Representation Theory in Complex Rank, II
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Global Theory: Chiral Homology
CHAPTER 4 Global Theory: Chiral Homology Qinovnik umiraet, i ordena ego ostats na lice zemli. Koz~ma Prutkov, “Mysli i Aforizmy”, 1860 † 4.1. The cookware This section collects some utensils and implements needed for the construction of chiral homology. A sensible reader should skip it, returning to the material when necessary. Here is the inventory together with a brief comment on the employment mode. (i) In 4.1.1–4.1.2 we consider homotopy direct limits of complexes and discuss their multiplicative properties. The material will be used in 4.2 where the de Rham complex of a D-module on Ran’s space R(X) is defined as the homotopy direct limit of de Rham complexes of the corresponding D-modules on all the Xn’s with respect to the family of all diagonal embeddings. See [BK], [Se]. (ii) In 4.1.3 and 4.1.4 we discuss the notion of the Dolbeault algebra which is an algebraic version of the ∂¯-resolution. Dolbeault resolutions are functorial and have nice multiplicative properties, so they are very convenient for computing the global de Rham cohomology of D-modules on R(X), in particular, the chiral homology of chiral algebras. An important example of Dolbeault algebras comes from the Thom-Sullivan construction; see [HS] §4. (iii) In 4.1.5 we recall the definitions of semi-free DG modules, semi-free com- mutative DG algebras, and the cotangent complex following [H] (see also [Dr1] and [KrM]). The original construction of the cotangent complex, due to Grothendieck [Gr1], [Il], was performed in the setting of simplicial algebras. -
Homological Stability of Diffeomorphism Groups 3
HOMOLOGICAL STABILITY OF DIFFEOMORPHISM GROUPS ALEXANDER BERGLUND AND IB MADSEN Abstract. In this paper we prove a stability theorem for block diffeomor- phisms of 2d-dimensional manifolds that are connected sums of Sd ×Sd. Com- bining this with a recent theorem of S. Galatius and O. Randal-Williams and Morlet’s lemma of disjunction, we determine the homology of the classifying space of their diffeomorphism groups relative to an embedded disk in a stable range. Dedicated to Dennis Sullivan on the occasion of his 70th birthday 1. Introduction The traditional method to obtain homotopical and homological information about diffeomorphism groups of high dimensional smooth manifolds is a two step procedure: the surgery exact sequence handles the larger group of block diffeomor- phisms and Waldhausen’s K-theory of spaces connects block diffeomorphisms and actual diffeomorphisms. In practice one is forced to retreat to rational information. The method was used by Farrell and Hsiang [17] to evaluate the rational homotopy groups of the orientation preserving diffeomorphism groups of spheres in a modest range of dimensions; Q, k ≡ 0 (4), n even n (1) πk(B Diff(S )) ⊗ Q = Q ⊕ Q, k ≡ 0 (4), n odd 0, otherwise. n n for 0 <k< 6 − 7. The orthogonal group SO(n + 1) is a subgroup of Diff(S ), and πkSO(n + 1) accounts for one Q when k ≡ 0 (4). The second Q in line two above is of a different nature. It comes from the connection between Waldhausen’s algebraic K-theory of spaces and the homotopy theory of the homogeneous space Diff(Sn)/ Diff(Sn) of block diffeomorphisms modulo diffeomorphisms. -
A Model Structure for the Goldman-Millson Theorem
A MODEL STRUCTURE FOR THE GOLDMAN–MILLSON THEOREM DANIEL ROBERT-NICOUD ABSTRACT. By a result of Vallette [Val14], we put a sensible model structure on the category of conilpotent Lie coalgebras. This gives us a powerful tool to study the subcategory of Lie algebras obtained by linear dualization, also known as the category of pronilpotent Lie algebras. This way, we recover weaker versions of the celebrated Goldman–Millson theorem and Dolgushev–Rogers theorem by purely homotopical methods. We explore the relations of this procedure with the existent literature, namely the works of Lazarev–Markl and Buijs–F´elix–Murillo–Tanr´e. CONTENTS 1. Introduction 1 2. Some notions of category theory 2 3. The Goldman–Millson theorem and the Dolgushev–Rogers theorem 7 4. The Vallette model structure on conilpotent coalgebras over a cooperad 9 5. A model structure for the Goldman–Millson theorem 13 6. Framings and the Dolgushev–Rogers theorem 17 References 20 1. INTRODUCTION The celebrated Goldman–Millson theorem [GM88] in one of its more recent versions states that every filtered quasi-isomorphism of differential graded Lie algebras — i.e. quasi-isomorphisms that are com- patible with certain filtrations of the algebras — induces an isomorphism between the moduli spaces of Maurer–Cartan elements of the algebras. Recently, this theorem was extended by V. A. Dolgushev and C. L. Rogers [DR15], who proved that such a morphism induces a weak equivalence of simplicial sets between the whole deformation ∞-groupoids of the algebras, of which the moduli space of Maurer– Cartan elements is the zeroth homotopy group. These theorems have a distinct homotopical flavor — after all, they say that the class of filtered quasi-isomorphisms is sent into the classe of weak equiva- lences of simplicial sets under a certain functor. -
Clifford Algebras and the Classical Dynamical Yang-Baxter Equation
Mathematical Research Letters 10, 253–268 (2003) CLIFFORD ALGEBRAS AND THE CLASSICAL DYNAMICAL YANG-BAXTER EQUATION A. Alekseev and E. Meinrenken Abstract. We describe a relationship of the classical dynamical Yang-Baxter equation with the following elementary problem for Clifford algebras: Given a vector space V with quadratic form QV , how is the exponential of an element in ∧2(V ) under exterior algebra multiplication related to its exponential under Clifford multiplication? 1. Introduction Let g be a real Lie algebra, equipped with a non-degenerate invariant qua- dratic form Q. Let Θ ∈∧3g be the cubic element defined by the quadratic form and the Lie algebra structure. An element r ∈∧2g is called a classical r-matrix for g if it satisfies the (modified) classical Yang-Baxter equation (CYBE) 1 [r, r]g = Θ 2 for some coupling constant ∈ R. Here [r, r]g is defined using the extension of the Lie bracket to the Schouten bracket on the exterior algebra, [·, ·]g : ∧kg × ∧lg →∧k+l−1g. Drinfeld [8] and Semenov-Tian-Shansky [20] gave a geometric interpretation of the CYBE in terms of Poisson-Lie group structures, and a classification of r-matrices for semi-simple Lie algebras and = 0 was obtained by Belavin-Drinfeld [6]. The CYBE admits an important generalization known as the classical dynam- ical Yang-Baxter equation (CDYBE). Let k ⊂ g be a Lie subalgebra. A classical dynamical r-matrix is a k-equivariant (meromorphic)function r : k∗ →∧2g satisfying the (modified)CDYBE [11] ∂r 1 ∧ e + [r, r]g = Θ. ∂µ i 2 i i i ∗ Here ei is a basis on k with dual basis e ∈ k , and µi are the corresponding coordinates on k∗. -
A Mathematical Approach to Wick Rotations
A mathematical approach to Wick rotations by Christer Helleland Thesis submitted in fulfilment of the requirements for the degree of PHILOSOPHIAE DOCTOR (PhD) ! Faculty of Science and Technology Department of Mathematics and Physics 2019 University of Stavanger NO-4036 Stavanger NORWAY www.uis.no ©2019 Christer Helleland ISBN: Click to enter ISBN. ISSN: Click to enter ISSN. PhD: Thesis UiS No. Click to enter PhD No. Contents List of Tables6 Acknowledgements8 Part 1. Introduction 9 Part 2. Preliminaries 12 1. Cartan involutions of linear Lie groups 12 2. Geometric invariant theory (GIT) 16 Part 3. Wick-rotations and real GIT 22 1. Introduction 22 2. Mathematical Preliminaries 23 2.1. Real form of a complex vector space 23 2.2. Real slices 24 2.3. Compatible real forms 25 3. Holomorphic Riemannian manifolds 27 3.1. Complexification of real manifolds 27 3.2. Complex differential geometry 28 3.3. Real slices from a frame-bundle perspective 29 4. Lie groups 31 4.1. Complex Lie groups and their real forms 31 4.2. Example: Split G2-holonomy manifolds 32 5. A standard Wick-rotation to a real Riemannian space 34 5.1. Minimal vectors and closure of real semi-simple orbits 34 5.2. Compatible triples and intersection of real orbits 36 5.3. The real Riemannian case 37 5.4. The adjoint action of the Lorentz groups O(n − 1; 1) 38 5.5. Uniqueness of real orbits and the class of complex Lie groups 41 6. Applications to the pseudo-Riemannian setting 45 6.1. Pseudo-Riemannian examples 47 Acknowledgements 48 Appendix A. -
On 2-Plectic Lie Groups
On 2-plectic Lie groups Mohammad Shafiee and Masoud Aminizadeh Abstract. A 2-plectic Lie group is a Lie group endowed with a 2-plectic structure which is left invariant. In this paper we provide some inter- esting examples of 2-plectic Lie groups. Also we study the structure of the set of Hamiltonian covectors and vectors of a 2-plectic Lie algebra. Moreover, the existence of i-isotropic and i-Lagrangian subgroups are in- vestigated. At last we obtain some results about the reduction of some 2-plectic structures. M.S.C. 2010: 53D05, 17B60. Key words: 2-plectic structure; quadratic Lie algebra; Hamiltonian vector. 1 Introduction A k-plectic manifold (M; !) is a smooth manifold M endowed with a (k + 1)-form ! which is closed and nondegenerate in the sense that ιX ! = 0, X 2 TM, implies that X = 0. In this case ! is called a k-plectic structure. These structures are general version of symplectic structures and they naturally appear in the Hamiltonian formulation of classical fields (see [4]and references therein ). Mathematically, in spite of general case, symplectic structures are very interesting and so they have been studied extensively. However, in recent years, 2-plectic structures also have been considered and these structures also are studied extensively ([1], [8]). An important class of 2-plectic manifolds are 2-plectic Lie groups. Similar to symplectic case, a 2-plectic Lie group is defined as follows. Definition 1.1. A 2-plectic Lie group (G; !) is a Lie group G endowed with a 2- plectic structure ! which is left invariant. -
Clifford Algebras and Lie Groups Lecture Notes, University of Toronto
Clifford algebras and Lie groups Eckhard Meinrenken Lecture Notes, University of Toronto, Fall 2009. Last revision: 12/2011 Contents Chapter 1. Symmetric bilinear forms 7 1. Quadratic vector spaces 7 2. Isotropic subspaces 9 3. Split bilinear forms 10 4. E.Cartan-Dieudonn´e'sTheorem 13 5. Witt's Theorem 16 6. Orthogonal groups for K = R; C 17 7. Lagrangian Grassmannians 23 Chapter 2. Clifford algebras 27 1. Exterior algebras 27 1.1. Definition 27 1.2. Universal property, functoriality 28 2. Clifford algebras 29 2.1. Definition and first properties 30 2.2. Universal property, functoriality 31 2.3. The Clifford algebras Cl(n; m) 32 2.4. The Clifford algebras Cl(n) 33 2.5. Symbol map and quantization map 34 2.6. Transposition 35 2.7. Chirality element 36 2.8. The trace and the super-trace 37 2.9. Extension of the bilinear form 38 2.10. Lie derivatives and contractions 38 2.11. The Lie algebra q(^2(V )) 40 2.12. A formula for the Clifford product 41 3. The Clifford algebra as a quantization of the exterior algebra 42 3.1. Differential operators 42 3.2. Graded Poisson algebras 44 3.3. Graded super Poisson algebras 45 3.4. Poisson structures on ^(V ) 46 Chapter 3. The spin representation 49 1. The Clifford group and the spin group 49 1.1. The Clifford group 49 1.2. The groups Pin(V ) and Spin(V ) 51 2. Clifford modules 54 2.1. Basic constructions 54 2.2. The spinor module SF 56 2.3. -
Semibounded Unitary Representations of Double Extensions of Hilbert–Loop Groups Tome 64, No 5 (2014), P
R AN IE N R A U L E O S F D T E U L T I ’ I T N S ANNALES DE L’INSTITUT FOURIER K. H. NEEB Semibounded Unitary Representations of Double Extensions of Hilbert–Loop Groups Tome 64, no 5 (2014), p. 1823-1892. <http://aif.cedram.org/item?id=AIF_2014__64_5_1823_0> © Association des Annales de l’institut Fourier, 2014, tous droits réservés. L’accès aux articles de la revue « Annales de l’institut Fourier » (http://aif.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://aif.cedram.org/legal/). Toute re- production en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement per- sonnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Ann. Inst. Fourier, Grenoble 64, 5 (2014) 1823-1892 SEMIBOUNDED UNITARY REPRESENTATIONS OF DOUBLE EXTENSIONS OF HILBERT–LOOP GROUPS by K. H. NEEB (*) Abstract. — A unitary representation π of a, possibly infinite dimensional, Lie group G is called semibounded if the corresponding operators idπ(x) from the derived representation are uniformly bounded from above on some non-empty open subset of the Lie algebra g of G. We classify all irreducible semibounded represen- tations of the groups Lbφ(K) which are double extensions of the twisted loop group Lφ(K), where K is a simple Hilbert–Lie group (in the sense that the scalar product on its Lie algebra is invariant) and φ is a finite order automorphism of K which leads to one of the 7 irreducible locally affine root systems with their canonical Z-grading. -
Dirac Operators in Representation Theory
Jing-Song Huang and Pavle Pandˇzi´c Dirac Operators in Representation Theory SPIN Springer’s internal project number, if known – Monograph – December 14, 2005 Springer Berlin Heidelberg NewYork Hong Kong London Milan Paris Tokyo Preface We begin with the origin of the Dirac equation and give a brief introduction of the Dirac operators due to Parthasarathy, Vogan and Kostant. Then we explain a conjecture of Vogan on Dirac cohomology, which we proved in [HP1], its applications and the organization of the book. 0.1 The Dirac equation. The Dirac equation has an interesting connection to E = mc2, the Einstein’s equation from his special theory of relativity. This equation relates the energy E of a particle at rest to its mass m through a conversion factor, the square of the speed c of light. However, this way of writing the equation obscures the underlying four-dimensional geometry. The relation for a particle in motion is a hyperbolic equation: E2 c2p p = (mc2)2. (0.1) − · Here the energy E is the first component of a vector (E,cp) = (E,cp1,cp2,cp3) and p is the vector that describes the momentum of the particle. This more general equation exhibits the mechanism of the conversion of mass into relative motion. 1 For describing relativistic spin 2 particles Dirac was to rewrite the quadratic Einstein relation (0.1) as a linear relation. This would seem impossible. But Dirac came up with a new idea by writing the relation as follows: 3 2 γ0E + c γj pj = mc I, (0.2) j=1 X where γ (k =0, 1, 2, 3) are 4 4 matrices and I is the 4 4 identity matrix. -
Local Commutative Algebra and Hochschild Cohomology Through the Lens of Koszul Duality
Local Commutative Algebra and Hochschild Cohomology Through the Lens of Koszul Duality by Benjamin Briggs A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mathematics University of Toronto c Copyright 2018 by Benjamin Briggs Abstract Local Commutative Algebra and Hochschild Cohomology Through the Lens of Koszul Duality Benjamin Briggs Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2018 This thesis splits into two halves, the connecting theme being Koszul duality. The first part concerns local commutative algebra. Koszul duality here manifests in the homotopy Lie algebra. In the second part, which is joint work with Vincent G´elinas,we study Hochschild cohomology and its characteristic action on the derived category. We begin by defining the homotopy Lie algebra π∗(φ) of a local homomorphism φ (or of a ring) in terms of minimal models, slightly generalising a classical theorem of Avramov. Then, starting with work of F´elixand Halperin, we introduce a notion of Lusternik-Schnirelmann category for local homomor- phisms (and rings). In fact, to φ we associate a sequence cat0(φ) ≥ cat1(φ) ≥ cat2(φ) ≥ · · · each cati(φ) being either a natural number or infinity. We prove that these numbers characterise weakly regular, com- plete intersection, and (generalised) Golod homomorphisms. We present examples which demonstrate how they can uncover interesting information about a homomorphism. We give methods for computing these numbers, and in particular prove a positive characteristic version of F´elixand Halperin's Mapping Theorem. A motivating interest in L.S. category is that finiteness of cat2(φ) implies the existence of certain six-term exact sequences of homotopy Lie algebras, following classical work of Avramov. -
Lie Algebras with Ad-Invariant Metrics. a Survey
LIE ALGEBRAS WITH AD-INVARIANT METRICS A SURVEY GABRIELA P. OVANDO Abstract. This is a survey work on Lie algebras with ad-invariant met- rics. We summarize main features, notions and constructions, in the aim of bringing into consideration the main research on the topic. We also give some list of examples in low dimensions. To the memory of our friend Sergio 1. Introduction An ad-invariant metric on a Lie algebra g is a non-degenerate symmetric bilinear form h , i which satisfies (1) h[x, y], zi + hy, [x, z]i = 0 for all x, y, z ∈ g. Lie algebras equipped with ad-invariant metrics are known as: “metric”[22], “metrised” [10], “orthogonal” [28], “quadratic” [27], “regular quadratic”[16], ”symmetric self-dual” [17], ”self-dual” [36] Lie algebras. Lie algebras which can be endowed with an ad-invariant metric are also known as “quadrable” or “metrisable”. They became relevant some years ago when they were useful in the formulation of some physical problems, for instance in the so known Adler-Kostant-Symes scheme. More recently they appeared in conformal field theory precisely as the Lie algebras for which a Sugawara construction exists [17]. They also constitute the basis for the construction of bi-algebras and they arXiv:1512.03997v2 [math.DG] 14 Jun 2017 give rise to interesting pseudo-Riemannian geometry [13]. For instance in [34] a result originally due to Kostant [26] was revalidated for pseudo-Riemannian metrics: it states that there exists a Lie group acting by isometries on a pseudo- Riemannian naturally reductive space (in particular symmetric spaces) whose Lie algebra can be endowed with an ad-invariant metric. -
A View of Symplectic Lie Algebras from Quadratic Poisson Algebras
Bolet´ınde Matem´aticas 26(1) 1{30 (2019) 1 A view of symplectic Lie algebras from quadratic Poisson algebras Una mirada a las ´algebrasde Lie simplecticas desde las ´algebrasde Poisson cuadr´aticas Andr´esRia~no1;a, Armando Reyes1;b Abstract. Using the concept of double extension, Benayadi [2] showed how to construct a new quadratic algebra (g(A);T ) given a quadratic algebra (A;B). With both algebras and an invertible skew-symmetric algebra D over A, he en- dowed (A;B) with a simplectic structure through a bilinear form !, obtaining a simplectic algebra (g(A);T; Ω). Our purpose in this short communication is to show the construction given by Benayadi and present the complete develop- ment of each one of his assertions. We remark that this communication does not have original results and it was made as a result of the undergraduated work titled “Construcci´onde ´algebrasde Lie simpl´ecticasdesde ´algebrasde Poisson cuadr´aticas"[5],which was awarded as the best mathematics under- graduated thesis in the XXVI Contest at Universidad Nacional de Colombia, Sede Bogot´a.The work was written by the first author under the direction of the second author. Keywords: Lie algebra, Poisson algebra, quadratic algebra, symplectic alge- bra. Resumen. A partir del concepto de doble extensi´on,Benayadi [2] mostr´o c´omoconstruir una nueva ´algebracuadr´atica(g(A);T ), a partir de un ´algebra cuadr´aticadada (A;B). Con estas dos ´algebrasy una derivaci´oninvertible anti-sim´etrica D sobre A, ´eldot´oa (A;B) de una estructura simpl´ectica a traves de una forma bilineal !, obteniendo as´ı una ´algebrasimpl´ectica (g(A);T; Ω).