McGraw (MUN) Innite Dimensional Lie Algebras Workshop July 2010 30 / 36 Eigenspaces Note: The eigenspace decomposition of T W is the direct sum of the subspaces Span{x (a) }. Since T X T W, the eigenspaces of O(m; n) with respect to T X direct sums of Span{x (a) }. Similarly, The eigenspace decomposition of T W is the direct sum of the subspaces Span{x (a+ε i ) i 1 i m}. Since T X T W, the eigenspaces of W (m; n) with respect to T X (viewed as a subgroup of T W ) is direct sums of Span{x (a+ε i ) i 1 i m}.

Normalizers of Tori Jason McGraw (MUN) Innite Dimensional Lie Algebras Workshop July 2010 31 / 36 Linear Automorphisms and Flag Structure Denition We denote by Aut 0 O(m; n) the subgroup of Aut c O(m; n) consisting of all ψ such m that ψ(x i ) = α i,j x j, α i,j F, 1 j m. The group Aut 0 O(m; n) is j=1 canonically isomorphic to a subgroup of GL(V ) = GL(m), which we denote by GL(m; n). If n i = n j for 1 i, j m then GL(m; n) = GL(m), otherwise it is properly contained in GL(m). The condition for a tuple (y 1,..., y n ) to be in A(m, n), y i = 0<a α i (a)x (a) with α i (p l ɛ j ) = 0 if n i + l > n j, (1) imposes a ag structure on the vector space V = Span{x 1,..., x m }.

Normalizers of Tori Jason McGraw (MUN) Innite Dimensional Lie Algebras Workshop July 2010 32 / 36 Normalizers of T X for X = W, S, H It is easy to see that for 1 i m the subspaces Span{x i } are eigenspaces of T W, T S and T H. If ψ N X (m;n) (T X ) then it sends an eigenspace of T X to an eigenspace. Since Aut c O(m; n) preserves the ltration and O(m; n) 1 = V, we have that if ψ N X (m;n) (T X ) then we have ψ(x i ) Span{x ji } for some 1 j i m. Hence the normalizers of T X in X (m; n) are isomorphic to a subset of m m monomial matrices in GL(m; n) X (m; n).

Normalizers of Tori Jason McGraw (MUN) Innite Dimensional Lie Algebras Workshop July 2010 33 / 36 Normalizers of T X for X = W, S, H Lemma Aut 0 O(m; n) S(m; n) = Aut 0 O(m; n) and Aut 0 O(m; n) H(m; n) = (Sp(m){F Id}) GL(m; n). Lemma Let X = W, S or H. If Q is a quasi-torus of X (m; n) then there is a ψ X (m; n) such that ψqψ 1 T X. Corollary Let X = W, S or H. If Q is a quasi-torus of Aut X (2) (m; n) then there is a ψ Aut X (2) (m; n) such that ψqψ 1 T X.

Normalizers of Tori Jason McGraw (MUN) Innite Dimensional Lie Algebras Workshop July 2010 34 / 36 Normalizer of T K For T K, the subspaces Span{x i } for 1 i 2r are eigenspaces and so is Span{x m, x i x i 1 i 2r}. If ψ N K(m;n) (T K ) then it sends an eigenspace of K(m; n) to an eigenspace. Since Aut c O(m; n) preserves the ltration, if ψ N K(m;n) (T K ) then for 1 i 2r we have for some 1 j i 2r or ψ(span{x i }) = Span{x ji } ψ(span{x m, x i x i 1 i r}). Since the dimension of Span{x m, x i x i 1 i r} is greater than one, ψ(x i ) Span{x ji }. Similarly, ψ(x m ) αx m + Span{x i x i 1 i r}.

Normalizers of Tori Jason McGraw (MUN) Innite Dimensional Lie Algebras Workshop July 2010 35 / 36 Normalizer of T K Recall that H(m; n) = {ψ Aut c O(m; n) ψ(ω H ) F ω H } and K(m; n) = {ψ Aut c O(m; n) ψ(ω K ) O(m; n) ω K }. Noticing that d(ω K ) = 2ω H and that if ψ N K(m;n) (T K ) then ψ leaves O(2r; (n 1,..., n 2r )) invariant we can prove the following lemma. Lemma If Q is a quasi-torus of K(m; n) then there is a ψ K(m; n) such that ψqψ 1 T K. Corollary If Q is a quasi-torus of Aut K(m; n) (1) then there is a ψ Aut K(m; n) (1) such that ψqψ 1 T K.

Normalizers of Tori Jason McGraw (MUN) Innite Dimensional Lie Algebras Workshop July 2010 36 / 36 Normalizer of T M The eigenspaces of T M are Span{x (a) }, Span{x (a+ε i ) i i = 1, 2}, Span{x (a+ε i ) i i = 1, 2}. Since the elements of an eigenspace of T M are homogeneous elements of the canonical Z-grading on M(2; n) and Aut M(2; n) preserves the canonical ltration, the normalizer of T M preserves the Z-grading. Since W (2; n) = i Z M 3i and the restriction of T M to W (2; n) is T W, we have that N Aut M(2;n) (T M ) Aut W M(2; n) and the restriction of N Aut M(2;n) (T M ) on W (2; n) is N Aut W (2;n) (T W ). Hence if we have a quasi-torus Q in N Aut M(2;n) (T M ) we can conjugate Q by an automorphism ψ in Aut W M(2; n) such that the restriction of ψ M Qψ 1 M is in T W. Lemma If Q is a quasi-torus of Aut M(2; n) then there is a ψ Aut M(2; n) such that ψqψ 1 T M.

Conclusion Jason McGraw (MUN) Innite Dimensional Lie Algebras Workshop July 2010 37 / 36 Conclusion Theorem If L = X (2) (m; n) is a simple graded Cartan type Lie algebra or a Melikian algebra and L = g G L g is a grading by a group G with no p-torsion then the grading is isomorphic to the eigenspace decomposition with respect to quasi-torus Q contained in T X.