Double Lie algebroids and second-order geometry. II. (English) Zbl 0971.58015

Mackenzie, K. C. H.

Adv. Math. 154, No. 1, 46-75 (2000).

From the author’s introduction: “In part I of this paper we proposed a Lie theory for double Lie groupoids based on the known Lie theory of ordinary Lie groupoids, on various constructions in Poisson geometry, and on the special features of the differential geometry of the tangent bundle [see the author, Adv. Math. 94, No. 2, 180-239 (1992; Zbl 0765.57025)]. That paper gave the first of the two steps involved in constructing the double Lie algebroid of a double Lie groupoid and one purpose of the present paper is to give the more difficult second step.
Ordinary Lie algebroids may be viewed both as generalizations of Lie algebras and therefore as vehicles for a generalized Lie theory [see the author, Bull. Lond. Math. Soc. 27, No. 2, 97-147 (1995; Zbl 0829.22001)] and as abstractions of the tangent bundle of an ordinary manifold see A. Weinstein [Fields Inst. Commun. 7, 207-231 (1996; Zbl 0829.22001)] and the author and P. Xu [Q. J. Math., Oxf. II. Ser. 49, No. 193, 59-85 (1998; Zbl 0926.58015)].
The differentiation process given here includes as special cases: (i) The process of passing from a double Lie group [see J.-H. Lu and A. Weinstein, J. Differ. Geom. 31, No. 2, 501-526 (1990; Zbl 0673.58019)] or matched pair of Lie groups [see S. Majid, Pac. J. Math. 141, No. 2, 311-332 (1990; Zbl 0735.17017)] to the corresponding double Lie algebra or matched pair of Lie algebras; (ii) the (related) process of obtaining a Lie bialgebra from a Poisson Lie group or, more generally, a Lie bialgebroid from a Poisson groupoid [the author and P. Xu, Duke Math. J. 73, No. 2, 415-452 (1994; Zbl 0844.22005)]; and (iii) the process of obtaining a pair of compatible partial connections from an affinoid [see A. Weinstein, Int. J. Math. 1, 343-360 (1990; Zbl 0725.58014)]. However, from our point of view the fundamental process is that of obtaining the double (iterated) tangent bundle \( T ^2 M = T(TM) \) from the double groupoid structure on \( M ^4, \) where elements of \( M ^4 \) are regarded as the corners of an empty square. All these processes yield double Lie algebroids and we show that the calculus possible for the double tangent bundle applies to them all.
In our previous paper [loc. cit.] it was shown how a single application of the Lie functor to a double Lie groupoid \( S \) produces an \({\mathcal L}A\)-groupoid, that is, a Lie groupoid object in the category of Lie algebroids. If one applies the Lie functor to, say, the vertical structure of \( S, \) then the \({\mathcal L}A\)-groupoid is vertically a Lie algebroid and horizontally a Lie groupoid. One may then take the Lie algebroid of this horizontal groupoid and obtain a double vector bundle whose horizontal structure is a Lie algebroid. Interchanging the order of the processes yields a second double vector bundle with the Lie algebroid structure now placed vertically. The two double vector bundles may be identified by a map derived from the canonical involution in the double tangent bundle of \(S\), and one thus obtains a double vector bundle all four sides of which have Lie algebroid structures; this is the double Lie algebroid of \(S\). This is the construction as outlined in the introduction to our previously mentioned paper [loc. cit.].
However, abstract \(\mathcal{L}A\)-groupoids also arise in nature without an underlying double Lie groupoid, most notably the cotangent \(\mathcal{L}A\)-groupoid of a Poisson groupoid or Poisson Lie group. We have therefore shown in Section 2 here that the Lie algebroid structure on an \(\mathcal{L}A\)-groupoid may be prolonged to the Lie algebroid of the groupoid structure. This extends, in particular, the construction of tangent Lie algebroid structures given by the author and P. Xu [loc. cit.].
This prolongation process goes back ultimately to the very classical description of the vector fields on a tangent bundle in terms of the vertical and complete lifts of the vector fields on the base manifold. Regarding a tangent bundle \( TM \) as the Lie algebroid of the pair (or coarse) groupoid \(M\times M\), the author and P. Xu [loc. cit.] extended this description: the vector fields on the Lie algebroid \(AG\) of any Lie groupoid \(G\rightrightarrows M \) are generated by those lifted from suitable vector fields on \( G \) and those core vector fields which come from sections of \( AG\). The bracket relations between vector fields obtained from these two processes now involve an operation by which suitable vector fields \( \xi \) on \( G \) induce covariant differential operators \( D _\xi \) on \( AG\); this is an extension of the intrinsic derivative as used by the author and Ping Xu in their mentioned paper.
In Section 2 we give a calculus of similar type for any \(\mathcal{L}A\)-groupoid. We must then prove that, in the case of the \( \mathcal{L}A\)-groupoids of a double Lie groupoid, the prolonged Lie algebroid structure on the double Lie algebroid of either \(\mathcal{L}A\)-groupoid coincides with the Lie algebroid of the Lie groupoid structure of the other. This result, Theorem 3.3, whose proof occupies most of Section 3, embodies not only the canonical isomorphisms arising from the structure of the double tangent bundle (see Examples 4.1 and 3.1), but canonical isomorphisms familiar in symplectic and Poisson geometry (see Example 4.4).
We feel that the basic simplicity of the construction given here, and the richness of the phenomena which it encompasses, establish beyond doubt that it is a natural and correct definition. We have argued this case in detail in a previous paper [see the author, Electron. Res. Announc. Am. Math. Soc. 4, No. 11, 74-87 (1998; Zbl 0924.58115)], which summarizes the results of the present paper, and of two subsequent ones [see the author, Int. J. Math. 10, No. 4, 435-456 (1999; Zbl 0961.58008); ‘Double Lie algebroids and the double of a Lie bialgebroid’, submitted for publication], and gives an overview of the background to all three”.

Reviewer: Mircea Craioveanu (Timişoara)

MSC:

58H05	Pseudogroups and differentiable groupoids
22A22	Topological groupoids (including differentiable and Lie groupoids)

Keywords:

Lie groupoid; Lie algebroid; double Lie groupoid; double Lie algebroid; Poisson Lie group; star section; Lie bialgebra; Lie bialgebroid; Poisson groupoid; affinoid; double tangent bundle; symplectic and Poisson geometry

Citations:

Zbl 0829.22001; Zbl 0765.57025; Zbl 0844.22007; Zbl 0926.58015; Zbl 0673.58018; Zbl 0735.17017; Zbl 0844.22005; Zbl 0725.58014; Zbl 0924.58115; Zbl 0961.58008; Zbl 0673.58019

References:

[1]	Brown, R.; Mackenzie, K. C.H., Determination of a double Lie groupoid by its core diagram, J. Pure Appl. Algebra, 80, 237-272 (1992) · Zbl 0766.22001
[2]	Coste, A.; Dazord, P.; Weinstein, A., Groupoı̈des symplectiques, Publications du Département de Mathématiques de l’Université de Lyon, I (1987), p. 1-65
[3]	Courant, T. J., Tangent Dirac structures, J. Phys. A, 23, 5153-5160 (1990) · Zbl 0715.58013
[4]	Drinfel’d, V. G., On Poisson homogeneous spaces of Poisson-Lie groups, Theoret. Math. Phys., 95, 226-227 (1993) · Zbl 0852.22018
[5]	Grabowski, J.; Urbański, P., Tangent and cotangent lifts and graded Lie algebras associated with Lie algebroids, Ann. Global Anal. Geom., 15, 447-486 (1997) · Zbl 0973.58006
[6]	Higgins, P. J.; Mackenzie, K. C.H., Algebraic constructions in the category of Lie algebroids, J. Algebra, 129, 194-230 (1990) · Zbl 0696.22007
[7]	Liu, Zhang-Ju; Weinstein, A.; Xu, Ping, Manin triples for Lie bialgebroids, J. Differential Geom., 45, 547-574 (1997) · Zbl 0885.58030
[8]	Lu, Jiang-Hua, Poisson homogeneous spaces and Lie algebroids associated to Poisson actions, Duke Math. J., 86, 261-304 (1997) · Zbl 0889.58036
[9]	Lu, Jiang-Hua; Weinstein, A., Poisson Lie groups, dressing transformations, and Bruhat decompositions, J. Differential Geom., 31, 501-526 (1990) · Zbl 0673.58018
[10]	Mackenzie, K., Lie Groupoids and Lie Algebroids in Differential Geometry. Lie Groupoids and Lie Algebroids in Differential Geometry, London Mathematical Society Lecture Note Series, 124 (1987), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0683.53029
[11]	Mackenzie, K. C.H., Double Lie algebroids and second-order geometry, I, Adv. Math., 94, 180-239 (1992) · Zbl 0765.57025
[12]	Mackenzie, K. C.H., Lie algebroids and Lie pseudoalgebras, Bull. London Math. Soc., 27, 97-147 (1995) · Zbl 0829.22001
[13]	Mackenzie, K. C.H., Drinfel’d doubles and Ehresmann doubles for Lie algebroids and Lie bialgebroids, Electron. Res. Announc. Amer. Math. Soc., 4, 74-87 (1998) · Zbl 0924.58115
[14]	Mackenzie, K. C.H., On symplectic double groupoids and the duality of Poisson groupoids, Internat. J. Math., 10, 435-456 (1999) · Zbl 0961.58008
[15]	K. C. H. Mackenzie, Double Lie algebroids and the double of a Lie bialgebroid, submitted for publication.; K. C. H. Mackenzie, Double Lie algebroids and the double of a Lie bialgebroid, submitted for publication.
[16]	K. C. H. Mackenzie, On Poisson actions, matched pairs and double structures, preprint, 1998.; K. C. H. Mackenzie, On Poisson actions, matched pairs and double structures, preprint, 1998.
[17]	Mackenzie, K. C.H., Affinoid structures and connections, Poisson Geometry. Poisson Geometry, Banach Center Publications, 51 (2000), Banach Center: Banach Center Warsaw, p. 175-186 · Zbl 1028.58023
[18]	Mackenzie, K. C.H.; Xu, Ping, Lie bialgebroids and Poisson groupoids, Duke Math. J., 73, 415-452 (1994) · Zbl 0844.22005
[19]	Mackenzie, K. C.H.; Xu, Ping, Classical lifting processes and multiplicative vector fields, Quart. J. Math. Oxford Ser. (2), 49, 59-85 (1998) · Zbl 0926.58015
[20]	Majid, S., Matched pairs of Lie groups associated to solutions of the Yang-Baxter equations, Pacific J. Math., 141, 311-332 (1990) · Zbl 0735.17017
[21]	Mokri, T., Matched pairs of Lie algebroids, Glasgow Math. J., 39, 167-181 (1997) · Zbl 0886.22012
[22]	Tulczyjew, W. M., Geometric Formulation of Physical Theories. Geometric Formulation of Physical Theories, Monographs and Textbooks in Physical Science, 11 (1989), Bibliopolis: Bibliopolis Naples · Zbl 0707.58001
[23]	Weinstein, A., Coisotropic calculus and Poisson groupoids, J. Math. Soc. Japan, 40, 705-727 (1988) · Zbl 0642.58025
[24]	Weinstein, A., Affine Poisson structures, Internat. J. Math., 1, 343-360 (1990) · Zbl 0725.58014
[25]	Weinstein, A., Lagrangian mechanics and groupoids, Fields Inst. Comm., 7, 207-231 (1996) · Zbl 0844.22007
[26]	Yano, K.; Ishihara, S., Tangent and Cotangent Bundles. Tangent and Cotangent Bundles, Pure and Applied Mathematics, 16 (1973), Dekker: Dekker New York · Zbl 0262.53024