Rigidity theorem

Survey on the work of A. Nijenhuis and R.W. Richardsen Jr.

Yuncherl Choi

December 2, 1999

Definition ([4]) A graded Lie algebra over a field K of characteristic ≠ 2 is a graded vector space E = ⊕_{α} E^{α} over K together with a bilinear map (a, b) ↦ [a, b] of E × E into E which satisfies the following conditions :

GLA1 : [E^{α}, E^{β}] ⊂ E^{α+β}

GLA2 : If a ∈ E^{α} and b ∈ E^{β}, then [a, b] = −(−1)^{αβ}[b, a].

GLA3 : If a ∈ E^{α}, b ∈ E^{β} and c ∈ E^{γ}, then

(−1)^{αγ}[[a, b], c] + (−1)^{βα}[[b, c], a] + (−1)^{γβ}[[c, a], b] = 0.

In the GLA3, the Jacobi identity has some equivalent forms :

[a, [b, c]] = [[a, b], c] + (−1)^{αβ}[b, [a, c]]
[a, [b, c]] − (−1)^{αβ}[b, [a, c]] = [[a, b], c]
(−1)^{αγ}[a, [b, c]] + (−1)^{βα}[b, [c, a]] + (−1)^{γβ}[c, [a, b]] = 0.

Example ([4]) Let V be a finite dimensional vector space over K. And let

C_{a}^{∗} = ⊕_{n}C_{a}^{n}, C^{∗} = ⊕_{n}C^{n}, and C_{l}^{∗} = ⊕_{n} C_{l}^{n}

be the cochain complexes where C_{a}^{n}, C^{n} and C_{l}^{n} are the sets of n-multilinear, left skew symmetric n-multilinear, skew symmetric n-multilinear maps from V into V respectively. The composition products ∘ on these complexes are defined as followings

(a) \underline{On C_{a}^{∗}} : f ∈ C_{a}^{m+1}, g ∈ C_{a}^{n+1} ⇒ f ∘ g ∈ C_{a}^{m+n+1}

(f ∘ g)(x_{1}, ⋯, x_{m+n+1})

= ∑_{i=1}^{m+1}(−1)^{n(i−1)}f(x_{1}, ⋯, x_{i−1}, g(x_{i}, ⋯, x_{i+n}), x_{i+n+1}, ⋯, x_{m+n+1}).

(b) \underline{On C^{∗}} : f ∈ C^{m+1}, g ∈ C^{n+1} ⇒ f ∘ g ∈ C^{m+n+1}

(f ∘ g)(x_{1}, ⋯, x_{m+n+1})

= ∑_{1} sgnσf(g(x_{σ(1)}, ⋯, x_{σ(n+1)}), x_{σ(n+2)}, ⋯, x_{σ(m+n)}, x_{m+n+1})

+ ∑_{2} sgnσf(x_{σ(1)}, ⋯, x_{σ(m)}, g(x_{σ(m+1)}, ⋯, x_{σ(m+n)}, x_{m+n+1}))

where the ∑_{1} means that σ(1) < ⋯ < σ(n) and σ(n + 2) < ⋯ < σ(m + n),

and the ∑_{2} means that σ(1) < ⋯ < σ(m) and σ(m + 1) < ⋯ < σ(m + n).

(c) \underline{On C_{l}^{∗}} : f ∈ C_{l}^{m+1}, g ∈ C_{l}^{n+1} ⇒ f ∘ g ∈ C_{l}^{m+n+1}

(f ∘ g)(x_{1}, ⋯, x_{m+n+1})

= ∑ sgnσf(g(x_{σ(1)}, ⋯, x_{σ(n+1)}), x_{σ(n+2)}, ⋯, x_{σ(m+n+1)})

where the ∑ means that σ(1) < ⋯ < σ(n+1) and σ(n+2) < ⋯ < σ(m+n+1).

With these composition products, C_{a}^{∗}, C^{∗}, and C_{l}^{∗} become the graded right symmetric algebras by reducing the degree 1, i.e., for f ∈ C^{m+1}, g ∈ C^{n+1}, h ∈ C^{∗},

(h∘f) ∘ g − (−1)^{mn}(h∘g) ∘ f = h ∘ (f ∘g) − (−1)^{mn}h ∘ (g∘f).

Now define the bracket by for f ∈ C^{m+1}, g ∈ C^{n+1}

[f, g] = −(−1)^{mn}f ∘ g + g ∘ f.

Then C_{a}^{∗}, C^{∗} and C_{l}^{∗} become the graded Lie algebras.

Definition ([4]) Let E = ⊕_{α}E^{α} be a GLA(graded Lie algebra) over ℝ or ℂ and let G be a Lie group whose Lie algebra is E^{0} and τ : G → GL(E) be a continuous homomorphism(i.e. a representation of G on E). Then (E, G, τ) is analytic graded Lie algebra if it satisfies the following conditions

(i) For each α, dim E^{α} < ∞.

(ii) dτ = ad_{E} : E^{0} → gl(E).

(iii) For each a ∈ G, τ(a) ∈ Aut(E).

Remark If we assume the base field of the vector space V is ℝ or ℂ, then C_{a}^{∗}, C^{∗} and C_{l}^{∗} are the analytic GLA with G = GL(V) and the representation τ which is defined by

(τ(a)f)(x_{1}, ⋯, x_{n}) = a^{−1}f(ax_{1}, ⋯, ax_{n}),

for f ∈ C_{a}^{n}(or C^{n}, C_{l}^{n} and a ∈ GL(V).

(1) τ(ab)f = τ(b)τ(a)f i.e., the action occur on the right side.

(2) dτ(a)(f) = [a, f]

dτ(a)(f) = \frac{d}{dt}|_{t=0} exp(−ta)f(exp(ta)x_{1}, ⋯, exp(ta)x_{n})
= −af(x_{1}, ⋯, x_{n})
+f(ax_{1}, x_{2}, ⋯, x_{n}) + ⋯ + f(x_{1}, ⋯, x_{n−1}, ax_{n})
= (−a ∘ f)(x_{1}, ⋯, x_{n}) + (f ∘ a)(x_{1}, ⋯, x_{n})
= [a, f](x_{1}, ⋯, x_{n})

Let’s denote M, the deformation space, as the set of elements of E^{1} satisfying [a, a] = 0, i.e.,

M = {a ∈ E^{1}|[a, a] = 0}.

For each μ ∈ M, we can define the derivation D_{μ} of E,

D_{μ}(f) = [μ, f]

from the Jacobi identity, i.e., for f ∈ E^{α}, g ∈ E

D_{μ}([f, g]) = [μ, [f, g]]
= [[μ, f], g] + (−1)^{α}[f, [μ, g]]
= [D_{μ}(f), g] + (−1)^{α}[f, D_{μ}(g)]

Moreover, we have D_{μ}∘D_{μ} = 0. Hence we obtain the graded supspace Z(μ), B(μ) and H(μ) of E defined by

Z(μ) = ker D_{μ}, B(μ) = imD_{μ}, H(μ) = Z/B.

In this case, Z(μ) is the subalgebra of E and B(μ) is the ideal of Z(μ).

Remark In the case C_{a}^{∗}, C^{∗} and C_{l}^{∗}, M is really the deformation space of associative, left symmetric and Lie algebras respectively. And D_{μ} is the coboundary map of the algebra defined by μ on V.

Now fix μ ∈ M and put D = D_{μ}. And some notations are given :

Notation Z : kernel of D B : image of D C : supplementaty subspace to Z in E H : supplementaty subspace to B in Z

Z^{α} = Z ∩ E^{α}, B^{α} = B ∩ E^{α}, C^{α} = C ∩ E^{α}, H^{α} = H ∩ E^{α}

N(V) : the neighborhood of the origin of the vecto space of V

With the above notations, we have a decomposition of E

E = B ⊕ H ⊕ C,

which is called Hodge decomposition of E. Of cource the decomposition in each degree is the same, i.e.,

E^{α} = B^{α} ⊕ H^{α} ⊕ C^{α},

Lemma ([4]) There exist N(μ, E^{1}), N(C^{0}), N(H^{1}), N(C^{1}) such that F : C^{0} × H^{1} × C^{1} → E^{1} which is defined by

F (a, h, b) = exp(a) · (μ + h + b)

is anaylitic and its restriction to N(C^{0})×N(H^{1})×N(C^{1}) is homeomorphic onto N(μ, E^{1}).

Proof.

dF_{(0,0,0)}(a, h, b) = \frac{d}{dt}|_{t=0}F (ta, th, tb)
= \frac{d}{dt}|_{t=0} exp(ta) · (μ + th + tb)
= [a, μ] + h + b
= −D(a) + h + b

Since D is an isomorphism of C^{0} onto B^{1} and E^{1} = B^{1} ⊕ H^{1} ⊕ C^{1}, dF_{(0,0,0)} is an isomorphism. By the inverse function theorem, F is locally diffeomorphic at (0, 0, 0). □

Theorem (Rigidity theorem [4]) Let E be the analytic GLA and μ be an element of M such that H^{1}(E, D_{μ}) = {0}, then μ is rigid, i.e., there exist N(μ, M) and N(C^{0}) such that μ′ ∈ N(μ, M) implies that μ′ = exp(a) · μ for some a ∈ N(C^{0})

Proof. From the lemma, there exist N(μ, E^{1}), N(C^{0}), N_{1}(C^{1}) such that

if μ′ ∈ N(μ, M), then μ′ = exp(a) · (μ + b)

where a ∈ N(C^{0}), b ∈ N_{1}(C^{1}). Since M is invariant under G, we have

μ′ ∈ M if and only if μ + b ∈ M.

Now define

P : C^{1} → E^{2}

by P (b) = [μ + b, μ + b]. Then we have (1) μ + b ∈ M if and only if P (b) = 0

(2) dP_{0}(b) = 2D(b)

∵ dP_{0}(b) = \frac{d}{dt}|_{t=0}P (tb) = \frac{d}{dt}|_{t=0}[μ + tb, μ + tb] = 2[μ, b] = 2D(b)

Therefore dP_{0} is injective, hence there exist N_{2}(C^{1}) such that P |_{N_{2}(C^{1})} is injective.

Put N(C^{1}) = N_{1}(C^{1})∩N_{2}(C^{1}), then we have that for b ∈ N(C^{1}), μ+b ∈ M if and only if b = 0. This proves. □

Fact Let S be an irreducible algebraic set in ℂ^{n}(resp. ℝ^{n}) and S_{0} be the set of simple points of S. Then

(a) S_{0} is a Zariski dense open subset of S (see [1] p.40)

(b) For any x ∈ S_{0}, there exist a neighborhood N(x, ℂ^{n})(resp. N(x, ℝ^{n})) such that

S ∩ N(x, ℂ^{n}) (resp.S ∩ N(x, ℝ^{n}))

is a complex submanifold of N(x, ℂ^{n})(resp. N(x, ℝ^{n})).

(c) For any x ∈ S_{0}, T_{x}S_{0} ≅ T_{x}S, hence dim S_{0} = dim S.

(d) For any Zariski open subset U ⊂ S, U is connected in the topology induced on U from the usual topology of ℂ^{n}.(In the case of ℝ^{n}, U has finite number of components)

Lemma ([5] p.104) If an algebraic group G acts on a reducible quasiprojective variety S, then the elements of G^{0}, the identity component, transform each irreducible component of S into itself.

Proposition ([4]) Let S be an algebraic set in ℂ^{n}(resp. ℝ^{n}) and G be a Lie subgroup of GL(ℂ^{n})(resp. GL(ℝ^{n})) such that G · S ⊂ S. Suppose there exists a point s_{0} ∈ S whose orbit Gx_{0} is an open subset of S. Then Gx_{0} is a Zariski open subset of S(resp. one component of a Zariski open subset of S).

Proof. May assume that G is connected. Each irreducible component of S is stable under the action of G. (∵ Let G̃ be an algebraic hull of G(Zariski closure of G), then G ⊂ G̃^{0}.

Now using the lemma above.) Now may assume S is irreducible. Let k be the dimension of S as an algebraic set, and 𝔤 be the Lie algebra of G. For any x ∈ ℂ^{n}, the orbit Gx is a complex submanifold of ℂ^{n}. Then T_{x}Gx = 𝔤(x) := {T (x)|T ∈ 𝔤} If x ∈ S, then dim 𝔤(x) ≤ k. Put U = {x ∈ S| dim 𝔤(x) = k}, then

(1) U is a Zariski dense open supset of S(see [1] p.40)

(2) U is nonempty (∵ given point x_{0})

(3) U ⊂ S_{0} where S_{0} is the set of simple points For x ∈ U, Gx is an open subset of U. Hence, the orbit of G divide U into disjoint open subsets. Since U is connected, there exist unique orbit of G on U. □

Theorem ([4]) Let E be the analytic GLA with the base field ℂ(resp. ℝ) and let μ ∈ M be such that H^{1}(E, D_{μ}) = {0}. Then the orbit Gμ is a Zarisk open subset of M(resp. is one component of a Zariski open subset of M).

References

[1] James E. Humphreys. Linear algebraic groups. Springer-Verlag, New York, 1975. Graduate Texts in Mathematics, No. 21.

[2] Albert Nijenhuis. On a class of common properties of some different types of algebras. Nieuw Arch. Wisk. (3), 17:17–46, 1969.

[3] Albert Nijenhuis. On a class of common properties of some different types of algebras. II. Nieuw Arch. Wisk. (3), 17:87–108, 1969.

[4] Albert Nijenhuis and Jr. Richardson, R. W. Cohomology and deformations in graded Lie algebras. Bull. Amer. Math. Soc., 72:1–29, 1966.

[5] A. L. Onishchik and È. B. Vinberg. Lie groups and algebraic groups. Springer-Verlag, Berlin, 1990. Translated from the Russian and with a preface by D. A. Leites.