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The Variety of Lagrangian Subalgebras of Real Semisimple Lie Algebras

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The Variety of Lagrangian Subalgebras of Real Semisimple Lie Algebras THE VARIETY OF LAGRANGIAN SUBALGEBRAS OF REAL SEMISIMPLE LIE ALGEBRAS A Dissertation Submitted to the Graduate School of the University of Notre Dame in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Oleksandra Lyapina Samuel R. Evens, Director Graduate Program in Mathematics Notre Dame, Indiana June 2009 THE VARIETY OF LAGRANGIAN SUBALGEBRAS OF REAL SEMISIMPLE LIE ALGEBRAS Abstract by Oleksandra Lyapina The purpose of our work is to generalize a result of Evens and Lu [9] from the complex case to the real one. In [9], Evens and Lu considered the variety L of complex Lagrangian subalgebras of g × g; where g is a complex semisimple Lie algebra. They described irreducible components, and showed that all irre- ducible components are smooth. In particular, some irreducible components are De Concini-Procesi compactifications of the adjoint group G associated to g; and its geometry at infinity plays a role in understanding L and in applications in Lie theory [10], [11]. In this work we consider an analogous problem, where g0 is a real semisimple Lie algebra. We describe the irreducible components of Λ; the real algebraic variety of Lagrangian subalgebras of g0 × g0. Let g be the complexification of g0: Denote by σ the complex conjugation on g with respect to g0: Let G be the adjoint group of g: Then σ can be lifted to an antiholomorphic involutive automorphism σ : G ! G; which we denote by σ too. The set of σ-fixed points Gσ = fg 2 G : σ(g) = gg is a real algebraic group σ with Lie algebra g0: We study G using a result of Steinberg [20] and the Atlas project software [23]. Our results also give a description of a compactification of Oleksandra Lyapina Gσ: Although our methods are similar to the methods of [9], the real structure introduces some additional difficulties. In memory of my mother Neonila Lyapina ii CONTENTS TABLES . iv ACKNOWLEDGMENTS . .v SYMBOLS . vi CHAPTER 1: INTRODUCTION . .1 CHAPTER 2: THE VARIETY OF LAGRANGIAN SUBALGEBRAS . .6 2.1 Evens and Lu's classification of (G × G)-orbits in L ........7 CHAPTER 3: PRELIMINARIES ON REAL SEMISIMPLE LIE ALGEBRAS 10 CHAPTER 4: AUTOMORPHISMS OF A REAL SEMISIMPLE LIE AL- GEBRA . 19 4.1 The group of quasi-inner automorphisms . 19 4.2 QInt g0-orbits through σ-stable parabolic subalgebras . 42 4.3 The group of outer automorphisms . 43 CHAPTER 5: (Gσ × Gσ)-ORBITS IN Λ . 47 5.1 σ-fixed points in L .......................... 47 5.2 (Gσ × Gσ)-orbits in Λ . 50 σ σ 5.3 Lagrangian subspaces in zS × zT ................... 52 5.4 Closures of (Gσ × Gσ)-orbits in Λ . 53 5.4.1 The variety of Lagrangian subalgebras in g0 × g0, where g0 is a compact real form . 59 5.4.2 Closures of (Gσ × Gσ)-orbits in Λ, continued. 60 5.5 Irreducible components . 61 BIBLIOGRAPHY . 66 iii TABLES 4.1 FUNDAMENTAL GROUPS OF SIMPLE LIE GROUPS OF AD- JOINT TYPE . 25 4.2 THE CONNECTED COMPONENTS OF SIMPLE REAL ALGE- BRAIC GROUPS . 29 iv ACKNOWLEDGMENTS I would like to express my infinite gratitude to my advisor, Professor Samuel Evens, for his guidance, encouragement and inspiration provided throughout my graduate studies at Notre Dame. I would like to thank to my undergraduate advisor Professor Eugene Karolin- sky. I wish to thank Eugene for many virtues I have learned from him. I would like to express my appreciation to the members of the doctoral com- mittee, J. Arlo Caine, William Dwyer, Michael Gekhtman, for reviewing the dis- sertation and helpful discussions. I also wish to thank all of my friends and colleagues for their help and support. I am especially grateful to my father, Yurij Lyapin, for his love and support. v SYMBOLS g a complex semisimple Lie algebra g0 a real form of g σ the complex conjugation on g with respect to g0 τ a Cartan involution on g0, a compact real form on g η a split real form on g compatible with σ l a Lagrangian subalgebra L the variety of complex Lagrangian subalgebras Λ the variety of real Lagrangian subalgebras vi CHAPTER 1 INTRODUCTION Let d be a 2n-dimensional Lie algebra over R or C equipped with a nonde- generate symmetric ad-invariant bilinear form h·; ·i (of signature (n; n) in the real case). Let D be a connected group with Lie D = d. Definition 1.0.1. A Lie subalgebra l ⊂ d is Lagrangian if l is maximal isotropic, i.e., hx; yi = 0 for all x; y 2 l; and dim l = n. The notion of Lagrangian subalgebra was introduced by Drinfeld in [7]. The main motivation for studying Lagrangian subalgebras is through the connection with the theory of Poisson Lie groups. Let us explain this connection in more detail. Definition 1.0.2. A Manin pair is a pair (d; g); where g is a Lagrangian subal- gebra of d: Now let us define the notion of Lie quasi-bialgebra following [6]. Definition 1.0.3. Let g be a Lie algebra, δ a g ^ g-valued 1-cocycle of g, and ' 2 V3 g. A triple (g; δ; ') is called a Lie quasi-bialgebra if 1 Alt(δ ⊗ id)δ(x) = ad ' for any x 2 g; 2 x 1 Alt(δ ⊗ id ⊗ id)' = 0; where adx(a ⊗ b ⊗ c) = [x ⊗ 1 ⊗ 1 + 1 ⊗ x ⊗ 1 + 1 ⊗ 1 ⊗ x; a ⊗ b ⊗ c] and Alt is the alternation map. Furthermore, if ' = 0 we say that (g; δ) is a Lie bialgebra. A Manin pair (d; g) defines a Lie quasi-bialgebra structure on g uniquely up to twisting t 2 V2 g [1]. Namely, (g; δ0;'0) is obtained by twisting (by t 2 V2 g) from (g; δ; ') if 0 δ (x) = δ(x) + adx t for all x 2 g; 1 '0 = ' + Alt(δ ⊗ id)t − CYB(t); 2 where CYB(t) = [t12; t13] + [t12; t23] + [t13; t23] (see [6]). Twisting is an equivalence relation. Definition 1.0.4. A Manin quasi-triple is (d; g; h); where (d; g) is a Manin pair and h is an isotropic linear subspace of d such that d = g ⊕ h: A Manin quasi-triple (d; g; h) defines uniquely a Lie quasi-bialgebra structure on g: This structure can be locally integrated to a quasi-Poisson structure πg;h on the Lie subgroup G of D with Lie G = g: Recall that g is a Lie bialgebra if and only if h is also a Lagrangian subalgebra of d: In the latter case (d; g; h) is called a Manin triple, and (G; πg;h) is a Poisson Lie group. In the case d is a complex or real reductive Lie algebra, all Manin triples were classified in [5]. 2 Now denote by L the variety of all Lagrangian subalgebras in d. Let M be a G-homogeneous space, i.e., the action of G on M is transitive. It follows from [7] and [14] that a (quasi-)Poisson (G; πg;h)-homogeneous structure on M is equivalent to a G-equivariant map P : M !L, m 7! lm such that lm \ g = gm, where gm is the Lie algebra of the stabilizer subgroup of G at m. Thus in order to describe the set of (quasi-)Poisson (G; πg;h)-homogeneous spaces up to local isomorphism it is enough to describe G-conjugacy classes of Lagrangian subalgebras of d. Now we consider L as an algebraic subvariety of the Grassmannian Gr(n; d). The Lie group G acts on L by the adjoint action. Evens and Lu [8] associated to each Manin triple (d; g; h) a (quasi-)Poisson structure Πg;h on L; making L into a (quasi-)Poisson variety. Moreover, each G-orbit in L is a (quasi-)Poisson submanifold and consequently a (G; πg;h)-homogeneous space. The (quasi-)Poisson structure Πg;h on L is constructed in such a way that for any (G; πg;h)-homogeneous space (M; π); the map P : M !L is a (quasi-)Poisson map onto the G-orbit of lm for any m 2 M: In many cases, the map P : M !L is a local diffeomorphism onto its image. Therefore, we can think of G-orbits in L as models for (G; πg;h)- homogeneous (quasi-)Poisson spaces. There are many examples of Lie algebras with a symmetric nondegenerate and ad-invariant bilinear form. The geometry on L depends on d and the bilinear form. Moreover, there are many Lagrangian splittings for a given d, resulting in different Poisson structures on L. It is also important to consider D-orbits in L. The D-orbits and their closures have an interesting geometric structure, which is crucial in understanding the geometry of L. 3 Now let g be a semisimple Lie algebra (real or complex) and d := g × g with the bilinear form h·; ·i given by 1 h(x ; x ); (y ; y )i = (hx ; y i − hx ; y i ); x ; x ; y ; y 2 g; 1 2 1 2 2 1 1 g 2 2 g 1 2 1 2 where h·; ·ig is the Killing form on g. The case of a complex semisimple Lie algebra g was considered in [13] and [8]. In [13], Karolinsky classified the Lagrangian subalgebras in g × g. Further, in [9], Evens and Lu studied the geometry of the variety L. In this paper we study the real algebraic variety of Lagrangian subalgebras in d = g × g, where g is a real semisimple Lie algebra. In future work, we plan to study the Poisson geometry of L defined by a Manin triple (d; g; h), where g is embedded diagonally into d.
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