Principal bundles admitting almost invariant structures

Infinite-dimensional super Lie groups

Let \Lambda be a finite abelian group. A dynamical system with transformation group \Lambda is a triple (A,\Lambda,\alpha) , consisting of a unital locally convex algebra A , the finite abelian group \Lambda and a group homomorphism \alpha:\Lambda\rightarrow\operatorname{Aut}(A) , which induces an action of \Lambda on A . In this paper we present a new, geometrically oriented approach to the noncommutative geometry of principal bundles with finite abelian structure group based on such dynamical systems.

1.1. Let G be a connected reductive algebraic group defined over a finite (prime) field Fp of p elements. For a given rational G-module M one can consider the restrictions of M to the restricted Lie algebra Lie(G) and to the finite group G(Fp) of Fp-rational points. Even though there is no direct functorial connection between the categories of restricted Lie(G)-modules and G(Fp)-modules, the ambient algebraic group G can serve as a bridge linking these two important module categories. Early results of Curtis in the 1960s provided a one-to-one correspondence between the simple restricted modules (over k = Fp) of Lie(G) and those of G(Fp). This correspondence is given by simply restricting certain classes of simple G-modules. More importantly, these results, along with work of Steinberg, allow one to easily transfer the questions pertaining to the computation of irreducible characters among these three categories. For over thirty years, this approach has been one of the main ideas used to relate the representation and cohomology theory of algebraic groups, Lie algebras, and finite groups of Lie type (in the defining characteristic) (see, e.g., [CPSvan], [Jan2], [Hum3], and [Jan1]). Given the previous results in this area, a natural course of investigation should be to find relationships between canonical modules, like projective modules, for Lie(G) and G(Fp). More precisely,

The study of mechanical systems on Lie algebroids permits an understanding of the dynamics described by a Lagrangian or Hamiltonian function for a wide range of mechanical systems in a unified framework. Systems defined in tangent bundles, Lie algebras, principal bundles, reduced systems, and constrained are included in such description. In this paper, we investigate how to derive the dynamics associated with a Lagrangian system defined on the set of admissible elements of a given Lie algebroid using Tulczyjew’s triple on Lie algebroids and constructing a Lagrangian Lie subalgebroid of a symplectic Lie algebroid, by building on the geometric formalism for mechanics on Lie algebroids developed by M. de Leon, J.C. Marrero and E. Martinez on “Lagrangian submanifolds and dynamics on Lie algebroids”.

In this paper we demonstrate that the exterior algebra of an Atiyah Lie algebroid generalizes the familiar notions of the physicist's BRST complex. To reach this conclusion, we develop a general picture of Lie algebroid isomorphisms as commutative diagrams between algebroids preserving the geometric structure encoded in their brackets. We illustrate that a necessary and sufficient condition for such a diagram to define a morphism of Lie algebroid brackets is that the two algebroids possess gauge-equivalent connections. This observation indicates that the aforementioned set of Lie algebroid isomorphisms should be regarded as equivalent to the set of local diffeomorphisms and gauge transformations. Moreover, a Lie algebroid isomorphism being a chain map in the exterior algebra sense ensures that isomorphic algebroids are cohomologically equivalent. The Atiyah Lie algebroids derived from principal bundles with common base manifolds and structure groups may therefore be divided into equivalence classes of isomorphic algebroids. Each equivalence class possesses a local representative which we refer to as the trivialized Lie algebroid, and we show that the exterior algebra of the trivialized algebroid gives rise to the BRST complex. We conclude by illustrating the usefulness of Lie algebroid cohomology in computing quantum anomalies, including applications to the chiral and Lorentz-Weyl (LW) anomalies. In particular, we pay close attention to the fact that the geometric intuition afforded by the Lie algebroid (which was absent in the naive BRST complex) provides hints of a deeper picture that simultaneously geometrizes the consistent and covariant forms of the anomaly. In the algebroid construction, the difference between the consistent and covariant anomalies is simply a different choice of basis.

The Frattini argument shows that, if N is a normal subgroup of a group G and if S is an intravariant subgroup of N, then G=~(S)N. Unfortunately, there appears to be no completely satisfactory Lie algebra analogue of the concept of an intravariant subgroup. We get around this difficulty by considering ways in which intravariant subgroups arise, and investigating the analogous situations for Lie algebras. In finite group theory, the best known examples of intravariant subgroups are 3s for suitable classes 3s In particular, Sylow p-subgroups are the projectors for the class of finite p-groups, a class which is minimal subject to the existence of projectors in all finite soluble groups. We show, subject to suitable conditions on N and 3s that if N is an ideal of the Lie algebra L and S is an 3s of N, then L=N+~(S). In particular, this holds if S is a Cartan subalgebra of N. All Lie algebras considered are finite-dimensional over some field F. We write U__< L, U~ L for U is a subalgebra, respectively ideal, of L. We denote the centraliser and normaliser of U in L by cgL(U ) andS(U) respectively./2 is the derived algebra and ~b(L) is the Frattini subalgebra of L. If M is an L-module, then ML={rn~M[mx=O for allx~L}.We denote by 9t and ~ the classes of all nilpotent and all soluble Lie algebras respectively. Let 3s be a homomorph of Lie algebras. The subalgebra S of L is called an 3s of L if

Owing to the existence of a dilatation generator of eigenvalues ± 2 , ± 1 , 0 \pm 2, \pm 1,0 the symplectic Lie algebra is considered as a | 2 | |2| -graded Lie algebra. The corresponding decomposition of the symplectic group Sp(2( n + 1 ), R ) {\text {Sp(2(}}n + 1{\text {),}}{\mathbf {R}}{\text {)}} makes the semidirect product (denoted L 0 {L^0} ) of the ( 2 n + 1 ) (2n + 1) -dimensional Weyl group by the conformal symplectic group CSp( 2 n , R ) {\text {CSp(}}2n,{\mathbf {R}}{\text {)}} appear as a privileged subgroup and permits one to construct a 2 n + 1 2n + 1 -dimensional homogeneous space possessing a natural contact form. Then Sp ( 2 ( n + 1 ) , R ) {\text {Sp}}(2(n + 1),{\mathbf {R}}) -valued Cartan connections on a L 0 {L^0} principal fibre bundle over a 2 n + 1 2n + 1 -dimensional manifold B 2 n + 1 {B_{2n + 1}} are constructed and called symplectic Cartan connections. The conditions for obtaining a unique symplectic Cartan connection are given. The existence of this unique Cartan connection is used to define the notion of contact structure over B 2 n + 1 {B_{2n + 1}} and it is shown that any L 0 {L^0} -structure of degree 2 2 over B 2 n + 1 {B_{2n + 1}} can be considered as a contact structure on it. Moreover it is shown that a contact structure can be associated in a canonical way to any contact manifold.

In this communication I would like to give a progress report on a program to construct graded geometric structures on graded differential manifolds. Most of the results so far are on the formal algebraic level and many important problems of analysis remain. Kostant I has given a definition of a graded differential manifold of dimension (n,m) and this is discussed in these proceedings by M. Batchelor. I only mention here that it consists of a pair (M,A) where M is an ordinary C ~ manifold of dimension n and A is a sheaf of Z2-graded commutative algebras whose underlying graded vector space V = V 0 + V I has dim V 0 = n, dim V 1 = m. A graded Lie group (G,A) then is a pair where G is a Lie group and the restricted dual A ° (vanishes on some ideal of finite codimension) has the structure of a graded Hopf algebra and can be represented as A ° = R(G) ^ E(G) where R(G) is the group ring, E(G) is the universal enveloping algebra of the graded Lie algebra G = G O + G I with G O the Lie algebra of G, and ^ denotes semidirect product. In analogy with the usual situation, graded frames can be introduced and the graded frame bundle constructed. This consists of a principal fibre bundle L(M,G) with group G = GL(n)×GL(m) such that the graded tangent bundle T(M,A) defined by Kostant is an associated vector bundle for L(M,G). The graded frame bundle (not really a bundle) is then given by the graded manifold (L(M,G), A8 H) where H ° = R(G) ^ E(G) , G = GL(n)× GL(m), G = gl(n,m) = End V. Now if G'C G is a closed Lie subgroup of G with a subHopf algebra H' C H , then (L(M,G'), A8 H') is called a graded G-struc ture if L(M,G') is reduced subbundle of L(M,G). There is a natural right action R a of H' defined on (L(M,G'), ASH'). Associated with the graded tangent bundle of a G-structure, there is the tangent sheaf derA + derH (der A = derivations of A), and this is a free A@ H module. A complement H to derH is called a horizontal subspace, and as usual H defines a connection if R * H = H. As underlying graded vector spaa ces H is isomorphic to V. In analogy with the ordinary ease, we study graded G-structures by studying graded Spencer cohomology. Let GC gl(n,m) = V8 V* be a graded Lie algebra. We define G (1) i i I I the first prolongation of G by all T E Hom(V,G) which satisfy T(u)v =(-i) luILvi T(V)U where u,v are homogeneous elements in V of degree lul, Ivl respectively. The k th prolongation is defined inductively by G (k) = (G(k-l)) (I) • We construct Z 2 gravector spaces C k+l'l = G (k) O Al(V *) where AI(V *) = AI(V~)0 Sl(V~) is the graded ded exterior algebra over V*. The cochain map ~:C k+l'/ >?,/+i is defined ~ by

A metric Lie algebra is a Lie algebra equipped with an invariant non-degenerate symmetric bilinear form. It is called indecomposable if it is not the direct sum of two metric Lie algebras. We are interested in describing the isomorphism classes of indecomposable metric Lie algebras. In the present paper we restrict ourselves to a certain class of solvable metric Lie algebras which includes all indecomposable metric Lie algebras with maximal isotropic centre. We will see that each metric Lie algebra belonging to this class is a twofold extension associated with an orthogonal representation of an abelian Lie algebra. We will describe equivalence classes of such extensions by a certain cohomology set. In particular we obtain a classification scheme for indecomposable metric Lie algebras with maximal isotropic centre and the classification of metric Lie algebras of index 2.

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

Cohen and Taylor [On a certain Lie algebra defined by a finite group, Am. Math. Mon. 114 (2007) 633–639] introduced certain Lie algebras constructed using finite groups called the Plesken Lie algebras. In this paper, we describe the representation of these Plesken Lie algebras, their irreducibility and the Plesken Lie algebra modules.

Using characters, one can set up the theory of representations for both finite and continuous groups. For continuous groups, another approach—the Lie theory—is also possible. It is shown that a similar theory, based on commutators, can be developed also in the case of finite groups. Essentially, the group algebra of a finite group is converted into a Lie algebra by replacing the usual associative product by the product x∘y=xy−yx. Then the resulting Lie algebra is a direct sum of special unitary Lie algebras.

String junctions for arbitrary Lie-algebra representations

This dissertation is devoted to the study of quantizations of nilpotent orbits and their covers, and their applications to the theory of unipotent representations. It combines the results from the papers written by the author during his PhD. In the first part, we construct a bijection between the sets of quantizations of a nilpotent orbit in a classical simple Lie algebra, and quantizations of its function algebra. In the second part, we study the geometry of nilpotent orbit covers, and the sets of quantizations of their function algebras. After that, we use the canonical quantizations of these function algebras to give new definitions of unipotent ideals and unipotent representations of a semisimple complex Lie algebra. Our definitions are vindicated by the many favorable properties which these ideals and bimodules enjoy. First of all, we show that both unipotent ideals and bimodules have nice geometric classifications. Unipotent ideals are classified by certain geometrically-defined equivalence classes of covers of nilpotent orbits. Unipotent bimodules are classified by irreducible representations of certain finite groups. We show that all unipotent ideals are maximal, and for classical groups all unipotent bimodules are unitary. In addition, we show that all unipotent bimodules are, as $G$-representations, of a very special form, proving a conjecture of Vogan. Finally, we show that all special unipotent bimodules are unipotent. The final assertion is proved using a certain refinement of Barbasch-Vogan-Lusztig-Spaltenstein duality, inspired by the symplectic duality. Along the way, we establish combinatorial algorithms (in classical and exceptional types) for computing the infinitesimal characters of unipotent ideals--Author's abstract

Finite and infinite dimensional Lie group structures on Riordan groups