DOCSLIB.ORG

On the Complexification of the Classical Geometries And

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On the Complexification of the Classical Geometries And On the complexication of the classical geometries and exceptional numb ers April Intro duction The classical groups On R Spn R and GLn C app ear as the isometry groups of sp ecial geome tries on a real vector space In fact the orthogonal group On R represents the linear isomorphisms of arealvector space V of dimension nleaving invariant a p ositive denite and symmetric bilinear form g on V The symplectic group Spn R represents the isometry group of a real vector space of dimension n leaving invariant a nondegenerate skewsymmetric bilinear form on V Finally GLn C represents the linear isomorphisms of a real vector space V of dimension nleaving invariant a complex structure on V ie an endomorphism J V V satisfying J These three geometries On R Spn R and GLn C in GLn Rintersect even pairwise in the unitary group Un C Considering now the relativeversions of these geometries on a real manifold of dimension n leads to the notions of a Riemannian manifold an almostsymplectic manifold and an almostcomplex manifold A symplectic manifold however is an almostsymplectic manifold X ie X is a manifold and is a nondegenerate form on X so that the form is closed Similarly an almostcomplex manifold X J is called complex if the torsion tensor N J vanishes These three geometries intersect in the notion of a Kahler manifold In view of the imp ortance of the complexication of the real Lie groupsfor instance in the structure theory and representation theory of semisimple real Lie groupswe consider here the question on the underlying geometrical structures on a complex vector space of dimension n giving rise to isometry groups which are the Lie theoretic complexications of the classical groups While the pro cedure is quite clear for c c the orthogonal group On COn R and the symplectic group Spn CSpn R it turns out that the socalled exceptional numbers E cf Wildb erger are quite useful to describ e the complexications of GLn CGLn R and accordingly of Un C In fact one nds that c GLn C GLn C GLn C GLn E and c Un E Un C GLn C After clarifying this task of so to say elementary linear algebra over the exceptional numbers we come to the relative notions of these groups over a given complex manifold The corresp onding notions are called a euclidean symplectic or exceptional structure on a complex manifold of dimension n They intersect in the notion of a Kahler manifold The question of existence of such structures are briey discussed The main p ointhowever is the nal example Consider a semisimple complex Lie algebra L and let a L be a semisimple element Denote further by G IntLGLL the adjoint group of L It is shown that the adjoint orbit GaL carries the structure of a Kahler manifold Elementary theory of exceptional vector spaces Euclidean structures Consider a complex vector space V of dimension n A euclidean structure on V is a symmetric and nondegenerate bilinear form g V V C Prop osition If g is a euclidean structure on V there exists an orthonormal basis v v on V n ie g v v Pro of Since g we can cho ose a vector v V so that g v v By rescaling v observe that z z is surjectiveonCwemay assume that g v v The orthogonal complement W v fw V j g v wg is a euclidean subspace of V ie the restriction of g to W W remains nondegenerate In fact if w W and supp ose that W w then since v w one nds that V w implying w since g is nondegenerate on V Nowby induction on the dimension the result follows Remark Recall that the rescaling argument do es not apply in the real ie in the classical case In fact in this case a nondegenerate bilinear form has an additional invariant its index that is the dimension of a maximal subspace of V on which g is p ositive denite n Let V C and dene the standard euclidean structure by n X g z w z w n Then any euclidean vector space is equivalenttoC g In particular the complex orthogonal group t On CfA GLn C j A A g isup to isomorphythe isometry group of such a structure Exceptional structures Consider a complex vector space V of nite dimension and an endo morphism J V V satisfying J Since the minimal p olynomial of J divides X X iX i J is necessarily semisimple ie V splits into the ieigenspaces V EigJi EigJ i WecallJ an exceptional structure on V ifthemultiplicities of the i and ieigenspace of J are equal dim EigJi dim EigJ i Prop osition Let J be an exceptional structure on a nite dimensional complex vector space V w w of V such Then V is even dimensional say of dimension n and there exists a basis v v n n that the matrix of J with resp ect to this basis is n I n i n Pro of Let v v w w be a basis such that the corresp onding matrix of J is n n i n Then the basis v v w w v w satises the assertion Recall the basic construction of the exceptional numb ers see Wildb erger Here E C C together with its vectorspace structure ov er C and its multiplicative structure resulting from j where j E The null numb ers were dened as the set of noninvertible elements E ie ze with z C or ze with z C Here e ij e ij The Calgebra E is isomorphic to the direct pro duct of Calgebras of C with itself via C E e e since e e e e and e e asoneveries immediately Observe furthermore that multiplication by j E gives E its canonical exceptional structure since je ie je ie n Nowwe can identify our complex vector space of dimension nwiththe Emo dul E in the obvious way Set for z w C and v V z jw v zv wJ v n Then it follows immediately that V E Remark Recall that in the real case a complex structure on a real vector space is dened only byan endomorphism J V V with J satisfying no additional prop erties on its eigenvalue multiplicities c c c c c Denoting by V the complexication of V ie V V C and by J V V the complexication R c c of J ie its unique complex linear extension from V to V then the eigenvalue multiplicities of J are automatically equal In fact the complex conjugation conj z z on C gives rise to an Rlinear map c c c c bar conj V V which is an isomorphism Since bar EigJ i EigJ i the multiplicities of i and i coincide n n n Let us briey lo ok at the isometries of C concerning the standard exceptional structure J C C given by the matrix I n with resp ect to the canonical basis of C GLn EfA GLn C j AI IAg n Lo oking at A GLn EasanElinear endomorphism F of E and representing it with resp ect to the null i shows that the matrix of F has to commute with ie it has b o ckform basis e e ineach factor E i A with A A GLn C This shows that A GLn E GLn C GLn C Symplectic structures Consider a complex vector space V of nite dimension A symplectic structure on V is a nondegenerate and skewsymmetric bilinear form V V C Prop osition Let b e a symplectic structure on V Then V is necessarily even dimensionalsay v v w w and of dimension nand there exists a symplectic basis v w of V ie n v w Pro of Since there exist vectors v w V so that v w One can assume that v w Let H spanv w Now considering the symplectic complement H fx V j x v x w g we see that V H H is direct sum decomp osition of symplectic vector spaces In fact H is obviously a symplectic subspace ie jH H is nondegenerate implying H H and therefore V H H as vector spaces for dimension reasons and nally H is also symplectic Indeed if x H and H x then since H x we nd that V x implying x Nowby induction the prop osition follows n The standard symplectic structure on C is given by x y x y g x y g x y n where g is the standard euclidean structure on C The complex symmetric group is t Sp n CfA GLn C j A IA I g and the prop osition shows that the isometry group of a complex symplectic vector space of dimension n is isomorphic to Spn C The unitary group over the exceptional numb ers Nowwewant to compute the pairwise intersection of the complex groups On C GL n E and Spn C in GLn C To that end consider n E E the complex linear conjugation z jw z jw on E Observe the free Emo dul E Denote by n n thate e ande e The canonical hermitean form h i E E E is dened by n X h i Decomp osition in complex and ctitious part yields the equation h i g j n n where g and are the standard euclidean and symplectic structures on E C A hermitean form h n is by denition a sesquilinear form on V E ie a Cbilinear form satisfying hv whv w and hw v hv w for all E and v w V According to the standard Cbasis e e je je of n n n n C E the matrices of g J and are resp ectively On the other hand the matrices according to the null basis e e e e n n is i i i i n n Dene nowbyUn E the complexlinear transformations of C E resp ecting the standard hermitean n form on E ie t Un EfA GLn E j A A g In fact such a transformation is necessary Elinear ie the matrix A commutes with I Since t t t A On C wehave A A and since A Sp n C wend A IA I ie IA A I AI We havenow seen that Un ESp n C GLn E On C On the other hand the equations h i g j g J showthatSpn C GLn E On C GLn ESpn C On CUn E The three complex n n E geometries on C intersect in the exceptional unitary group U A B To identify Un E let us nally recall what it means that a matrix X in GLn C is C D t symplectic ie X IX X t t i A C C A t t ii B D D B t t iii A D C B n n Representing the linear transformation X E E with resp ect to the null basis yields
Load more

Recommended publications
  • Math 651 Homework 1 - Algebras and Groups Due 2/22/2013
    Math 651 Homework 1 - Algebras and Groups Due 2/22/2013 1) Consider the Lie Group SU(2), the group of 2 × 2 complex matrices A T with A A = I and det(A) = 1. The underlying set is z −w jzj2 + jwj2 = 1 (1) w z with the standard S3 topology. The usual basis for su(2) is 0 i 0 −1 i 0 X = Y = Z = (2) i 0 1 0 0 −i (which are each i times the Pauli matrices: X = iσx, etc.). a) Show this is the algebra of purely imaginary quaternions, under the commutator bracket. b) Extend X to a left-invariant field and Y to a right-invariant field, and show by computation that the Lie bracket between them is zero. c) Extending X, Y , Z to global left-invariant vector fields, give SU(2) the metric g(X; X) = g(Y; Y ) = g(Z; Z) = 1 and all other inner products zero. Show this is a bi-invariant metric. d) Pick > 0 and set g(X; X) = 2, leaving g(Y; Y ) = g(Z; Z) = 1. Show this is left-invariant but not bi-invariant. p 2) The realification of an n × n complex matrix A + −1B is its assignment it to the 2n × 2n matrix A −B (3) BA Any n × n quaternionic matrix can be written A + Bk where A and B are complex matrices. Its complexification is the 2n × 2n complex matrix A −B (4) B A a) Show that the realification of complex matrices and complexifica- tion of quaternionic matrices are algebra homomorphisms.
    [Show full text]
  • Symplectic Linear Algebra Let V Be a Real Vector Space. Definition 1.1. A
    INTRODUCTION TO SYMPLECTIC TOPOLOGY C. VITERBO 1. Lecture 1: Symplectic linear algebra Let V be a real vector space. Definition 1.1. A symplectic form on V is a skew-symmetric bilinear nondegen- erate form: (1) ω(x, y)= ω(y, x) (= ω(x, x) = 0); (2) x, y such− that ω(x,⇒ y) = 0. ∀ ∃ % For a general 2-form ω on a vector space, V , we denote ker(ω) the subspace given by ker(ω)= v V w Vω(v, w)=0 { ∈ |∀ ∈ } The second condition implies that ker(ω) reduces to zero, so when ω is symplectic, there are no “preferred directions” in V . There are special types of subspaces in symplectic manifolds. For a vector sub- space F , we denote by F ω = v V w F,ω(v, w) = 0 { ∈ |∀ ∈ } From Grassmann’s formula it follows that dim(F ω)=codim(F ) = dim(V ) dim(F ) Also we have − Proposition 1.2. (F ω)ω = F (F + F )ω = F ω F ω 1 2 1 ∩ 2 Definition 1.3. A subspace F of V,ω is ω isotropic if F F ( ω F = 0); • coisotropic if F⊂ω F⇐⇒ | • Lagrangian if it is⊂ maximal isotropic. • Proposition 1.4. (1) Any symplectic vector space has even dimension (2) Any isotropic subspace is contained in a Lagrangian subspace and Lagrangians have dimension equal to half the dimension of the total space. (3) If (V1,ω1), (V2,ω2) are symplectic vector spaces with L1,L2 Lagrangian subspaces, and if dim(V1) = dim(V2), then there is a linear isomorphism ϕ : V V such that ϕ∗ω = ω and ϕ(L )=L .
    [Show full text]
  • CLIFFORD ALGEBRAS and THEIR REPRESENTATIONS Introduction
    CLIFFORD ALGEBRAS AND THEIR REPRESENTATIONS Andrzej Trautman, Uniwersytet Warszawski, Warszawa, Poland Article accepted for publication in the Encyclopedia of Mathematical Physics c 2005 Elsevier Science Ltd ! Introduction Introductory and historical remarks Clifford (1878) introduced his ‘geometric algebras’ as a generalization of Grassmann alge- bras, complex numbers and quaternions. Lipschitz (1886) was the first to define groups constructed from ‘Clifford numbers’ and use them to represent rotations in a Euclidean ´ space. E. Cartan discovered representations of the Lie algebras son(C) and son(R), n > 2, that do not lift to representations of the orthogonal groups. In physics, Clifford algebras and spinors appear for the first time in Pauli’s nonrelativistic theory of the ‘magnetic elec- tron’. Dirac (1928), in his work on the relativistic wave equation of the electron, introduced matrices that provide a representation of the Clifford algebra of Minkowski space. Brauer and Weyl (1935) connected the Clifford and Dirac ideas with Cartan’s spinorial represen- tations of Lie algebras; they found, in any number of dimensions, the spinorial, projective representations of the orthogonal groups. Clifford algebras and spinors are implicit in Euclid’s solution of the Pythagorean equation x2 y2 + z2 = 0 which is equivalent to − y x z p = 2 p q (1) −z y + x q ! " ! " # $ so that x = q2 p2, y = p2 + q2, z = 2pq. If the numbers appearing in (1) are real, then − this equation can be interpreted as providing a representation of a vector (x, y, z) R3, null ∈ with respect to a quadratic form of signature (1, 2), as the ‘square’ of a spinor (p, q) R2.
    [Show full text]
  • Patterns of Maximally Entangled States Within the Algebra of Biquaternions
    J. Phys. Commun. 4 (2020) 055018 https://doi.org/10.1088/2399-6528/ab9506 PAPER Patterns of maximally entangled states within the algebra of OPEN ACCESS biquaternions RECEIVED 21 March 2020 Lidia Obojska REVISED 16 May 2020 Siedlce University of Natural Sciences and Humanities, ul. 3 Maja 54, 08-110 Siedlce, Poland ACCEPTED FOR PUBLICATION E-mail: [email protected] 20 May 2020 Keywords: bipartite entanglement, biquaternions, density matrix, mereology PUBLISHED 28 May 2020 Original content from this Abstract work may be used under This paper proposes a description of maximally entangled bipartite states within the algebra of the terms of the Creative Commons Attribution 4.0 biquaternions–Ä . We assume that a bipartite entanglement is created in a process of splitting licence. one particle into a pair of two indiscernible, yet not identical particles. As a result, we describe an Any further distribution of this work must maintain entangled correlation by the use of a division relation and obtain twelve forms of biquaternions, attribution to the author(s) and the title of representing pure maximally entangled states. Additionally, we obtain other patterns, describing the work, journal citation mixed entangled states. Finally, we show that there are no other maximally entangled states in Ä and DOI. than those presented in this work. 1. Introduction In 1935, Einstein, Podolski and Rosen described a strange phenomenon, in which in its pure state, one particle behaves like a pair of two indistinguishable, yet not identical, particles [1]. This phenomenon, called by Einstein ’a spooky action at a distance’, has been investigated by many authors, who tried to explain this strange correlation [2–9].
    [Show full text]
  • Majorana Spinors
    MAJORANA SPINORS JOSE´ FIGUEROA-O'FARRILL Contents 1. Complex, real and quaternionic representations 2 2. Some basis-dependent formulae 5 3. Clifford algebras and their spinors 6 4. Complex Clifford algebras and the Majorana condition 10 5. Examples 13 One dimension 13 Two dimensions 13 Three dimensions 14 Four dimensions 14 Six dimensions 15 Ten dimensions 16 Eleven dimensions 16 Twelve dimensions 16 ...and back! 16 Summary 19 References 19 These notes arose as an attempt to conceptualise the `symplectic Majorana{Weyl condition' in 5+1 dimensions; but have turned into a general discussion of spinors. Spinors play a crucial role in supersymmetry. Part of their versatility is that they come in many guises: `Dirac', `Majorana', `Weyl', `Majorana{Weyl', `symplectic Majorana', `symplectic Majorana{Weyl', and their `pseudo' counterparts. The tra- ditional physics approach to this topic is a mixed bag of tricks using disparate aspects of representation theory of finite groups. In these notes we will attempt to provide a uniform treatment based on the classification of Clifford algebras, a work dating back to the early 60s and all but ignored by the theoretical physics com- munity. Recent developments in superstring theory have made us re-examine the conditions for the existence of different kinds of spinors in spacetimes of arbitrary signature, and we believe that a discussion of this more uniform approach is timely and could be useful to the student meeting this topic for the first time or to the practitioner who has difficulty remembering the answer to questions like \when do symplectic Majorana{Weyl spinors exist?" The notes are organised as follows.
    [Show full text]
  • Maxwell's Equations
    Maxwell’s equations Daniel Henry Gottlieb August 1, 2004 Abstract We express Maxwell’s equations as a single equation, first using the divergence of a special type of matrix field to obtain the four current, and then the divergence of a special matrix to obtain the Electromagnetic field. These two equations give rise to a remarkable dual set of equations in which the operators become the matrices and the vectors become the fields. The decoupling of the equations into the wave equation is very simple and natural. The divergence of the stress energy tensor gives the Lorentz Law in a very natural way. We compare this approach to the related descriptions of Maxwell’s equations by biquaternions and Clifford algebras. 1 Introduction Maxwell’s equations have been expressed in many forms in the century and a half since their dis- covery. The original equations were 16 in number. The vector forms, written below, consist of 4 equations. The differential form versions consists of two equations; see [Misner, Thorne and Wheeler(1973); see equations 4.10, 4.11]. See also [Steven Parrott(1987) page 98 -100 ] The ap- plication of quaternions, and their complexification, the biquaternions, results in a version of one equation. William E. Baylis (1999) equation 3.8, is an example. In this work, we obtain one Maxwell equation, (10), representing the electromagnetic field as a matrix and the divergence as a vector multiplying the field matrix. But we also obtain a remarkable dual formulation of Maxwell’s equation, (15), wherein the operator is now the matrix and the field is now the vector.
    [Show full text]
  • 5 Spinor Calculus
    5 Spinor Calculus 5.1 From triads and Euler angles to spinors. A heuristic introduction. As mentioned already in Section 3.4.3, it is an obvious idea to enrich the Pauli algebra formalism by introducing the complex vector space V(2; C) on which the matrices operate. The two-component complex vectors are traditionally called spinors28. We wish to show that they give rise to a wide range of applications. In fact we shall introduce the spinor concept as a natural answer to a problem that arises in the context of rotational motion. In Section 3 we have considered rotations as operations performed on a vector space. Whereas this approach enabled us to give a group-theoretical definition of the magnetic field, a vector is not an appropriate construct to account for the rotation of an orientable object. The simplest mathematical model suitable for this purpose is a Cartesian (orthogonal) three-frame, briefly, a triad. The problem is to consider two triads with coinciding origins, and the rotation of the object frame is described with respect to the space frame. The triads are represented in terms of their respective unit vectors: the space frame as Σs(x^1; x^2; x^3) and the object frame as Σc(e^1; e^2; e^3). Here c stands for “corpus,” since o for “object” seems ambiguous. We choose the frames to be right-handed. These orientable objects are not pointlike, and their parametrization offers novel problems. In this sense we may refer to triads as “higher objects,” by contrast to points which are “lower objects.” The thought that comes most easily to mind is to consider the nine direction cosines e^i · x^k but this is impractical, because of the six relations connecting these parameters.
    [Show full text]
  • Symmetries of the Space of Linear Symplectic Connections
    Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 002, 30 pages Symmetries of the Space of Linear Symplectic Connections Daniel J.F. FOX Departamento de Matem´aticas del Area´ Industrial, Escuela T´ecnica Superior de Ingenier´ıa y Dise~noIndustrial, Universidad Polit´ecnica de Madrid, Ronda de Valencia 3, 28012 Madrid, Spain E-mail: [email protected] Received June 30, 2016, in final form January 07, 2017; Published online January 10, 2017 https://doi.org/10.3842/SIGMA.2017.002 Abstract. There is constructed a family of Lie algebras that act in a Hamiltonian way on the symplectic affine space of linear symplectic connections on a symplectic manifold. The associated equivariant moment map is a formal sum of the Cahen{Gutt moment map, the Ricci tensor, and a translational term. The critical points of a functional constructed from it interpolate between the equations for preferred symplectic connections and the equations for critical symplectic connections. The commutative algebra of formal sums of symmetric tensors on a symplectic manifold carries a pair of compatible Poisson structures, one induced from the canonical Poisson bracket on the space of functions on the cotangent bundle poly- nomial in the fibers, and the other induced from the algebraic fiberwise Schouten bracket on the symmetric algebra of each fiber of the cotangent bundle. These structures are shown to be compatible, and the required Lie algebras are constructed as central extensions of their linear combinations restricted to formal sums of symmetric tensors whose first order term is a multiple of the differential of its zeroth order term.
    [Show full text]
  • Classification of Left Octonion Modules
    Classification of left octonion modules Qinghai Huo, Yong Li, Guangbin Ren Abstract It is natural to study octonion Hilbert spaces as the recently swift development of the theory of quaternion Hilbert spaces. In order to do this, it is important to study first its algebraic structure, namely, octonion modules. In this article, we provide complete classification of left octonion modules. In contrast to the quaternionic setting, we encounter some new phenomena. That is, a submodule generated by one element m may be the whole module and may be not in the form Om. This motivates us to introduce some new notions such as associative elements, conjugate associative elements, cyclic elements. We can characterize octonion modules in terms of these notions. It turns out that octonions admit two distinct structures of octonion modules, and moreover, the direct sum of their several copies exhaust all octonion modules with finite dimensions. Keywords: Octonion module; associative element; cyclic element; Cℓ7-module. AMS Subject Classifications: 17A05 Contents 1 introduction 1 2 Preliminaries 3 2.1 The algebra of the octonions O .............................. 3 2.2 Universal Clifford algebra . ... 4 3 O-modules 6 4 The structure of left O-moudles 10 4.1 Finite dimensional O-modules............................... 10 4.2 Structure of general left O-modules............................ 13 4.3 Cyclic elements in left O-module ............................. 15 arXiv:1911.08282v2 [math.RA] 21 Nov 2019 1 introduction The theory of quaternion Hilbert spaces brings the classical theory of functional analysis into the non-commutative realm (see [10, 16, 17, 20, 21]). It arises some new notions such as spherical spectrum, which has potential applications in quantum mechanics (see [4, 6]).
    [Show full text]
  • Unifying the Hyperbolic and Spherical 2-Body Problem with Biquaternions
    Unifying the Hyperbolic and Spherical 2-Body Problem with Biquaternions Philip Arathoon December 2020 Abstract The 2-body problem on the sphere and hyperbolic space are both real forms of holo- morphic Hamiltonian systems defined on the complex sphere. This admits a natural description in terms of biquaternions and allows us to address questions concerning the hyperbolic system by complexifying it and treating it as the complexification of a spherical system. In this way, results for the 2-body problem on the sphere are readily translated to the hyperbolic case. For instance, we implement this idea to completely classify the relative equilibria for the 2-body problem on hyperbolic 3-space for a strictly attractive potential. Background and outline The case of the 2-body problem on the 3-sphere has recently been considered by the author in [1]. This treatment takes advantage of the fact that S3 is a group and that the action of SO(4) on S3 is generated by the left and right multiplication of S3 on itself. This allows for a reduction in stages, first reducing by the left multiplication, and then reducing an intermediate space by the residual right-action. An advantage of this reduction-by-stages is that it allows for a fairly straightforward derivation of the relative equilibria solutions: the relative equilibria may first be classified in the intermediate reduced space and then reconstructed on the original phase space. For the 2-body problem on hyperbolic space the same idea does not apply. Hyperbolic 3-space H3 cannot be endowed with an isometric group structure and the symmetry group SO(1; 3) does not arise as a direct product of two groups.
    [Show full text]
  • Quaternion Algebraic Geometry
    Quaternion Algebraic Geometry Dominic Widdows June 5, 2006 Abstract QUATERNION ALGEBRAIC GEOMETRY DOMINIC WIDDOWS St Anne’s College, Oxford Thesis submitted Hilary Term, 2000, in support of application to supplicate for the degree of D.Phil. This thesis is a collection of results about hypercomplex and quaternionic manifolds, focussing on two main areas. These are exterior forms and double complexes, and the ‘algebraic geometry’ of hypercomplex manifolds. The latter area is strongly influenced by techniques from quaternionic algebra. A new double complex on quaternionic manifolds is presented, a quaternionic version of the Dolbeault complex on a complex manifold. It arises from the decomposition of real-valued exterior forms on a quaternionic manifold M into irreducible representations of Sp(1). This decomposition gives a double complex of differential forms and operators ∼ as a result of the Clebsch-Gordon formula Vr⊗V1 = Vr+1⊕Vr−1 for Sp(1)-representations. The properties of the double complex are investigated, and it is established that it is elliptic in most places. Joyce has created a new theory of quaternionic algebra [J1] by defining a quaternionic tensor product for AH-modules (H-modules equipped with a special real subspace). The 1 theory can be described using sheaves over CP , an interpretation due to Quillen [Q]. AH-modules and their quaternionic tensor products are classified. Stable AH-modules are described using Sp(1)-representations. This theory is especially useful for describing hypercomplex manifolds and forming close analogies with complex geometry. Joyce has defined and investigated q-holomorphic functions on hypercomplex manifolds. There is also a q-holomorphic cotangent space which again arises as a result of the Clebsch-Gordon formula.
    [Show full text]
  • 1 Symplectic Vector Spaces
    Symplectic Manifolds Tam´asHausel 1 Symplectic Vector Spaces Let us first fix a field k. Definition 1.1 (Symplectic form, symplectic vector space). Given a finite dimensional k-vector space W , a symplectic form on W is a bilinear, alternating non degenerate form on W , i.e. an element ! 2 ^2W ∗ such that if !(v; w) = 0 for all w 2 W then v = 0. We call the pair (W; !) a symplectic vector space. Example 1.2 (Standard example). Consider a finite dimensional k-vector space V and define W = V ⊕ V ∗. We can introduce a symplectic form on W : ! : W × W ! k ((v1; α1); (v2; α2)) 7! α1(v2) − α2(v1) Definition 1.3 (Symplectomorphism). Given two symplectic vector spaces (W1;!1) and (W2;!2), we say that a linear map f : W1 ! W2 is a symplectomorphism if it is an isomorphism of vector ∗ spaces and, furthermore, f !2 = !1. We denote by Sp(W ) the group of symplectomorphism W ! W . Let us consider a symplectic vector space (W; !) with even dimension 2n (we will see soon that it is always the case). Then ! ^ · · · ^ ! 2 ^2nW ∗ is a volume form, i.e. a non-degenerate top form. For f 2 Sp(W ) we must have !n = f ∗!n = det f · !n so that det f = 1 and, in particular, Sp(W ) ≤ SL(W ). We also note that, for n = 1, we have Sp(W ) =∼ SL(W ). Definition 1.4. Let U be a linear subspace of a symplectic vector space W . Then we define U ? = fv 2 W j !(v; w) = 0 8u 2 Ug: We say that U is isotropic if U ⊆ U ?, coisotropic if U ? ⊆ U and Lagrangian if U ? = U.
    [Show full text]