Some Graded Lie Algebra Structures Associated wit h Lie Algebras and Lie Algebroids Qunfeng Yang A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduat e Department of Mat hematics University of Toronto @Copyright by Qunfeng Yang 1999 National Library Bibliothèque nationale du Canada Acquisitions and Acquisitions et Bibliographie Services services bibliographiques 395 Wellington Street 395, rue Wellington Ottawa ON KIA ON4 0th- ON KI A ON4 Canada Çanada The authar has granted a non- L'auteur a accordé une licence non exclusive licence allowing the exclusive permettant à la National Libmy of Canada to Bibliothèque nationale du Canada de reproduce, loan, distribute or sell reproduire, prêter, distribuer ou copies of this thesis in microform, vendre des copies de cette thèse sous paper or electronic formats. la forme de microfiche/nlm, de reproduction sur papier ou sur format électronique. The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial extracts fkom it Ni la thèse ni des extraits substantiels may be printed or otherwise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation. Some Graded Lie Algebra Structures Associat ed wit h Lie Algebras and Lie Algebroids Ph.D Dissertation, 1999 Qunfeng Yang Depart ment of Mat hematics University of Toront O Abstract The main objects of this thesis are graded Lie algebras associated with a Lie algebra or a Lie dgebroid such as the Fkolicher-Nijenhuis algebra, the Kodaira-Spencer algebra and the newly constructed Gelfand-Dorfman algebra and generalized Nijenhuis-Richardson dgebra. Main results are summarized as follows: We introduce a derived bracket which contains the Rolicher-Nijenhuis bracket as a special case asd prove an interesting formula for this derived bracket. We develop a rigorous rnechonism for the Kodaira-Spencer alge- bra, reveal its relation with R-matrices in the sense of M. A. Semenov-Tian-Shansky and construct fiom it a new example of the knit product structures of graded Lie dgebras. For a given Lie algebra, we construct a new graded Lie algebra cded the Gelfand-Dorfman algebra which provides for r-matrices a graded Lie algebra background and includes the well-known Schouten-Nijenhuis algebra of the Lie algebra as a subalgebra. We establish an anti-homomorphism fiom th% graded Lie algebra to the Nijenhuis-Richardson dge- bra of the dual space of the Lie algebra, which sheds new light on our understanding of Drinfeld's construction of Lie algebra stmctures on the dual space with r-matrices. In addition, we generalize the Nijenhuis-Richardson algebra from the vector space case to the vector bundle case so that Lie algebroids on a vector bundle are defined by this general- ized Nijenhuis-Richardson algebra. We prove that this generalized Nijenhuis-Richardson algebra is isomorphic to both the linear Schouten-Nijenhuis algebra on the dual bundle of the vector bundle and the derivation dgebra associated with the exterior algebra bundle of this dual bundle. A concept of a 2n-ary Lie dgebroid is proposed as an application of these isomorphisms. Acknowledgments First, 1w ould like to thank my supervisor, Professor Velimir Jurdjevic, for his guidance and constant support over the past several years, which made an absolute difference in my life. 1 am gratefd to him for showing me a living example of many great virtues. Professor Albert Nijenhuis, Professor Ragna-OlafB uchweitz and Professor Jim Stash- eff read earlier drafts of this thesis and offered a number of constructive comrnents and criticisms. In particular, Professor Buchweitz drew my attention to the work of Kodaira and Spencer; Professor Nijenhuis and Professor Stashef generously shared their insights into topics which I om concerned in this thesis. 1 am deeply grateful to all of them. 1 dso want to thank Professor Rick Sharpe who patiently and kindly answered a lot of questions of mine during the last several yeoss. My friends Mr. Chen Yuming, Mr. Lu Hong, Mr. Ra0 Aibin, Dr. Sun Heng, and Dr. Wang Yan deserve a mention for their friendship and moral support over a long period of time. In this category, 1 should also mite down the names of Mr. Deng Wei, Ms. Gu Hong, Mr. Yang Bing, Mr. Zhang Hui, Ms. Liu Xiaoling, Mr. Li Gufeng, Mr. Zhang Jianxin, Mr. Pan Min and Dr. Deng Minghua who provided me the hancial support for my trip to Canada more thân four years ago. Without this help, the present thesis would have never been writ ten. Thanks also go to aU the staff in out department of mathematics, especially Ms. Ida Bulat, for their warm-hearted assistance during my studies in Toronto. Contents Abstract ii Acknowledgments iii 1 Introduction 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 MainResults 1 . . . . . . . . . . . . . . . . . . . . 1.1.1 The Frolicher-Nijenhuis Algebra 2 . . . . . . . . . . . . . . . . . . . . . 1.1.2 The Kodaira-Spencer Algebra 2 . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Knit Product Structures 3 1.1.4 The Gelfand-Dorfman Algebra . . . . . . . . . . . . . . . . . . . . . 3 1.1.5 The Generalized Nijenhuis-Richardson Algebra . . . . . . . . . . . . 4 . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Techniques behind the Results 5 1.3 The Structure of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Cast of Chasacters 7 Preliminaries 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Graded Lie Algebras 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 BasicDefinitions 8 . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 The Derivation Algebra 9 . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 The Right Pre-Lie Algebra 10 . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 The Semidirect Product 11 2.1 -5 The Knit Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 TheShdeAlgebra 12 . . . . . . . . . . . . . . . . 2.3 Classicd Examples of the Graded Lie Algebra 14 2.3.1 The Schouten-Nijenhuis Algebra for a Lie Algebra . . . . . . . . . . 15 . . . . . . . . . . 2.3.2 The Schouten-Nijenhuis Algebra over a Manifold 15 . . . . . . . . . . . . . . . . . . 2.3.3 The Nijenhuis-Richardson Algebra 16 2.3.4 The Lie hduced Algebra . . . . . . . . . . . . . . . . . . . . . . . . 17 3 One Structure and Some Operators 19 . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Semidirect Product Structure 19 3.2 Two Operators and the Cup Algebra . . . . . . . . . . . . . . . . . . . . . 22 3.2.1 Operators 6 and O . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 The Cup Algebra 24 4 The Fkolicher-Nijenhuis Algebra 28 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Nijenhuis' Idea 28 4.2 A Derived Bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 5 The Kodaira-Spencer Algebra 38 . . . . . . . . . . . . . . . . 5.1 Construction of the Kodaira-Spencer Algebra 38 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 A Second Knit Product 42 6 The Gelfand-Dorfman Algebra 46 . . . . . . . . . . . . . . . . 6.1 Construction of the Gelfand-Dorfman Algebra 46 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The Cyclic Subalgebra 48 . . . . . . . . . . . . . . . . . . . . . . . . 6.3 The Schouten-Nijenhuis Algebra 54 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Drinfeld's Construction 58 7 The Generalized Nijenhuis-Richardson Algebra 63 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Building the Algebra 63 . . . . . . . . . . . . . . . . . . . . 7.2 The Linear Schouten-Nijenhuis Algebra 68 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 The Derivation Algebra 77 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Zn-ary Lie Algebroids 83 References Appendix: Glossary of Important Symbols Chapter 1 Introduction Various constructions and algebra structures can be described in terms of degree 1, bradset-square O elements of gaded Lie algebras. Such descriptions usually provide new perspectives when we are dealing with some problems associated with these construc- tions and structures. This is clear in algebraic deformation theory ([GS], see also [LMS]). The Gerstanhaber algebra and the Nijenhuis-Richardson algebra are powerful tools in the study of deformations of associative and Lie dgebras respectively ([Gl] and [NR2]). It is also clear in differential geometry. Examples here includes the characterization of Pois- son structures on a manifold through the Schouten-Nijenhuis algebra over the manifold ([VI]) and the Newlander-Nirenberg theorem in tenns of the Frolicher-Nijenhuis algebra over a manifold ([NP?] and [FN1,2]). We will construct in this thesis two new graded Lie dgebra structures which are called the Gelfand-Dorfman algebra for a Lie algebra and the generalized Nijenhuis-Richardson algebra over a vector bundle and provides some new insights into the well-known Fkolicher-Nijenhuis algebra and the Kodaira-Spencer algebra. These graded Lie algebras describe such important mathematical objects as r-matrices, Lie algebroids, R-matrices and Nijenhuis operators in the above-mentioned manner. 1.1 Main Results We Est our main results in order of their appearasce in the body of this thesis. 1.1.1 The Fkolicher-Nijenhuis Algebra The Frclicher-Nijenhuis algebra on Alt(V, V) for a Lie algebra V was studied in [N2]. Its degree 1, bracket-square O elements are sometimes called Nijenhuis operators. A Nijenhuis operator induces a second Lie algebra structure on V and this new Lie algebra structure plays an important role in the bihamiltonain method of studying completely integrable Harniltonian systcms ([Dl and [K-SM], see also [MM]). In Chapter 4, we will introduce a bracket on AN(V, V) which is derived from the Nijenhuis-Richardson bracket on Alt(V, V)[1] (a graded vector space obtained by shifting Alt(V, V) down by 1 degree) and which contains the Rolicher-Nijenhuis bracket as a special case. We particulady focus on a formula associated with this derived bracket (Theorem 4.5). Such a formula is established in [N2] for the Ftiilicher-Nijenhuis bracket to express the Fkolicher-Nijenhuis bracket for the new Lie algebra on V induced by a Nijenhuis operator in terms of the Frolicher-Nijenhuis bracket for the original Lie algebra on V. While it is not diffcult to realize that Nijenhuis' formula holds for our more general derived bracket, the proof of this formula in our thesis is new. 1.1.2 The Kodaira-Spencer Algebra In Chapter 5, we establish the Kodaira-Spencer algebra on Alt(V, V) for a Lie algebra V. It provides a graded Lie algebra description of both the classical and the modified classical Yang-Baxter equations associated with the Lie algebra V in the sense of Semenov-Tias- Shansky ([STS]). Some interesting results of the Kodaira-Spencer algebra follow from our approach to its constructon. For example, we easily have that an interesting operator 8 is a homomorphism from the Kodaira-Spencer algebra to the Nijenhuis-Richardson algebra of the underlying vector space of the Lie algebra V ( see (5.4)). The fact that R-matrices, as solutions to Yang-Baxter equations, define new Lie algebra structures on V becomes a direct consequence of this homomorphism. The Kodaira-Spencer algebra was originally defined on the graded vector space of vector-valued differential forms on a manifold ([KSI and [BM]). To my knowledge, the version we consider in this thesis has not been studied before. 1.1.3 Knit Product Structures A knit product is a graded Lie algebra structure on the direct surn of two graded Lie al- gebras when they have mutual representations on each other satisfying certain conditions (2.5). In Chapter 4, we have a more clear (compared with [N2]) and more straight (compared with [Mi]) exposition of the knit product of the Nijenhuis-Richardson alge- bra Alt (V,V ) [l] ([NR2]) and the Fkolicher-Nijenhuis algebra Alt ( V, V) (Theorem 4.4). In addition, we show in Chapter 5 there exists a knit product structure between the Nijenhuis-Richardson dgebra and the Kodaira-Spencer algebra (Theorem 5.6). As far as 1 know, this is only the second example of a knit product of graded Lie dgebras. We point out constructions similar to the knit product have been studied for some other algebra structures in mathematics. For exmple, Majid considered the Lie algebra case and coined the name a matched pair ([Ml) for two Lie algebras fiom the direct sum of which a new Lie algebra can be constructed. Mokri studied a matched pair of Lie algebroids ([Mo]). The newly constructed structure is called a twilled extension for Lie algebras by Kosmann-Schwanbach and Magri ([K-SM]a)n d for Lie-Reinhart algebras by Huebschmann ([HI)T. he name of a knit product for graded Lie algebras is given by Michor ([Mi]). 1.1.4 The Gelfand-Dorfman Algebra The first new graded Lie algebra we construct in this thesis is the Gelfand-Dorfman algebra /\ V @ V for a Lie algebra V. Its degree 1, bracket-square O elements are general (not necessarily anti-symmetric) r-matrices of the Lie dgebra V ([Dr1,2]). In Chapter 6, besides the construction of the Gelfand-Dorfman algebra (Theorem 6.1), we establish two results. First, we show that the Gelfand-Dorfman algebra contains a subalgebra isomorphic to the Schouten-Nijenhuis algebra (Theorem 6.7). This is a natural result since anti-symmetric r-matrices are degree 1, bracket square O elements of the Schouten-Nijenhuis algebra. Second, we establish an anti-homomorphism fiom the Gelfand-Dorfman dgebra to the Nijenhuis-Richardson algebra Alt(VmV, *)[l] for the vector space V* (Theorem 6 -9). This anti-homomorphism generalizes a construction of Drinfeld in the Poisson-Lie group theory (see Proposition 6.8). 1.1.5 The Generalized Nijenhuis-Richardson Algebra To describe the generalized Nijenhuis-Richardson algebra, it is convenient to recd the notion of a Lie algebroid ([Ma1,2]) first. A Lie algebroid over a smooth manifold M is a vector bundle A over M together with a Lie algebra structure on the space r(A)o f smooth sections of A and a bundle map p : A -t TM such that p defines a Lie algebra homomorphism from r(A)t o X(M), the Lie algebra of vector fields over M, and there holds for f E C"(M) and tl,6 E l'(A), the following derivation law, A Lie algebroid is a generalization of a Lie algebra. The natural question is : what is the graded Lie algebra on a vector bundle which defines Lie algebroid structures? In Chapter 7, we constmct such a graded Lie algebra LR(A) for a vector bundle A through a generalization of the Nijenhuis- Richatdson algebra from the vector space case to the vector bundle one (Theorem 7.3). It is known that Lie algebroids on a vector bundle A are in one-one correspondence with linear Poisson structures on its dual bundle A* ([Cl and [CDW], see also [Wl]). This is sometimes called the generalized Lie-Poisson construction. In Chapter 7, we point out that linear polyvector fields on a vector bundle constitute a subalgebra of the Schouten-Nijenhuis algebra over the bundle (considered as a manifold), which will be called the linear Schouten-Nijenhuis dgebra over the bundle, and prove that the general- ized Nijenhuis-Richardson algebra for a vector bundle is isomorphic to the linear Schouten- Nijenhuis algebra over its dual bundle (Theorem 7.10). Since the degree 1, bracket-square O elements of the Iinear Schouten-Nijenhuis algebra dehe linear Poisson structures, our result extends the generalized Lie-Poisson construction. In the course of developing The- orem 7.10, we dso give a different proof of the following result: the Schouten-Nijenhuis algebra over a manifold N is a subalgebra of the Nijenhuis-Richardson algebra for the vector space COD(N)( [CKMV]). We also extend another correspondence in the Lie algebroid theory, the correspondence between Lie algebroids on a vector bundle A and 1-difierentials of sections of the exterior algebra bundle of its dual bundle A* ([K-SMa]n d [XI). We establish an isomorphisrn between the generalized Nijenhuis-Richardson algebra for A and the derintion dgebra