DOCSLIB.ORG

The Adjoint Representation of a Reductive Group and Hyperplane Arrangements

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Adjoint Representation of a Reductive Group and Hyperplane Arrangements REPRESENTATION THEORY An Electronic Journal of the American Mathematical Society Volume 3, Pages 444{456 (November 9, 1999) S 1088-4165(99)00066-7 THE ADJOINT REPRESENTATION OF A REDUCTIVE GROUP AND HYPERPLANE ARRANGEMENTS J. MATTHEW DOUGLASS Abstract. Let G be a connected reductive algebraic group with Lie algebra g defined over an algebraically closed field, k,withchark= 0. Fix a parabolic subgroup of G with Levi decomposition P = LU where U is the unipotent radical of P .Letu=Lie(U)andletzdenote the center of Lie(L). Let T be a maximal torus in L with Lie algebra t. Then the root system of (g; t) is a subset of t∗ and by restriction to z,therootsoftin u determine an arrangement of hyperplanes in z we denote by z. In this paper we construct an isomorphism A P z z of graded k[z]-modules HomG(g∗;k[G (z+u)]) ∼= D( ), where D( )isthe z × A PA k[z]-module of derivations of . We also show that HomG(g∗;k[G (z+u)]) P A × z and k[z] HomG(g∗;k[G u]) are isomorphic graded k[z]-modules, so D( ) ⊗ ×P A and k[z] HomG(g∗;k[G u]) are isomorphic, graded k[z]-modules. It follows immediately⊗ that z is a× free hyperplane arrangement. This result has been proved using case-by-caseA arguments by Orlik and Terao. By keeping track of the gradings involved, and recalling that g affords a self-dual representation of G, we recover a result of Sommers, Trapa, and Broer which states that the degrees in which the adjoint representation of G occurs as a constituent of the graded, rational G-module k[G P u] are the exponents of z. This result has also been proved, again using case-by-case× arguments, by SommersA and Trapa and independently by Broer. 1. Introduction Suppose k is an algebraically closed field with characteristic zero and G is a connected, reductive, algebraic group defined over k with Lie algebra g.Let denote the cone of nilpotent elements in g.ThenGacts on g by the adjointN representation and is a closed, G-invariant subvariety, so G acts on k[ ], the ring of regular functionsN on .Since is a cone, k[ ] inherits a gradingN from k[g] and each homogeneous component,N Nk[ ] ,isaG-stableN subspace. Kostant [5] N j proved that g occurs as a constituent of k[ ]j if and only if j is an exponent of the Weyl group of G. Precisely, he proved theN equality of polynomials in q: dim Hom (g,k[ ] )qj =qn1 + +qnl G N j ··· j 0 X≥ where n1,...,nl is the multiset of exponents of W (see [7, 6.2] for a definition of the exponents{ of}W ). § Sommers and Trapa [9] and also Broer [4] have generalized this result as follows. Fix a maximal torus, T , and a parabolic subgroup, P ,ofGwith T P .LetU denote the unipotent radical of P and let L be a Levi subgroup of P⊆ containing Received by the editors March 8, 1999 and, in revised form September 28, 1999. 1991 Mathematics Subject Classification. Primary 22E46. c 1999 American Mathematical Society 444 ADJOINT REPRESENTATION AND ARRANGEMENTS 445 T . We follow the convention that lower case Fraktur letters will denote the Lie algebra of the group denoted by the same upper case Roman letter, so for example t = Lie(T ), and u = Lie(U). Let z denote the center of l,soz t.Thenzacts on u by restricting the adjoint representation of g and the weights⊆ of z on u are the restrictions to z of the weights of t on u. The kernels of these weights determine a hyperplane arrangement in z we denote by (z, z). Orlik and Terao [8] have proven, using case-by-case arguments, that z is a freeA hyperplane arrangement and Orlik and Solomon [6] have computed theA exponents of z in all cases. Now P acts on u and so we can form the associatedA fibre bundle, G P u,over G/P with fibre u.ThenGacts on G P u by left multiplication and× so G acts on k[G P u], the ring of global regular× functions on G P u. Moreover, k[G P u] inherits× a grading from the scalar action of k on u, k[G× P u] k[G ×P u] , ∼= j 0 j and each homogeneous component is a G-stable subspace.× Sommers≥ and× Trapa [9], and independently Broer [4], have proven that g occurs asL a constituent of P z k[G u]j if and only if j is an exponent of . Precisely, they prove the equality of polynomials× in q: A P j nz nz (1.1) dim Hom (g,k[G u] )q =q 1 + +q r G × j ··· j 0 X≥ z z z where n1,...,nr is the multiset of exponents of as defined in [7, Definition 4.25]. { } A In the special case when P is a Borel subgroup of G,thenz=t, the exponents t P of are the exponents of the Weyl group of G,andk[ ]j and k[G u]j afford equivalentA representations of G for j 0, so we recover Kostant’sN result.× The proofs given by Sommers and≥ Trapa and by Broer of the equality (1.1) are both case-by-case arguments, with separate arguments for each type of root system. In this paper, our main result is that there is an isomorphism of graded k[z]- P z z modules Γ: HomG(g∗,k[G (z+u)]) D( ), where D( )isthek[z]-module of derivations of z. We also× show that there→ isA an isomorphismA of graded k[z]-modules A P P z between HomG(g∗,k[G (z+u)]) and k[z] HomG(g∗,k[G u]). Thus D( )and P× ⊗ × A k[z] HomG(g∗,k[G u]) are isomorphic, graded k[z]-modules, so by keeping track of the⊗ gradings involved,× and recalling that g affords a self-dual representation of G, we see that the isomorphism Γ explains the computation, (1.1), of Sommers, Trapa, and Broer. P z Our proof that HomG(g∗,k[G (z+u)]) and D( ) are isomorphic, graded k[z]- modules does not involve any case-by-case× computations.A We define Γ explicitly and then show, using that the set of semisimple elements is dense in g and some results on the P -orbit structure of z + u, that it’s a bijection. P z Since k[z] HomG(g∗,k[G u]) and D( ) are isomorphic k[z]-modules, it fol- lows immediately⊗ that the arrangement× z Ais free. We thus obtain a proof, free of case-by-case considerations, of the resultA of Orlik and Terao [8] that the restrictions of the reflection arrangements arising from Weyl groups are free. After learning of our ideas about the definition of Γ, Broer [3] has given another proof that it’s surjective, the difficult part of showing that it’s an isomorphism. His proof is elegant, using Saito’s criterion, but relies on a previous result of his concerning sums of exponents. Our proof that Γ is surjective is completely different than Broer’s and uses only some basic algebraic geometry and properties of root systems and algebraic groups. 446 J. MATTHEW DOUGLASS Moreover, our technique can be extended to give a new proof of a theorem of Broer’s [2, Theorem 1] that characterizes when the restriction mapping from T MorG(g,V)toMorW(t,V ) is an isomorphism for a rational G-module V .Ina subsequent paper we hope to give a parabolic analog of Broer’s theorem, where g is replaced by AdG(z + u) (the closure of a Dixmier sheet in g), t is replaced by z, and V T is replaced by V L. The rest of this paper is organized as follows. In 2 we collect several results about hyperplane arrangements in the form we need later;§ in 3 we discuss associ- ated fibre bundles over G/P ;in 4 we collect some results about§ the adjoint action of U on z + u;in 5 we construct§ Γ and prove it’s an isomorphism; and in 6we explain how the fact§ that Γ is an isomorphism implies the results discussed above.§ Finally, we use the following notation throughout this paper. If X and Y are varieties defined over k,thenk[X] denotes the k-algebra of global regular functions on X and Mor(X, Y ) denotes the set of all morphisms from X to Y . If a group, G, acts on X and Y ,thenMorG(X, Y ) denotes the set of all G-equivariant morphisms from X to Y .IfVand W are k-vector spaces, then V W and Hom(V,W)denote V W and Hom (V,W) respectively. If V and W ⊗afford representations of G, ⊗k k then HomG(V,W) denotes the set of all G-equivariant linear transformations from V to W . 2. Hyperplane arrangements In this paper a “hyperplane arrangement” is a pair, (V, ), where V is a finite dimensional k-vector space and is a finite multiset of hyperplanesA in V . This slight generalization of the usual meaningA has no effect on the invariants of arrangements we consider. Suppose (V, ) is an arrangement with dim V = n.Let µ 1 j m A {j| ≤ ≤ } be a multiset of linear functionals so that = ker µ1,...,ker µ . Fix a basis, A { m } e1,...,en ,ofV and let x1,...,xn be the dual basis of V ∗.Thenfor1 i m { } n { } ≤ ≤ we may write µj = i=1 ξi;j xi where the ξi;j ’s are in k.Wemayidentifyk[V] with the polynomial ring k[x1,...,xn]andif∂i denotes partial differentiation with P respect to xi,thenDerk[V], the k[V ]-module of k-linear derivations of k[V ], has basis ∂1,...,∂n .
Load more

Recommended publications
  • DIFFERENTIAL GEOMETRY FINAL PROJECT 1. Introduction for This
    DIFFERENTIAL GEOMETRY FINAL PROJECT KOUNDINYA VAJJHA 1. Introduction For this project, we outline the basic structure theory of Lie groups relating them to the concept of Lie algebras. Roughly, a Lie algebra encodes the \infinitesimal” structure of a Lie group, but is simpler, being a vector space rather than a nonlinear manifold. At the local level at least, the Fundamental Theorems of Lie allow one to reconstruct the group from the algebra. 2. The Category of Local (Lie) Groups The correspondence between Lie groups and Lie algebras will be local in nature, the only portion of the Lie group that will be of importance is that portion of the group close to the group identity 1. To formalize this locality, we introduce local groups: Definition 2.1 (Local group). A local topological group is a topological space G, with an identity element 1 2 G, a partially defined but continuous multiplication operation · :Ω ! G for some domain Ω ⊂ G × G, a partially defined but continuous inversion operation ()−1 :Λ ! G with Λ ⊂ G, obeying the following axioms: • Ω is an open neighbourhood of G × f1g Sf1g × G and Λ is an open neighbourhood of 1. • (Local associativity) If it happens that for elements g; h; k 2 G g · (h · k) and (g · h) · k are both well-defined, then they are equal. • (Identity) For all g 2 G, g · 1 = 1 · g = g. • (Local inverse) If g 2 G and g−1 is well-defined in G, then g · g−1 = g−1 · g = 1. A local group is said to be symmetric if Λ = G, that is, every element g 2 G has an inverse in G.A local Lie group is a local group in which the underlying topological space is a smooth manifold and where the associated maps are smooth maps on their domain of definition.
    [Show full text]
  • 10 Group Theory and Standard Model
    Physics 129b Lecture 18 Caltech, 03/05/20 10 Group Theory and Standard Model Group theory played a big role in the development of the Standard model, which explains the origin of all fundamental particles we see in nature. In order to understand how that works, we need to learn about a new Lie group: SU(3). 10.1 SU(3) and more about Lie groups SU(3) is the group of special (det U = 1) unitary (UU y = I) matrices of dimension three. What are the generators of SU(3)? If we want three dimensional matrices X such that U = eiθX is unitary (eigenvalues of absolute value 1), then X need to be Hermitian (real eigenvalue). Moreover, if U has determinant 1, X has to be traceless. Therefore, the generators of SU(3) are the set of traceless Hermitian matrices of dimension 3. Let's count how many independent parameters we need to characterize this set of matrices (what is the dimension of the Lie algebra). 3 × 3 complex matrices contains 18 real parameters. If it is to be Hermitian, then the number of parameters reduces by a half to 9. If we further impose traceless-ness, then the number of parameter reduces to 8. Therefore, the generator of SU(3) forms an 8 dimensional vector space. We can choose a basis for this eight dimensional vector space as 00 1 01 00 −i 01 01 0 01 00 0 11 λ1 = @1 0 0A ; λ2 = @i 0 0A ; λ3 = @0 −1 0A ; λ4 = @0 0 0A (1) 0 0 0 0 0 0 0 0 0 1 0 0 00 0 −i1 00 0 01 00 0 0 1 01 0 0 1 1 λ5 = @0 0 0 A ; λ6 = @0 0 1A ; λ7 = @0 0 −iA ; λ8 = p @0 1 0 A (2) i 0 0 0 1 0 0 i 0 3 0 0 −2 They are called the Gell-Mann matrices.
    [Show full text]
  • Representation Theory
    M392C NOTES: REPRESENTATION THEORY ARUN DEBRAY MAY 14, 2017 These notes were taken in UT Austin's M392C (Representation Theory) class in Spring 2017, taught by Sam Gunningham. I live-TEXed them using vim, so there may be typos; please send questions, comments, complaints, and corrections to [email protected]. Thanks to Kartik Chitturi, Adrian Clough, Tom Gannon, Nathan Guermond, Sam Gunningham, Jay Hathaway, and Surya Raghavendran for correcting a few errors. Contents 1. Lie groups and smooth actions: 1/18/172 2. Representation theory of compact groups: 1/20/174 3. Operations on representations: 1/23/176 4. Complete reducibility: 1/25/178 5. Some examples: 1/27/17 10 6. Matrix coefficients and characters: 1/30/17 12 7. The Peter-Weyl theorem: 2/1/17 13 8. Character tables: 2/3/17 15 9. The character theory of SU(2): 2/6/17 17 10. Representation theory of Lie groups: 2/8/17 19 11. Lie algebras: 2/10/17 20 12. The adjoint representations: 2/13/17 22 13. Representations of Lie algebras: 2/15/17 24 14. The representation theory of sl2(C): 2/17/17 25 15. Solvable and nilpotent Lie algebras: 2/20/17 27 16. Semisimple Lie algebras: 2/22/17 29 17. Invariant bilinear forms on Lie algebras: 2/24/17 31 18. Classical Lie groups and Lie algebras: 2/27/17 32 19. Roots and root spaces: 3/1/17 34 20. Properties of roots: 3/3/17 36 21. Root systems: 3/6/17 37 22. Dynkin diagrams: 3/8/17 39 23.
    [Show full text]
  • Special Unitary Group - Wikipedia
    Special unitary group - Wikipedia https://en.wikipedia.org/wiki/Special_unitary_group Special unitary group In mathematics, the special unitary group of degree n, denoted SU( n), is the Lie group of n×n unitary matrices with determinant 1. (More general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special case.) The group operation is matrix multiplication. The special unitary group is a subgroup of the unitary group U( n), consisting of all n×n unitary matrices. As a compact classical group, U( n) is the group that preserves the standard inner product on Cn.[nb 1] It is itself a subgroup of the general linear group, SU( n) ⊂ U( n) ⊂ GL( n, C). The SU( n) groups find wide application in the Standard Model of particle physics, especially SU(2) in the electroweak interaction and SU(3) in quantum chromodynamics.[1] The simplest case, SU(1) , is the trivial group, having only a single element. The group SU(2) is isomorphic to the group of quaternions of norm 1, and is thus diffeomorphic to the 3-sphere. Since unit quaternions can be used to represent rotations in 3-dimensional space (up to sign), there is a surjective homomorphism from SU(2) to the rotation group SO(3) whose kernel is {+ I, − I}. [nb 2] SU(2) is also identical to one of the symmetry groups of spinors, Spin(3), that enables a spinor presentation of rotations. Contents Properties Lie algebra Fundamental representation Adjoint representation The group SU(2) Diffeomorphism with S 3 Isomorphism with unit quaternions Lie Algebra The group SU(3) Topology Representation theory Lie algebra Lie algebra structure Generalized special unitary group Example Important subgroups See also 1 of 10 2/22/2018, 8:54 PM Special unitary group - Wikipedia https://en.wikipedia.org/wiki/Special_unitary_group Remarks Notes References Properties The special unitary group SU( n) is a real Lie group (though not a complex Lie group).
    [Show full text]
  • Group Theory - QMII 2017
    Group Theory - QMII 2017 Reminder Last time we said that a group element of a matrix lie group can be written as an exponent: a U = eiαaX ; a = 1; :::; N: We called Xa the generators, we have N of them, they span a basis for the Lie algebra, and they can be found by taking the derivative with respect to αa at α = 0. The generators are closed under the Lie product (∼ [·; ·]), and related by the structure constants [Xa;Xb] = ifabcXc: (1) We end with the Jacobi identity [Xa; [Xb;Xc]] + [Xb; [Xc;Xa]] + [Xc; [Xa;Xb]] = 0: (2) 1 Representations of Lie Algebra 1.1 The adjoint rep. part- I One of the most important representations is the adjoint. There are two equivalent ways to defined it. Here we follow the definition by Georgi. By plugging Eq. (1) into the Jacobi identity Eq. (2) we get the following: fbcdfade + fabdfcde + fcadfbde = 0 (3) Proof: 0 = [Xa; [Xb;Xc]] + [Xb; [Xc;Xa]] + [Xc; [Xa;Xb]] = ifbcd [Xa;Xd] + ifcad [Xb;Xd] + ifabd [Xc;Xd] = − (fbcdfade + fcadfbde + fabdfcde) Xe (4) 1 Now we can define a set of matrices Ta s.t. [Ta]bc = −ifabc: (5) Then by using the relation of the structure constants Eq. (3) we get [Ta;Tb] = ifabcTc: (6) Proof: [Ta;Tb]ce = [Ta]cd[Tb]de − [Tb]cd[Ta]de = −facdfbde + fbcdfade = fcadfbde + fbcdfade = −fabdfcde = fabdfdce = ifabd[Td]ce (7) which means that: [Ta;Tb] = ifabdTd: (8) Therefore the structure constants themselves generate a representation. This representation is called the adjoint representation. The adjoint of su(2) We already found that the structure constants of su(2) are given by the Levi-Civita tensor.
    [Show full text]
  • Groups and Representations the Material Here Is Partly in Appendix a and B of the Book
    Groups and representations The material here is partly in Appendix A and B of the book. 1 Introduction The concept of symmetry, and especially gauge symmetry, is central to this course. Now what is a symmetry: you have something, e.g. a vase, and you do something to it, e.g. turn it by 29 degrees, and it still looks the same then we call the operation performed, i.e. the rotation, a symmetryoperation on the object. But if we first rotate it by 29 degrees and then by 13 degrees and it still looks the same then also the combined operation of a rotation by 42 degrees is a symmetry operation as well. The mathematics that is involved here is that of groups and their representations. The symmetry operations form the group and the objects the operations work on are in representations of the group. 2 Groups A group is a set of elements g where there exist an operation ∗ that combines two group elements and the results is a third: ∃∗ : 8g1; g2 2 G : g1 ∗ g2 = g3 2 G (1) This operation must be associative: 8g1; g2; g3 2 G :(g1 ∗ g2) ∗ g3 = g1 ∗ (g2 ∗ g3) (2) and there exists an element unity: 91 2 G : 8g 2 G : g ∗ 1 = 1 ∗ g = g (3) and for every element in G there exists an inverse: 8g 2 G : 9g−1 2 G : g ∗ g−1 = g−1 ∗ g = 1 (4) This is the general definition of a group. Now if all elements of a group commute, i.e.
    [Show full text]
  • Characterization of SU(N)
    University of Rochester Group Theory for Physicists Professor Sarada Rajeev Characterization of SU(N) David Mayrhofer PHY 391 Independent Study Paper December 13th, 2019 1 Introduction At this point in the course, we have discussed SO(N) in detail. We have de- termined the Lie algebra associated with this group, various properties of the various reducible and irreducible representations, and dealt with the specific cases of SO(2) and SO(3). Now, we work to do the same for SU(N). We de- termine how to use tensors to create different representations for SU(N), what difficulties arise when moving from SO(N) to SU(N), and then delve into a few specific examples of useful representations. 2 Review of Orthogonal and Unitary Matrices 2.1 Orthogonal Matrices When initially working with orthogonal matrices, we defined a matrix O as orthogonal by the following relation OT O = 1 (1) This was done to ensure that the length of vectors would be preserved after a transformation. This can be seen by v ! v0 = Ov =) (v0)2 = (v0)T v0 = vT OT Ov = v2 (2) In this scenario, matrices then must transform as A ! A0 = OAOT , as then we will have (Av)2 ! (A0v0)2 = (OAOT Ov)2 = (OAOT Ov)T (OAOT Ov) (3) = vT OT OAT OT OAOT Ov = vT AT Av = (Av)2 Therefore, when moving to unitary matrices, we want to ensure similar condi- tions are met. 2.2 Unitary Matrices When working with quantum systems, we not longer can restrict ourselves to purely real numbers. Quite frequently, it is necessarily to extend the field we are with with to the complex numbers.
    [Show full text]
  • Covariant Dirac Operators on Quantum Groups
    Covariant Dirac Operators on Quantum Groups Antti J. Harju∗ Abstract We give a construction of a Dirac operator on a quantum group based on any simple Lie alge- bra of classical type. The Dirac operator is an element in the vector space Uq (g) ⊗ clq(g) where the second tensor factor is a q-deformation of the classical Clifford algebra. The tensor space Uq (g) ⊗ clq(g) is given a structure of the adjoint module of the quantum group and the Dirac operator is invariant under this action. The purpose of this approach is to construct equivariant Fredholm modules and K-homology cycles. This work generalizes the operator introduced by Bibikov and Kulish in [1]. MSC: 17B37, 17B10 Keywords: Quantum group; Dirac operator; Clifford algebra 1. The Dirac operator on a simple Lie group G is an element in the noncommutative Weyl algebra U(g) cl(g) where U(g) is the enveloping algebra for the Lie algebra of G. The vector space g generates a Clifford⊗ algebra cl(g) whose structure is determined by the Killing form of g. Since g acts on itself by the adjoint action, U(g) cl(g) is a g-module. The Dirac operator on G spans a one dimensional invariant submodule of U(g)⊗ cl(g) which is of the first order in cl(g) and in U(g). Kostant’s Dirac operator [10] has an additional⊗ cubical term in cl(g) which is constant in U(g). The noncommutative Weyl algebra is given an action on a Hilbert space which is also a g-module.
    [Show full text]
  • Chapter 3 Adjoint Representations and the Derivative Of
    Chapter 3 Adjoint Representations and the Derivative of exp 3.1 The Adjoint Representations Ad and ad Given any two vector spaces E and F ,recallthatthe vector space of all linear maps from E to F is denoted by Hom(E,F). The vector space of all invertible linear maps from E to itself is a group denoted GL(E). When E = Rn,weoftendenoteGL(Rn)byGL(n, R) (and if E = Cn,weoftendenoteGL(Cn)byGL(n, C)). The vector space Mn(R)ofalln n matrices is also ⇥ denoted by gl(n, R)(andMn(C)bygl(n, C)). 121 122 CHAPTER 3. ADJOINT REPRESENTATIONS AND THE DERIVATIVE OF exp Then, GL(gl(n, R)) is the vector space of all invertible linear maps from gl(n, R)=Mn(R)toitself. For any matrix A MA(R)(orA MA(C)), define the 2 2 maps LA :Mn(R) Mn(R)andRA :Mn(R) Mn(R) by ! ! LA(B)=AB, RA(B)=BA, for all B Mn(R). 2 Observe that LA RB = RB LA for all A, B Mn(R). ◦ ◦ 2 For any matrix A GL(n, R), let 2 AdA :Mn(R) Mn(R)(conjugationbyA) ! be given by 1 AdA(B)=ABA− for all B Mn(R). 2 Observe that AdA = LA R 1 and that AdA is an ◦ A− invertible linear map with inverse Ad 1. A− 3.1. ADJOINT REPRESENTATIONS Ad AND ad 123 The restriction of AdA to invertible matrices B GL(n, R) yields the map 2 AdA : GL(n, R) GL(n, R) ! also given by 1 AdA(B)=ABA− for all B GL(n, R).
    [Show full text]
  • A. Riemannian Geometry and Lie Groups
    Cheap Orthogonal Constraints in Neural Networks A. Riemannian Geometry and Lie Groups In this section we aim to give a short summary of the basics of classical Riemannian geometry and Lie group theory needed for our proofs in following sections. The standard reference for classical Riemannian geometry is do Carmo’s book (do Carmo, 1992). An elementary introduction to Lie group theory with an emphasis on concrete examples from matrix theory can be found in (Hall, 2015). A.1. Riemannian geometry A Riemannian manifold is a smooth manifold M equipped with a smooth metric h·; ·ip : TpM × TpM! R which is a positive definite inner product for every p 2 M. We will omit the dependency on the point p whenever it is clear from the context. We will work with finite dimensional manifolds. Given a metric, we can define the length of a curve γ on the R b p 0 0 manifold as L(γ) := a hγ (t); γ (t)i dt. The distance between two points is the infimum of the lengths of the piece-wise smooth curves on M connecting them. When the manifold is connected, this defines a distance function that turns the manifold into a metric space. An affine connection r on a smooth manifold is a bilinear application that maps two vector fields X; Y to a new one rX Y such that it is linear in X, and linear and Leibnitz in Y . Connections give a notion of variation of a vector field along another vector field. In particular, the covariant derivative is the restriction of the connection to a curve.
    [Show full text]
  • Loop Quantum Gravity: an Inside View
    Loop Quantum Gravity: An Inside View T. Thiemann∗ MPI f. Gravitationsphysik, Albert-Einstein-Institut, Am M¨uhlenberg 1, 14476 Potsdam, Germany and Perimeter Institute for Theoretical Physics, 31 Caroline Street N, Waterloo, ON N2L 2Y5, Canada Preprint AEI-2006-066 Abstract This is a (relatively) non – technical summary of the status of the quantum dynamics in Loop Quantum Gravity (LQG). We explain in detail the historical evolution of the subject and why the results obtained so far are non – trivial. The present text can be viewed in part as a response to an article by Nicolai, Peeters and Zamaklar. We also explain why certain no go conclusions drawn from a mathematically correct calculation in a recent paper by Helling et al are physically incorrect. arXiv:hep-th/0608210v1 29 Aug 2006 ∗[email protected],[email protected] 1 Contents 1 Introduction 3 2 Classical preliminaries 6 3 Canonical quantisation programme 8 4 Status of the quantisation programme for Loop Quantum Gravity (LQG) 12 4.1 CanonicalquantumgravitybeforeLQG . .............. 12 4.2 Thenewphasespace................................ ......... 13 4.3 Quantumkinematics ............................... .......... 16 4.3.1 Elementaryfunctions. .......... 16 4.3.2 Quantum ∗ algebra ..................................... 17 − 4.3.3 Representations of A ..................................... 18 4.3.4 The kinematical Hilbert space and its properties . ................. 19 4.4 Quantumdynamics................................. ......... 21 4.4.1 Reduction of Gauss – and spatial diffeomorphism constraint............... 21 4.4.1.1 Gaussconstraint. .. .. .. .. .. .. .. .. ..... 21 4.4.1.2 Spatial diffeomorphism constraint . .......... 22 4.4.2 Reduction of the Hamiltonian constraint . ............... 23 4.4.2.1 Hamiltonian constraint . ....... 25 4.4.2.2 Master Constraint Programme . ....... 32 4.4.2.2.1 Graph changing Master constraint .
    [Show full text]
  • THE ADJOINT REPRESENTATION L-FUNCTION for GL(N)
    Pacific Journal of Mathematics THE ADJOINT REPRESENTATION L-FUNCTION FOR GL.n/ YUVAL ZVI FLICKER Vol. 154, No. 2 June 1992 PACIFIC JOURNAL OF MATHEMATICS Vol. 154, No. 2, 1992 THE ADJOINT REPRESENTATION L-FUNCTION FOR GL(n) YUVAL Z. FLICKER Ideas underlying the proof of the "simple" trace formula are used to show the following. Let F be a global field, and A its ring of adeles. Let π be a cuspidal representation of GL(«, A) which has a supercuspidal component, and ω a unitary character of Ax/Fx . Let So be a complex number such that for every separable extension E of F of degree n, the L-function L(s, ωoNorm^) over E vanishes at s = s0 to the order m > 0. Then the product L-function L(s, π <g> ω x π) vanishes at s = So to the order m . This result is a reflection of the fact that the tensor product of a finite dimensional representation with its contragredient contains a copy of the trivial representation. Let F be a global field, A its ring of adeles and Ax its group of ideles. Denote by G the group scheme GL(n) over F, and put G = G(F), G = G(A), and Z ~ Fx , Z ~ Ax for the correspond- ing centers. Fix a unitary character e of Z/Z, and signify by π a cuspidal representation of G whose central character is ε. For al- most all F-places v the component πv of π at υ is unramified and is determined by a semi-simple conjugacy class t(πv) in G = G(C) with eigenvalues (zj(πυ); 1 < i < n).
    [Show full text]