Extensions of Lie-Rinehart algebras and the Chern-Weil construction.

*(English)*Zbl 0946.17008
McCleary, John (ed.), Higher homotopy structures in topology and mathematical physics. Proceedings of an international conference, June 13-15, 1996, Poughkeepsie, NY, USA, to honor the 60th birthday of Jim Stasheff. Providence, RI: American Mathematical Society. Contemp. Math. 227, 145-176 (1999).

The Chern-Weil homomorphism of principal bundles belongs to the greatest discoveries of differential geometry and topology of the 20th century and was an inspiration to many generalizations. The author points out the generalizations in the direction of

(a) regular Lie algebroids [J. Kubarski, The Chern-Weil homomorphism of regular Lie algebroids, Publ. Dépt. Math. Nouv. Sér., Univ. Claude-Bernard Lyon 1, 1-69 (1991), see also Characteristic classes of some Pradines-type groupoids and a generalization of the Bott vanishing theorem, Differ. Geom. Appl., Proc. Conf., Brno, Czechoslovakia, Commun., 189-198 (1987; Zbl 0641.57012)],

(b) Lie algebra extensions [N. Teleman, Global Anal. Appl., Vol. III, Intern. Sem. Course Trieste 1972, 195-202 (1974; Zbl 0306.57025); Atti Accad. Naz. Lincei., Rend., Cl. Sci. Fis. Mat. Natur. (8) 52, 498-506 (1972; Zbl 0283.53018) and 708-711 (1972; Zbl 0283.53019)],

(c) extensions of principal bundles [K. Mackenzie, Ann. Global Anal. Geom. 6, 141-163 (1988; Zbl 0627.55010)].

Lie algebroids appear as the infinitesimal objects of principal bundles, vector bundles, transversally complete foliations, Poisson manifolds, Jacobi manifolds, etc.

J. Huebschmann examines an algebraic equivalent of Lie algebroids, called Lie-Rinehart algebras. A Lie-Rinehart algebra over \(A\) \((A\) is some commutative \(R\)-algebra where \(R\) is a commutative unital ring) is a pair \((L, \omega)\) where \(L\) is an \(A\)-module and an \(R\)-Lie algebra and \(\omega:L \to \text{Der}(A)\) is an action of the \(R\)-Lie algebra \(L\) on \(A\), such that \(\omega\) is \(A\)-linear and fulfills the axiom \([a\cdot \alpha,\beta] =a\cdot [\alpha,\beta]- \omega(\alpha) (a)\cdot \beta\). These objects have appeared during the last 50 years under more then ten names, for example as: Lie pseudoalgèbre (Herz), \((R,C)\)-Lie algebra (Rinehart), Lie algebra extensions (Teleman), Lie-Rinehart algebra (Huebschmann), etc.

N. Teleman (1972) constructed the Chern-Weil homomorphism for an arbitrary extension of such algebras \[ e:0\to L'\to L @>\pi>> L''\to 0 \] under the assumption \(\mathbb{Q}\subset R\). J. Huebschmann in the reviewed work loses this assumption. In his previous fundamental paper devoted to Lie-Rinehart algebras [J. Reine Angew. Math. 408, 57-113 (1990; Zbl 0699.53037)] the author pointed out that \(A\) can not possess a unit, however, the whole technical apparatus used in his papers needs the unitality of \(A\). For example, there does not exist – in general – (used by the author) a mapping \(M\otimes_R N\to M\otimes_AN\), \(m \otimes_R n\mapsto m\otimes_An\), since in the tensor product \(M\otimes_AN\) over the nonunital ring \(A\) the equality \((r\cdot m)\otimes_A n=m\otimes_A (r\cdot n)\), \(r\in R\), may not hold. For the classical general definition of an \(R\)-algebra, not necessarily with 1, see for example [N. Jacobson, Structure of rings, AMS, Providence, R. I. (1956; Zbl 0073.02002)].

Now, we concentrate on the assumption of the unitality of \(A\). The author obtains a number of important theorems concerning exact sequences of Lie-Rinehart algebras \(e:0\to L'\to L\to L''\to 0\) (called extensions). Such a sequence fulfills all postulates for the sequences investigated by Teleman (except of the assumption \(\mathbb{Q}\subset R)\).

The original contribution of the author is to define the characteristic homomorphism of \(e\) by using the developed machine of coalgebras.

In the context of the Atiyah sequence of a principal bundle, the author obtains an equivalence of the Chern-Weil homomorphim of a principal bundle \(P(W;G)\) with the homomorphism of the induced Atiyah extension \[ e_P:0\to P \otimes_{Ad}{\mathfrak g}\to TP/ G\to TW\to 0 \] under the assumption of the compactness of the structure Lie group (Reviewer’s remarks: (a) the assumption of the connectedness of the total space \(P\) is needed, (b) the above equivalence holds for arbitrary Lie group \(G\) [see J. Kubarski, 1991 (loc. cit.), an. I. W. Bel’ko, Characteristic classes of transitive Lie algebroids (in Russian), Minsk, Belarus (1994) (Preprint), and Vestsi Akad. Navuk Belarusi, Ser. Fiz.-Mat. Navuk 1997, No. 1, 50-55 (1997; Zbl 0914.57017)]).

The author inserts 4 examples of extensions \(e\) which can be nonintegrable, that is, not covered by the classical theory of principal bundles. In the last the author gives an example of a \(TC\)-foliation having a nonintegral Lie algebroid and a nontrivial Chern-Weil homomorphism. Namely, his example concerns a foliation of \(SU(2) \times SU(2)\) by a one-parametric subgroup dense in the maximal torus \(S^1\times S^1 \subset SU(2) \times SU(2)\). (Reviewer’s remark: this example is a particular case considered by J. Kubarski (1991) of the \(TC\)-foliation of left cosets of a nonclosed Lie subgroup \(H\) in a compact, connected and semisimple Lie grou \(G\).)

At the end the author finds in the theory of Lie-Rinehart algebras a wide class of extensions admitting a flat partial connection and proves some generalization of the Bott Vanishing Theorem [R. Bott, Proc. Symp. Pure Math. 16, 127-131 (1970; Zbl 0206.50501), P. Molino, C. R. Acad. Sci., Paris, Sér. A 272, 779-781 (1971; Zbl 0211.26701), F. Kamber and Ph. Tondeur, Foliated bundles and characteristic classes. Springer Verlag, Lecture Notes Math. 493 (1975; Zbl 0308.57011)], some version of the Bott Vanishing Theorem for Lie algebroids was investigated in J. Kubarski [Trans. Am. Math. Soc. 348, 2151-2167 (1996; Zbl 0858.22009)].

For the entire collection see [Zbl 0904.00041].

(a) regular Lie algebroids [J. Kubarski, The Chern-Weil homomorphism of regular Lie algebroids, Publ. Dépt. Math. Nouv. Sér., Univ. Claude-Bernard Lyon 1, 1-69 (1991), see also Characteristic classes of some Pradines-type groupoids and a generalization of the Bott vanishing theorem, Differ. Geom. Appl., Proc. Conf., Brno, Czechoslovakia, Commun., 189-198 (1987; Zbl 0641.57012)],

(b) Lie algebra extensions [N. Teleman, Global Anal. Appl., Vol. III, Intern. Sem. Course Trieste 1972, 195-202 (1974; Zbl 0306.57025); Atti Accad. Naz. Lincei., Rend., Cl. Sci. Fis. Mat. Natur. (8) 52, 498-506 (1972; Zbl 0283.53018) and 708-711 (1972; Zbl 0283.53019)],

(c) extensions of principal bundles [K. Mackenzie, Ann. Global Anal. Geom. 6, 141-163 (1988; Zbl 0627.55010)].

Lie algebroids appear as the infinitesimal objects of principal bundles, vector bundles, transversally complete foliations, Poisson manifolds, Jacobi manifolds, etc.

J. Huebschmann examines an algebraic equivalent of Lie algebroids, called Lie-Rinehart algebras. A Lie-Rinehart algebra over \(A\) \((A\) is some commutative \(R\)-algebra where \(R\) is a commutative unital ring) is a pair \((L, \omega)\) where \(L\) is an \(A\)-module and an \(R\)-Lie algebra and \(\omega:L \to \text{Der}(A)\) is an action of the \(R\)-Lie algebra \(L\) on \(A\), such that \(\omega\) is \(A\)-linear and fulfills the axiom \([a\cdot \alpha,\beta] =a\cdot [\alpha,\beta]- \omega(\alpha) (a)\cdot \beta\). These objects have appeared during the last 50 years under more then ten names, for example as: Lie pseudoalgèbre (Herz), \((R,C)\)-Lie algebra (Rinehart), Lie algebra extensions (Teleman), Lie-Rinehart algebra (Huebschmann), etc.

N. Teleman (1972) constructed the Chern-Weil homomorphism for an arbitrary extension of such algebras \[ e:0\to L'\to L @>\pi>> L''\to 0 \] under the assumption \(\mathbb{Q}\subset R\). J. Huebschmann in the reviewed work loses this assumption. In his previous fundamental paper devoted to Lie-Rinehart algebras [J. Reine Angew. Math. 408, 57-113 (1990; Zbl 0699.53037)] the author pointed out that \(A\) can not possess a unit, however, the whole technical apparatus used in his papers needs the unitality of \(A\). For example, there does not exist – in general – (used by the author) a mapping \(M\otimes_R N\to M\otimes_AN\), \(m \otimes_R n\mapsto m\otimes_An\), since in the tensor product \(M\otimes_AN\) over the nonunital ring \(A\) the equality \((r\cdot m)\otimes_A n=m\otimes_A (r\cdot n)\), \(r\in R\), may not hold. For the classical general definition of an \(R\)-algebra, not necessarily with 1, see for example [N. Jacobson, Structure of rings, AMS, Providence, R. I. (1956; Zbl 0073.02002)].

Now, we concentrate on the assumption of the unitality of \(A\). The author obtains a number of important theorems concerning exact sequences of Lie-Rinehart algebras \(e:0\to L'\to L\to L''\to 0\) (called extensions). Such a sequence fulfills all postulates for the sequences investigated by Teleman (except of the assumption \(\mathbb{Q}\subset R)\).

The original contribution of the author is to define the characteristic homomorphism of \(e\) by using the developed machine of coalgebras.

In the context of the Atiyah sequence of a principal bundle, the author obtains an equivalence of the Chern-Weil homomorphim of a principal bundle \(P(W;G)\) with the homomorphism of the induced Atiyah extension \[ e_P:0\to P \otimes_{Ad}{\mathfrak g}\to TP/ G\to TW\to 0 \] under the assumption of the compactness of the structure Lie group (Reviewer’s remarks: (a) the assumption of the connectedness of the total space \(P\) is needed, (b) the above equivalence holds for arbitrary Lie group \(G\) [see J. Kubarski, 1991 (loc. cit.), an. I. W. Bel’ko, Characteristic classes of transitive Lie algebroids (in Russian), Minsk, Belarus (1994) (Preprint), and Vestsi Akad. Navuk Belarusi, Ser. Fiz.-Mat. Navuk 1997, No. 1, 50-55 (1997; Zbl 0914.57017)]).

The author inserts 4 examples of extensions \(e\) which can be nonintegrable, that is, not covered by the classical theory of principal bundles. In the last the author gives an example of a \(TC\)-foliation having a nonintegral Lie algebroid and a nontrivial Chern-Weil homomorphism. Namely, his example concerns a foliation of \(SU(2) \times SU(2)\) by a one-parametric subgroup dense in the maximal torus \(S^1\times S^1 \subset SU(2) \times SU(2)\). (Reviewer’s remark: this example is a particular case considered by J. Kubarski (1991) of the \(TC\)-foliation of left cosets of a nonclosed Lie subgroup \(H\) in a compact, connected and semisimple Lie grou \(G\).)

At the end the author finds in the theory of Lie-Rinehart algebras a wide class of extensions admitting a flat partial connection and proves some generalization of the Bott Vanishing Theorem [R. Bott, Proc. Symp. Pure Math. 16, 127-131 (1970; Zbl 0206.50501), P. Molino, C. R. Acad. Sci., Paris, Sér. A 272, 779-781 (1971; Zbl 0211.26701), F. Kamber and Ph. Tondeur, Foliated bundles and characteristic classes. Springer Verlag, Lecture Notes Math. 493 (1975; Zbl 0308.57011)], some version of the Bott Vanishing Theorem for Lie algebroids was investigated in J. Kubarski [Trans. Am. Math. Soc. 348, 2151-2167 (1996; Zbl 0858.22009)].

For the entire collection see [Zbl 0904.00041].

Reviewer: J.Kubarski (Łódź)

##### MSC:

17B65 | Infinite-dimensional Lie (super)algebras |

53C05 | Connections, general theory |

55R40 | Homology of classifying spaces and characteristic classes in algebraic topology |

57R30 | Foliations in differential topology; geometric theory |

22E65 | Infinite-dimensional Lie groups and their Lie algebras: general properties |