DOCSLIB.ORG

Notes for Math 261A Lie Groups and Lie Algebras March 28, 2007

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Notes for Math 261A Lie Groups and Lie Algebras March 28, 2007 Notes for Math 261A Lie groups and Lie algebras March 28, 2007 Contents Contents 1 How these notes came to be 4 Dependence of results and other information 5 Lecture 1 6 Lecture 2 9 Tangent Lie algebras to Lie groups 9 Lecture 3 12 Lecture 4 15 Lecture 5 19 Simply Connected Lie Groups 19 Lecture 6 - Hopf Algebras 24 The universal enveloping algebra 27 Lecture 7 29 Universality of Ug 29 Gradation in Ug 30 Filtered spaces and algebras 31 Lecture 8 - The PBW Theorem and Deformations 34 Deformations of associative algebras 35 Formal deformations of associative algebras 37 Formal deformations of Lie algebras 38 Lecture 9 40 Lie algebra cohomology 41 Lecture 10 45 Lie algebra cohomology 45 H2(g, g) and Deformations of Lie algebras 46 2 Lecture 11 - Engel’s Theorem and Lie’s Theorem 50 The radical 55 Lecture 12 - Cartan Criterion, Whitehead and Weyl Theorems 56 Invariant forms and the Killing form 56 Lecture 13 - The root system of a semisimple Lie algebra 63 Irreducible finite dimensional representations of sl(2) 67 Lecture 14 - More on Root Systems 69 Abstract Root systems 70 The Weyl group 72 Lecture 15 - Dynkin diagrams, Classification of root systems 77 Construction of the Lie algebras An, Bn, Cn, and Dn 82 Isomorphisms of small dimension 83 Lecture 16 - Serre’s Theorem 85 Lecture 17 - Constructions of Exceptional simple Lie Algebras 90 Lecture 18 - Representations of Lie algebras 96 Highest weights 99 Lecture 19 - The Weyl character formula 104 Lecture 20 - Compact Lie groups 114 Lecture 21 - An overview of Lie groups 118 Lie groups in general 118 Lie groups and Lie algebras 120 Lie groups and finite groups 121 Lie groups and Algebraic groups (over R) 122 Important Lie groups 123 Lecture 22 - Clifford algebras 126 Lecture 23 132 Clifford groups, Spin groups, and Pin groups 133 Lecture 24 139 Spin representations of Spin and Pin groups 139 Triality 142 More about Orthogonal groups 143 3 Lecture 25 - E8 145 Lecture 26 150 Construction of E8 152 Lecture 27 156 Construction of the Lie group of E8 159 Real forms 160 Lecture 28 162 Working with simple Lie groups 164 Every possible simple Lie group 166 Lecture 29 168 Lecture 30 - Irreducible unitary representations of SL2(R) 175 Finite dimensional representations 175 The Casimir operator 178 Lecture 31 - Unitary representations of SL2(R) 181 Background about infinite dimensional representations 181 The group SL2(R) 182 Solutions to (some) Exercises 187 References 195 Index 197 How these notes came to be 4 How these notes came to be Among the Berkeley professors, there was once Allen Knutson, who would teach Math 261. But it happened that professor Knutson was on sabbatical at UCLA, and eventually went there for good. During this turbulent time, Maths 261AB were cancelled two years in a row. The last of these four semesters (Spring 2006), some graduate students gath- ered together and asked Nicolai Reshetikhin to teach them Lie theory in a giant reading course. When the dust settled, there were two other pro- fessors willing to help in the instruction of Math 261A, Vera Serganova and Richard Borcherds. Thus Tag Team 261A was born. After a few lectures, professor Reshetikhin suggested that the stu- dents write up the lecture notes for the benefit of future generations. The first four lectures were produced entirely by the “editors”. The re- maining lectures were LATEXed by Anton Geraschenko in class and then edited by the people in the following table. The columns are sorted by lecturer. Nicolai Reshetikhin Vera Serganova Richard Borcherds 1 Anton Geraschenko 11 Sevak Mkrtchyan 21 Hanh Duc Do 2 Anton Geraschenko 12 Jonah Blasiak 22 An Huang 3 Nathan George 13 Hannes Thiel 23 Santiago Canez 4 Hans Christianson 14 Anton Geraschenko 24 Lilit Martirosyan 5 Emily Peters 15 Lilit Martirosyan 25 Emily Peters 6 Sevak Mkrtchyan 16 Santiago Canez 26 Santiago Canez 7 Lilit Martirosyan 17 Katie Liesinger 27 Martin Vito-Cruz 8 David Cimasoni 18 Aaron McMillan 28 Martin Vito-Cruz 9 Emily Peters 19 Anton Geraschenko 29 Anton Geraschenko 10 Qingtau Chen 20 Hanh Duc Do 30 Lilit Martirosyan 31 Sevak Mkrtchyan Richard Borcherds then edited the last third of the notes. The notes were further edited (and often expanded or rearranged) by Crystal Hoyt, Sevak Mkrtchyan, and Anton Geraschenko. Send corrections and comments to [email protected]. Dependence of results and other information 5 Dependence of results and other informa- tion Within a lecture, everything uses the same counter, with the exception of exercises. Thus, item a.b is the b-th item in Lecture a, whether it is a theorem, lemma, example, equation, or anything else that deserves a number and isn’t an exercise. Lecture 1 6 Lecture 1 Definition 1.1. A Lie group is a smooth manifold G with a group structure such that the multiplication µ : G G G and inverse map ι : G G are smooth maps. × → → ◮ Exercise 1.1. If we assume only that µ is smooth, does it follow that ι is smooth? n Example 1.2. The group of invertible endomorphisms of C , GLn(C), is a Lie group. The automorphisms of determinant 1, SLn(C), is also a Lie group. Example 1.3. If B is a bilinear form on Cn, then we can consider the Lie group A GL (C) B(Av, Aw)= B(v,w) for all v,w Cn . { ∈ n | ∈ } If we take B to be the usual dot product, then we get the group On(C). T 0 1m If we let n =2m be even and set B(v,w)= v 1m 0 w, then we get Sp (C). − 2m Example 1.4. SU SL (C)isa real form (look in lectures 27,28, and n ⊆ n 29 for more on real forms). Example 1.5. We’d like to consider infinite matrices, but the multi- plication wouldn’t make sense, so we can think of GLn GLn+1 via A 0 ⊆ A ( 0 1 ), then define GL as GLn. That is, invertible infinite ∞ n matrices7→ which look like the identity almost everywhere. S Lie groups are hard objects to work with because they have global characteristics, but we’d like to know about representations of them. Fortunately, there are things called Lie algebras, which are easier to work with, and representations of Lie algebras tell us about representations of Lie groups. Definition 1.6. A Lie algebra is a vector space V equipped with a Lie bracket [ , ] : V V V , which satisfies × → 1. Skew symmetry: [a, a]=0 for all a V , and ∈ 2. Jacobi identity: [a, [b, c]]+[b, [c, a]]+[c, [a, b]] = 0 for all a, b, c V . ∈ A Lie subalgebra of a Lie algebra V is a subspace W V which is closed ⊆ under the bracket: [W, W ] W . ⊆ Lecture 1 7 Example 1.7. If A is a finite dimensional associative algebra, you can set [a, b] = ab ba. If you start with A = Mn, the algebra of n n matrices, then− you get the Lie algebra gl . If you let A M be× the n ⊆ n algebra of matrices preserving a fixed flag V V V Cn, then 0 ⊂ 1 ⊂ ··· k ⊆ you get parabolicindexparabolic subalgebras Lie sub-algebras of gln. Example 1.8. Consider the set of vector fields on Rn, Vect(Rn)= ℓ = i ∂ { Σe (x) [ℓ1,ℓ2]= ℓ1 ℓ2 ℓ2 ℓ1 . ∂xi | ◦ − ◦ } ◮ Exercise 1.2. Check that [ℓ1,ℓ2] is a first order differential operator. Example 1.9. If A is an associative algebra, we say that ∂ : A A is a derivation if ∂(ab)=(∂a)b + a∂b. Inner derivations are those→ of the form [d, ] for some d A; the others are called outer derivations. We · ∈ denote the set of derivations of A by (A), and you can verify that it is n D n a Lie algebra. Note that Vect(R ) above is just (C∞(R )). D The first Hochschild cohomology, denoted H1(A, A), is the quotient (A)/ inner derivations . D { } Definition 1.10. A Lie algebra homomorphism is a linear map φ : L → L′ that takes the bracket in L to the bracket in L′, i.e. φ([a, b]L) = [φ(a),φ(b)]L′ . A Lie algebra isomorphism is a morphism of Lie algebras that is a linear isomorphism.1 A very interesting question is to classify Lie algebras (up to isomor- phism) of dimension n for a given n. For n = 2, there are only two: the trivial bracket [ , ] = 0, and [e1, e2] = e2. For n = 3, it can be done without too much trouble. Maybe n = 4 has been done, but in general, it is a very hard problem. k k If ei is a basis for V , with [ei, ej] = cijek (the cij are called the structure{ } constants of V ), then the Jacobi identity is some quadratic k relation on the cij, so the variety of Lie algebras is some quadratic surface in C3n. Given a smooth real manifold M n of dimension n, we can construct Vect(M n), the set of smooth vector fields on M n. For X Vect(M n), ∈ we can define the Lie derivative LX by (LX f)(m)= Xm f, so LX acts n · · on C∞(M ) as a derivation. ◮ Exercise 1.3. Verify that [LX , LY ]= LX LY LY LX is of the form n ◦ − ◦ LZ for a unique Z Vect(M ). Then we put a Lie algebra structure on n ∈ n Vect(M )= (C∞(M )) by [X,Y ]= Z. D 1The reader may verify that this implies that the inverse is also a morphism of Lie algebras.
Load more

Recommended publications
  • On the Left Ideal in the Universal Enveloping
    proceedings of the american mathematical society Volume 121, Number 4, August 1994 ON THE LEFT IDEAL IN THE UNIVERSALENVELOPING ALGEBRA OF A LIE GROUP GENERATED BY A COMPLEX LIE SUBALGEBRA JUAN TIRAO (Communicated by Roe W. Goodman) Abstract. Let Go be a connected Lie group with Lie algebra go and 'et n be a Lie subalgebra of the complexification g of go . Let C°°(Go)* be the annihilator of h in C°°(Go) and let s/ =^{Ccc(G0)h) be the annihilator of C00(Go)A in the universal enveloping algebra %(g) of g. If h is the complexification of the Lie algebra ho of a Lie subgroup Ho of Go then sf = %({g)h whenever Ho is closed, is a known result, and the point of this paper is to prove the converse assertion. The paper has two distinct parts, one for C°° , the other for holomorphic functions. In the first part the Lie algebra ho of the closure of Ha is characterized as the annihilator in go of C°°(Go)h , and it is proved that ho is an ideal in /i0 and that ho = ho © v where v is an abelian subalgebra of Ao . In the second part we consider a complexification G of Go and assume that h is the Lie algebra of a closed connected subgroup H of G. Then we establish that s/{cf(G)h) = %?(g)h if and only if G/H has many holomorphic functions. This is the case if G/H is a quasi-affine variety. From this we get that if H is a unipotent subgroup of G or if G and H are reductive groups then s/(C°°(G0)A) = iY(g)h .
    [Show full text]
  • Lie Groups and Lie Algebras
    2 LIE GROUPS AND LIE ALGEBRAS 2.1 Lie groups The most general definition of a Lie group G is, that it is a differentiable manifold with group structure, such that the multiplication map G G G, (g,h) gh, and the inversion map G G, g g−1, are differentiable.× → But we shall7→ not need this concept in full generality.→ 7→ In order to avoid more elaborate differential geometry, we will restrict attention to matrix groups. Consider the set of all invertible n n matrices with entries in F, where F stands either for the real (R) or complex× (C) numbers. It is easily verified to form a group which we denote by GL(n, F), called the general linear group in n dimensions over the number field F. The space of all n n matrices, including non- invertible ones, with entries in F is denoted by M(n,×F). It is an n2-dimensional 2 vector space over F, isomorphic to Fn . The determinant is a continuous function det : M(n, F) F and GL(n, F) = det−1(F 0 ), since a matrix is invertible → −{ } 2 iff its determinant is non zero. Hence GL(n, F) is an open subset of Fn . Group multiplication and inversion are just given by the corresponding familiar matrix 2 2 2 2 2 operations, which are differentiable functions Fn Fn Fn and Fn Fn respectively. × → → 2.1.1 Examples of Lie groups GL(n, F) is our main example of a (matrix) Lie group. Any Lie group that we encounter will be a subgroups of some GL(n, F).
    [Show full text]
  • Introduction to Representations Theory of Lie Groups
    Introduction to Representations Theory of Lie Groups Raul Gomez October 14, 2009 Introduction The purpose of this notes is twofold. The first goal is to give a quick answer to the question \What is representation theory about?" To answer this, we will show by examples what are the most important results of this theory, and the problems that it is trying to solve. To make the answer short we will not develop all the formal details of the theory and we will give preference to examples over proofs. Few results will be proved, and in the ones were a proof is given, we will skip the technical details. We hope that the examples and arguments presented here will be enough to give the reader and intuitive but concise idea of the covered material. The second goal of the notes is to be a guide to the reader interested in starting a serious study of representation theory. Sometimes, when starting the study of a new subject, it's hard to understand the underlying motivation of all the abstract definitions and technical lemmas. It's also hard to know what is the ultimate goal of the subject and to identify the important results in the sea of technical lemmas. We hope that after reading this notes, the interested reader could start a serious study of representation theory with a clear idea of its goals and philosophy. Lets talk now about the material covered on this notes. In the first section we will state the celebrated Peter-Weyl theorem, which can be considered as a generalization of the theory of Fourier analysis on the circle S1.
    [Show full text]
  • Kac-Moody Algebras and Applications
    Kac-Moody Algebras and Applications Owen Barrett December 24, 2014 Abstract This article is an introduction to the theory of Kac-Moody algebras: their genesis, their con- struction, basic theorems concerning them, and some of their applications. We first record some of the classical theory, since the Kac-Moody construction generalizes the theory of simple finite-dimensional Lie algebras in a closely analogous way. We then introduce the construction and properties of Kac-Moody algebras with an eye to drawing natural connec- tions to the classical theory. Last, we discuss some physical applications of Kac-Moody alge- bras,includingtheSugawaraandVirosorocosetconstructions,whicharebasictoconformal field theory. Contents 1 Introduction2 2 Finite-dimensional Lie algebras3 2.1 Nilpotency.....................................3 2.2 Solvability.....................................4 2.3 Semisimplicity...................................4 2.4 Root systems....................................5 2.4.1 Symmetries................................5 2.4.2 The Weyl group..............................6 2.4.3 Bases...................................6 2.4.4 The Cartan matrix.............................7 2.4.5 Irreducibility...............................7 2.4.6 Complex root systems...........................7 2.5 The structure of semisimple Lie algebras.....................7 2.5.1 Cartan subalgebras............................8 2.5.2 Decomposition of g ............................8 2.5.3 Existence and uniqueness.........................8 2.6 Linear representations of complex semisimple Lie algebras...........9 2.6.1 Weights and primitive elements.....................9 2.7 Irreducible modules with a highest weight....................9 2.8 Finite-dimensional modules............................ 10 3 Kac-Moody algebras 10 3.1 Basic definitions.................................. 10 3.1.1 Construction of the auxiliary Lie algebra................. 11 3.1.2 Construction of the Kac-Moody algebra................. 12 3.1.3 Root space of the Kac-Moody algebra g(A) ..............
    [Show full text]
  • Lie Groups and Lie Algebras
    Lie groups and Lie algebras Jonny Evans March 10, 2016 1 Lecture 1: Introduction To the students, past, present and future, who have/are/will taken/taking/take this course and to those interested parties who just read the notes and gave me feedback: thank you for providing the background level of enthusiasm, pedantry and general con- fusion needed to force me to improve these notes. All comments and corrections are very welcome. 1.1 What is a representation? Definition 1.1. A representation ρ of a group G on an n-dimensional vector space V over a field k is an assignment of a k-linear map ρ(g): V ! V to each group element g such that • ρ(gh) = ρ(g)ρ(h) • ρ(1G) = 1V . Here are some equivalent definitions. A representation is... 1. ...an assignment of an n-by-n matrix ρ(g) to each group element g such that ρ(gh) = ρ(g)ρ(h) (i.e. the matrices multiply together like the group elements they represent). This is clearly the same as the previous definition once you have picked a basis of V to identify linear maps with matrices; 2. ...a homomorphism ρ: G ! GL(V ). The multiplicativity condition is now hiding in the definition of homomorphism, which requires ρ(gh) = ρ(g)ρ(h); 3. ...an action of G on V by linear maps. In other words an action ρ~: G × V ! V such that, for each g 2 G, the map ρ(g): V ! V defined by ρ(g)(v) =ρ ~(g; v) is linear.
    [Show full text]
  • Effective Sato-Tate Conjecture for Abelian Varieties and Applications
    EFFECTIVE SATO–TATE CONJECTURE FOR ABELIAN VARIETIES AND APPLICATIONS ALINA BUCUR, FRANCESC FITE,´ AND KIRAN S. KEDLAYA Abstract. From the generalized Riemann hypothesis for motivic L-functions, we derive an effective version of the Sato–Tate conjecture for an abelian variety A defined over a number field k with connected Sato–Tate group. By effective we mean that we give an upper bound on the error term in the count predicted by the Sato–Tate measure that only depends on certain invariants of A. We discuss three applications of this conditional result. First, for an abelian variety defined over k, we consider a variant of Linnik’s problem for abelian varieties that asks for an upper bound on the least norm of a prime whose normalized Frobenius trace lies in a given interval. Second, for an elliptic curve defined over k with complex multiplication, we determine (up to multiplication by a nonzero constant) the asymptotic number of primes whose Frobenius trace attain the integral part of the Hasse–Weil bound. Third, for a pair of abelian varieties A and A′ defined over k with no common factors up to k-isogeny, we find an upper bound on the least norm of a prime at which the respective Frobenius traces of A and A′ have opposite sign. Contents 1. Introduction 1 Notations and terminology. 4 Acknowledgements 4 2. Conjectural framework 5 2.1. Sato–Tate groups 5 2.2. Motivic L-functions 5 3. Effective Sato–Tate Conjecture 7 3.1. Lie group theory background 8 3.2. A multivariate Vinogradov function 9 3.3.
    [Show full text]
  • 22 Representations of SL2
    22 Representations of SL2 Finite dimensional complex representations of the following are much the same: SL2(R), sl2R, sl2C (as a complex Lie algebra), su2R, and SU2. This is because finite dimensional representations of a simply connected Lie group are in bi- jection with representations of the Lie algebra. Complex representations of a REAL Lie algebra L correspond to complex representations of its complexifica- tion L ⊗ C considered as a COMPLEX Lie algebra. Note that complex representations of a COMPLEX Lie algebra L⊗C are not the same as complex representations of the REAL Lie algebra L⊗C =∼ L+L. The representations of the real Lie algebra correspond roughly to (reps of L)⊗(reps of L). Strictly speaking, SL2(R) is not simply connected, which is not important for finite dimensional representations. 2 Set Ω = 2EF + 2F E + H ∈ U(sl2R). The main point is that Ω commutes with sl2R. You can check this by brute force: [H, Ω] = 2 ([H, E]F + E[H, F ]) + ··· | {z } 0 [E, Ω] = 2[E, E]F + 2E[F, E] + 2[E, F ]E + 2F [E, E] + [E, H]H + H[E, H] = 0 [F, Ω] = Similar Thus, Ω is in the center of U(sl2R). In fact, it generates the center. This does not really explain where Ω comes from. Why does Ω exist? The answer is that it comes from a symmetric invariant bilinear form on the Lie algebra sl2R given by (E, F ) = 1, (E, E) = (F, F ) = (F, H) = (E, H) = 0, (H, H) = 2. This bilinear form is an invariant map L ⊗ L → C, where L = sl2R, which by duality gives an invariant element in L ⊗ L, which turns out to be 2E ⊗ F + 2F ⊗ E + H ⊗ H.
    [Show full text]
  • 2018-06-108.Pdf
    NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY Feature S E European Tensor Product and Semi-Stability M M Mathematical Interviews E S Society Peter Sarnak Gigliola Staffilani June 2018 Obituary Robert A. Minlos Issue 108 ISSN 1027-488X Prague, venue of the EMS Council Meeting, 23–24 June 2018 New books published by the Individual members of the EMS, member S societies or societies with a reciprocity agree- E European ment (such as the American, Australian and M M Mathematical Canadian Mathematical Societies) are entitled to a discount of 20% on any book purchases, if E S Society ordered directly at the EMS Publishing House. Bogdan Nica (McGill University, Montreal, Canada) A Brief Introduction to Spectral Graph Theory (EMS Textbooks in Mathematics) ISBN 978-3-03719-188-0. 2018. 168 pages. Hardcover. 16.5 x 23.5 cm. 38.00 Euro Spectral graph theory starts by associating matrices to graphs – notably, the adjacency matrix and the Laplacian matrix. The general theme is then, firstly, to compute or estimate the eigenvalues of such matrices, and secondly, to relate the eigenvalues to structural properties of graphs. As it turns out, the spectral perspective is a powerful tool. Some of its loveliest applications concern facts that are, in principle, purely graph theoretic or combinatorial. This text is an introduction to spectral graph theory, but it could also be seen as an invitation to algebraic graph theory. The first half is devoted to graphs, finite fields, and how they come together. This part provides an appealing motivation and context of the second, spectral, half. The text is enriched by many exercises and their solutions.
    [Show full text]
  • Math 210C. Representations of Sl2 1. Introduction in This Handout, We
    Math 210C. Representations of sl2 1. Introduction In this handout, we work out the finite-dimensional k-linear representation theory of sl2(k) for any field k of characteristic 0. (There are also infinite-dimensional irreducible k-linear representations, but here we focus on the finite-dimensional case.) This is an introduction to ideas that are relevant in the general classification of finite-dimensional representations of \semisimple" Lie algebras over fields of characteristic 0, and is a crucial technical tool for our later work on the structure of general connected compact Lie groups (especially to explain the ubiquitous role of SU(2) in the general structure theory). When k is fixed during a discussion, we write sl2 to denote the Lie algebra sl2(k) of traceless 2×2 matrices over k (equipped with its usual Lie algebra structure via the commutator inside − + the associative k-algebra Mat2(k)). Recall the standard k-basis fX ; H; X g of sl2 given by 0 0 1 0 0 1 X− = ;H = ;X+ = ; 1 0 0 −1 0 0 satisfying the commutation relations [H; X±] = ±2X±; [X+;X−] = H: For an sl2-module V over k (i.e., k-vector space V equipped with a map of Lie algebras sl2 ! Endk(V )), we say it is irreducible if V 6= 0 and there is no nonzero proper sl2- submodule. We say that V is absolutely irreducible (over k) if for any extension field K=k the scalar extension VK is irreducible as a K-linear representation of the Lie algebra sl2(K) = K ⊗k sl2 over K.
    [Show full text]
  • Unité, Généralité Et Méthode Axiomatique Dans L’Article De Weyl De 1925-1926 Sur Les Groupes De Lie
    introduction first part second part third part / conclusion Unité, généralité et méthode axiomatique dans l’article de Weyl de 1925-1926 sur les groupes de Lie Christophe Eckes Université Paris VII 12 mars introduction first part second part third part / conclusion introduction - We should not reduce Weyl’s paper to this result. - how to describe the interplay between different mathematical domains (algebra, analysis situs) and different methods (Cartan’s, Hurwitz’s and Frobenius’s methods) ? - which relation can we find between the study of special cases (SL(n,C), O(n), Sp(2n,C)) and the general theory of semi-simple complex Lie groups ? introduction first part second part third part / conclusion General remarks Weyl’s article on the representation of semisimple groups is well known, because of one result in particular: The complete reducibility theorem for semisimple complex Lie groups. - how to describe the interplay between different mathematical domains (algebra, analysis situs) and different methods (Cartan’s, Hurwitz’s and Frobenius’s methods) ? - which relation can we find between the study of special cases (SL(n,C), O(n), Sp(2n,C)) and the general theory of semi-simple complex Lie groups ? introduction first part second part third part / conclusion General remarks Weyl’s article on the representation of semisimple groups is well known, because of one result in particular: The complete reducibility theorem for semisimple complex Lie groups. - We should not reduce Weyl’s paper to this result. introduction first part second part third part / conclusion General remarks Weyl’s article on the representation of semisimple groups is well known, because of one result in particular: The complete reducibility theorem for semisimple complex Lie groups.
    [Show full text]
  • Arxiv:1503.03840V1 [Math.SG] 12 Mar 2015 T21-82-0-1Fdradb H Uoensinefo Science European the by CAST
    A NOTE ON SYMPLECTIC AND POISSON LINEARIZATION OF SEMISIMPLE LIE ALGEBRA ACTIONS EVA MIRANDA Abstract. In this note we prove that an analytic symplectic action of a semisimple Lie algebra can be locally linearized in Darboux coordinates. This result yields simultaneous analytic linearization for Hamiltonian vector fields in a neighbourhood of a common zero. We also provide an example of smooth non-linearizable Hamiltonian action with semisim- ple linear part. The smooth analogue only holds if the semisimple Lie algebra is of compact type. An analytic equivariant b-Darboux theorem for b-Poisson manifolds and an analytic equivariant Weinstein splitting theorem for general Poisson manifolds are also obtained in the Poisson setting. 1. Introduction A classical result due to Bochner [1] establishes that a compact Lie group action on a smooth manifold is locally equivalent, in the neighbourhood of a fixed point, to the linear action. This result holds in the k category. It is interesting to comprehend to which extent we can expect a simC ilar behaviour for non-compact groups. As observed in [9] if the Lie group is connected the linearization problem can be formulated in the following terms: find a linear system of coordinates for the vector fields corresponding to the one parameter subgroups of G or, more generally, consider the representation of a Lie algebra and find coordinates on the manifold that simultaneously linearize the vector fields arXiv:1503.03840v1 [math.SG] 12 Mar 2015 in the image of the representation that vanish at a point. This is the point of view that we adopt in this note when we refer to linearization.
    [Show full text]
  • Notices of the American Mathematical Society ABCD Springer.Com
    ISSN 0002-9920 Notices of the American Mathematical Society ABCD springer.com Highlights in Springer’s eBook Collection of the American Mathematical Society September 2009 Volume 56, Number 8 Bôcher, Osgood, ND and the Ascendance NEW NEW 2 EDITION EDITION of American Mathematics forms bridges between From the reviews of the first edition The theory of elliptic curves is Mathematics at knowledge, tradition, and contemporary 7 Chorin and Hald provide excellent distinguished by the diversity of the Harvard life. The continuous development and explanations with considerable insight methods used in its study. This book growth of its many branches permeates and deep mathematical understanding. treats the arithmetic theory of elliptic page 916 all aspects of applied science and 7 SIAM Review curves in its modern formulation, technology, and so has a vital impact on through the use of basic algebraic our society. The book will focus on these 2nd ed. 2009. X, 162 p. 7 illus. number theory and algebraic geometry. aspects and will benefit from the (Surveys and Tutorials in the Applied contribution of world-famous scientists. Mathematical Sciences) Softcover 2nd ed. 2009. XVIII, 514 p. 14 illus. Invariant Theory ISBN 978-1-4419-1001-1 (Graduate Texts in Mathematics, 2009. XI, 263 p. (Modeling, Simulation & 7 Approx. $39.95 Volume 106) Hardcover of Tensor Product Applications, Volume 3) Hardcover ISBN 978-0-387-09493-9 7 $59.95 Decompositions of ISBN 978-88-470-1121-2 7 $59.95 U(N) and Generalized Casimir Operators For access check with your librarian page 931 Stochastic Partial Linear Optimization Mathematica in Action Riverside Meeting Differential Equations The Simplex Workbook The Power of Visualization A Modeling, White Noise Approach G.
    [Show full text]