Deformations of subalgebras of Lie algebras.

*(English)*Zbl 0215.38603From the introduction: In this paper we shall be concerned with “deformations” of subalgebras of Lie algebras. More generally, we are interested in deformations of subalgebras in which the ambient Lie algebra is also allowed to vary. Precisely, we consider the following situation. Let \(\mathcal M\) be the algebraic set of all Lie algebra multiplications on a finite-dimensional vector space \(V\) (taken over \(\mathbb R\) or \(\mathbb C\) for simplicity), and \(\Gamma_n(V)\) the Grassmann manifold of all \(n\)-dimensional subspaces of \(V\). Let \(\mathcal S\) be the algebraic subset of \(\mathcal M\times \Gamma_n(V)\) consisting of all pairs \((\eta,W)\) such that \(W\) is a subalgebra of the Lie algebra \((V,\eta)\). Let \(\mathfrak g= (V, \mu)\) be a Lie algebra and \(\mathfrak h\) a subalgebra of \(\mathfrak g\). We are interested in geometric properties of \(\mathcal S\) in a neighborhood of \((\mu,\mathfrak h)\).

Our main result, Theorem 2.5, states that if the Lie algebra cohomology space \(H^2(\mathfrak h,\mathfrak g/\mathfrak h)\) vanishes, then a neighborhood of \((\mu,\mathfrak h)\) in \(\mathcal S\) is isomorphic (as an analytic space) to the product of a neighborhood of \(\mu\) in \(\mathcal M\) and an open ball in \(\mathbb R^k\) (or \(\mathbb C^k)\), where \(k=\dim Z^1(\mathfrak h,\mathfrak g/\mathfrak h)\).

As easy consequences of Theorem 2.5, we obtain the results discussed in (a)–(c) below. These results complement and extend the results of two earlier papers [S. Page and author, Trans. Am. Math. Soc. 127, 302–312 (1967; Zbl 0163.03003); the author, Ill. J. Math. 11, 92–110 (1967; Zbl 0147.28202)].

(a) Let \(\mathfrak h\subset \mathfrak g\) be as above. Then \(\mathfrak h\) is a weakly stable subalgebra of \(\mathfrak g\) if, roughly speaking, for every one-parameter family \((\mathfrak g_t) = (V,\mu_t)\) of Lie algebra structures on \(V\) with \(\mathfrak g_0 = \mathfrak g\), there exists a one-parameter family \((\mathfrak h_t)\) of subspaces of \(V\) with \(\mathfrak h_0 = \mathfrak h\) such that \(\mathfrak h_t\) is a subalgebra of \(\mathfrak g_t\) for \(t\) sufficiently small. It follows from Theorem 2.5 that if \(H^2(\mathfrak h\subset \mathfrak g) =0\), then \(\mathfrak h\) is weakly stable.

(b) Let \(\mathfrak h = (U,\eta)\) and \(\mathfrak g = (V,\mu)\) be Lie algebras, and \(\mathcal N\) (resp. \(\mathcal M)\) the set of all Lie multiplications on \(U\) (resp. \(V)\). A homomorphism \(\rho: \mathfrak h\to \mathfrak g\) is stable if, for every \(\eta'\in\mathcal N\) near \(\eta\) and every \(\mu'\in\mathcal M\) near \(\mu\), there exists a homomorphism \(\rho': (U,\eta')\to(V,\mu')\) which is near \(\rho\). We show that \(\rho\) is stable if \(H^2(\mathfrak h, \mathfrak g) = 0\). If \(\mathfrak h\) is a subalgebra of \(\mathfrak g\) and \(\rho\) the inclusion map, we obtain a strengthened form of Theorem 6.2 of the first cited paper on stable subalgebras.

(c) Let \(\mathfrak h\subset \mathfrak g\) be Lie algebras. If \((\mathfrak h_t)\) is a one-parameter family of subalgebras of \(\mathfrak g\) with \(\mathfrak h_0 = \mathfrak h\), then it was shown in the second cited paper that the “initial tangent vector” of the family \((\mathfrak h_t)\) is an element of \(Z^1(\mathfrak h,\mathfrak g/\mathfrak h)\). We show that if \(H^2(\mathfrak h,\mathfrak g/\mathfrak h)=0\), then every \(\alpha\in Z^1(\mathfrak h,\mathfrak g/\mathfrak h)\) occurs as the initial tangent vector of a one-parameter family of subalgebras. In a sense, the elements of \(H^2(\mathfrak h,\mathfrak g/\mathfrak h)\) occur as “obstructions” to finding one-parameter families of subalgebras with a given initial tangent vector. [This result was also obtained by A. Nijenhuis, Nederl. Akad. Wet., Proc., Ser. A 71, 119–136 (1968; Zbl 0153.36202).]

In a slightly different setting we also obtain a result on “relatively stable” subalgebras of Lie algebras. This result has applications to the study of the variation of isotropy subalgebras for differentiable transformation groups which are discussed in §10.

Our proofs use only elementary methods, primarily the implicit function theorem. The proofs carry over with no essential changes to the case of subalgebras of associative algebras. Here, Lie algebra cohomology is replaced by the cohomology of associative algebras. Our results are also valid with only minor modifications for Lie and associative algebras over algebraically closed fields.

Our main result, Theorem 2.5, states that if the Lie algebra cohomology space \(H^2(\mathfrak h,\mathfrak g/\mathfrak h)\) vanishes, then a neighborhood of \((\mu,\mathfrak h)\) in \(\mathcal S\) is isomorphic (as an analytic space) to the product of a neighborhood of \(\mu\) in \(\mathcal M\) and an open ball in \(\mathbb R^k\) (or \(\mathbb C^k)\), where \(k=\dim Z^1(\mathfrak h,\mathfrak g/\mathfrak h)\).

As easy consequences of Theorem 2.5, we obtain the results discussed in (a)–(c) below. These results complement and extend the results of two earlier papers [S. Page and author, Trans. Am. Math. Soc. 127, 302–312 (1967; Zbl 0163.03003); the author, Ill. J. Math. 11, 92–110 (1967; Zbl 0147.28202)].

(a) Let \(\mathfrak h\subset \mathfrak g\) be as above. Then \(\mathfrak h\) is a weakly stable subalgebra of \(\mathfrak g\) if, roughly speaking, for every one-parameter family \((\mathfrak g_t) = (V,\mu_t)\) of Lie algebra structures on \(V\) with \(\mathfrak g_0 = \mathfrak g\), there exists a one-parameter family \((\mathfrak h_t)\) of subspaces of \(V\) with \(\mathfrak h_0 = \mathfrak h\) such that \(\mathfrak h_t\) is a subalgebra of \(\mathfrak g_t\) for \(t\) sufficiently small. It follows from Theorem 2.5 that if \(H^2(\mathfrak h\subset \mathfrak g) =0\), then \(\mathfrak h\) is weakly stable.

(b) Let \(\mathfrak h = (U,\eta)\) and \(\mathfrak g = (V,\mu)\) be Lie algebras, and \(\mathcal N\) (resp. \(\mathcal M)\) the set of all Lie multiplications on \(U\) (resp. \(V)\). A homomorphism \(\rho: \mathfrak h\to \mathfrak g\) is stable if, for every \(\eta'\in\mathcal N\) near \(\eta\) and every \(\mu'\in\mathcal M\) near \(\mu\), there exists a homomorphism \(\rho': (U,\eta')\to(V,\mu')\) which is near \(\rho\). We show that \(\rho\) is stable if \(H^2(\mathfrak h, \mathfrak g) = 0\). If \(\mathfrak h\) is a subalgebra of \(\mathfrak g\) and \(\rho\) the inclusion map, we obtain a strengthened form of Theorem 6.2 of the first cited paper on stable subalgebras.

(c) Let \(\mathfrak h\subset \mathfrak g\) be Lie algebras. If \((\mathfrak h_t)\) is a one-parameter family of subalgebras of \(\mathfrak g\) with \(\mathfrak h_0 = \mathfrak h\), then it was shown in the second cited paper that the “initial tangent vector” of the family \((\mathfrak h_t)\) is an element of \(Z^1(\mathfrak h,\mathfrak g/\mathfrak h)\). We show that if \(H^2(\mathfrak h,\mathfrak g/\mathfrak h)=0\), then every \(\alpha\in Z^1(\mathfrak h,\mathfrak g/\mathfrak h)\) occurs as the initial tangent vector of a one-parameter family of subalgebras. In a sense, the elements of \(H^2(\mathfrak h,\mathfrak g/\mathfrak h)\) occur as “obstructions” to finding one-parameter families of subalgebras with a given initial tangent vector. [This result was also obtained by A. Nijenhuis, Nederl. Akad. Wet., Proc., Ser. A 71, 119–136 (1968; Zbl 0153.36202).]

In a slightly different setting we also obtain a result on “relatively stable” subalgebras of Lie algebras. This result has applications to the study of the variation of isotropy subalgebras for differentiable transformation groups which are discussed in §10.

Our proofs use only elementary methods, primarily the implicit function theorem. The proofs carry over with no essential changes to the case of subalgebras of associative algebras. Here, Lie algebra cohomology is replaced by the cohomology of associative algebras. Our results are also valid with only minor modifications for Lie and associative algebras over algebraically closed fields.