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Let k be an algebraically closed field of characteristic 0. We study some cohomological properties of Lie subalgebras of the Witt algebra W = Der (k [ t , t − 1 ]) and the one-sided Witt algebra W ≥ − 1 = Der (k [ t ]). In the first part of the paper, we consider finite codimension subalgebras of W ≥ − 1. We compute derivations and one-dimensional extensions of such subalgebras. These correspond to Ext U (L) 1 (M , L) , where L is a subalgebra of W ≥ − 1 and M is a one-dimensional representation of L. We find that these subalgebras exhibit a kind of rigidity: their derivations and extensions are controlled by the full one-sided Witt algebra. As an application of these computations, we prove that any isomorphism between finite codimension subalgebras of W ≥ − 1 extends to an automorphism of W ≥ − 1. The second part of the paper is devoted to explaining the observed rigidity. We define a notion of "completely non-split extension" and prove that W ≥ − 1 is the universal completely non-split extension of any of its subalgebras of finite codimension. In some sense, this means that even when studying subalgebras of W ≥ − 1 as abstract Lie algebras, they remember that they are contained in W ≥ − 1. We also consider subalgebras of infinite codimension, explaining the similarities and differences between the finite and infinite codimension situations. Almost all of the results above are also true for subalgebras of the Witt algebra. We summarise results for W at the end of the paper. [ABSTRACT FROM AUTHOR]

We study sympathetic (i.e., perfect and complete) Lie algebras. Among other topics they arise in the study of adjoint Lie algebra cohomology. Here a motivation comes from a conjecture of Pirashvili, which says that a finite-dimensional complex perfect Lie algebra is semisimple if and only if its adjoint cohomology vanishes. We prove several general results for sympathetic Lie algebras and for the adjoint Lie algebra cohomology of arbitrary finite-dimensional Lie algebras in characteristic zero using a result of Hochschild and Serre. Moreover, for certain semidirect products we obtain explicit results for the adjoint cohomology. [ABSTRACT FROM AUTHOR]

In this paper we generalize the notion of the Taylor spectrum to modules over an arbitrary Lie algebra and study it for finite-dimensional modules. We show that the spectrum can be described as the set of simple submodules in the case of nilpotent and semisimple Lie algebras. We also show that this result does not hold for solvable Lie algebras and obtain a precise description of the spectrum in the case of Borel subalgebras of semisimple Lie algebras. [ABSTRACT FROM AUTHOR]

We give an explicit description of the inner cohomology of an adèlic locally symmetric space of a given level structure attached to the general linear group of prime rank n with coefficients in a locally constant sheaf of complex vector spaces. We show that for all primes n the inner cohomology vanishes in all degrees for nonconstant sheaves, otherwise the quotient module of the inner cohomology classes that are not cuspidal is trivial in all degrees for primes n = 2 , 3 , and for all primes n ≥ 5 the description of the same at degrees where it is nontrivial is given in terms of algebraic Hecke characters. [ABSTRACT FROM AUTHOR]

We compute the Poisson cohomology associated with several three dimensional Lie algebras. Together with existing results and the classification of three dimensional Lie algebras, this provides the Poisson cohomology of all linear Poisson structures in dimension 3. [ABSTRACT FROM AUTHOR]

Abstract Each sub-Riemannian geometry with bracket generating distribution enjoys a background structure determined by the distribution itself. At the same time, those geometries with constant sub-Riemannian symbols determine a unique Cartan connection leading to their principal invariants. We provide cohomological description of the structure of these curvature invariants in the cases where the background structure is one of the parabolic geometries. As an illustration, constant curvature models are discussed for certain sub-Riemannian geometries. [ABSTRACT FROM AUTHOR]

Starting with Lie’s classification of finite-dimensional transitive Lie algebras of vector fields on C 2 we construct transitive Lie algebras of vector fields on the bundle C 2 × C by lifting the Lie algebras from the base. There are essentially three types of transitive lifts and we compute all of them for the Lie algebras from Lie’s classification. The simplest type of lift is encoded by Lie algebra cohomology. [ABSTRACT FROM AUTHOR]

We study the derived representation scheme DRepn(A) parametrizing the n-dimensional representations of an associative algebra A over a field of characteristic zero. We show that the homology of DRepn(A) is isomorphic to the Chevalley-Eilenberg homology of the current Lie coalgebra gln*(C) defined over a Koszul dual coalgebra of A. We extend this isomorphism to representation schemes of Lie algebras: for a finite-dimensional reductive Lie algebra g, we define the derived affine scheme DRepg(a) parametrizing the representations (in g) of a Lie algebra a; we show that the homology of DRepg(a) is isomorphic to the Chevalley-Eilenberg homology of the Lie coalgebra g*(C), where C is a cocommutative DG coalgebra Koszul dual to the Lie algebra a. We construct a canonical DG algebra map Φg(a): DRepg(a)G -> DReph(a)W, which is a homological extension of the classical restriction homomorphism. We call Φg(a) a derived Harish-Chandra homomorphism. We conjecture that, for a two-dimensional abelian Lie algebra a, the derived Harish-Chandra homomorphism is a quasi-isomorphism, and provide some evidence for this conjecture. For any complex Lie algebra g, we compute the Euler characteristic of DRepg(a)G in terms of matrix integrals over G and compare it to the Euler characteristic of DReph(a)W. This yields an interesting combinatorial identity, which we prove for gl_n and sl_n (for all n). Our identity is analogous to the classical Macdonald identity, and our quasi-isomorphism conjecture is analogous to the strong Macdonald conjecture proved by S.Fishel, I.Grojnowski and C. Teleman. We explain this analogy by giving a new homological interpretation of Macdonald's conjectures in terms of derived representation schemes, parallel to our Harish-Chandra quasi-isomorphism conjecture. [ABSTRACT FROM AUTHOR]

All classes of integrable cocycles inH2(L,L) are obtained for Lie algebra of typeG2over an algebraically closed field of characteristic 2. It is proved that there exist only two orbits of classes of integrable cocycles with respect to automorphism group. The global deformation is shown to exist for any nontrivial class of integrable cocycles. These deformations are isomorphic to one of the two algebras of Cartan type, one of which beingS(3:1,ω) while the otherH(4:1,ω). [ABSTRACT FROM AUTHOR]

We study the critical numbers of the Rankin–Selberg convolution of arbitrary pairs of cohomological cuspidal automorphic representations and we parametrize these critical numbers by certain 1-dimensional subrepresentations attached to the corresponding pair of finite dimensional representations of the related general linear groups. [ABSTRACT FROM AUTHOR]

We show that the theory of Lie algebra cohomology can be recast in a topological setting and that classical results, such as the Shapiro lemma and the van Est isomorphism, carry over to this augmented context. [ABSTRACT FROM AUTHOR]

We develop a relative version of Kostant's harmonic theory and use this to prove a relative version of Kostant's theorem on Lie algebra (co)homology. These are associated to two nested parabolic subalgebras in a semisimple Lie algebra. We show how relative homology groups can be used to realize representations with lowest weight in one (regular or singular) affine Weyl orbit. In the regular case, we show how all the weights in the orbit can be realized as relative homology groups (with different coefficients). These results are motivated by applications to differential geometry and the construction of invariant differential operators. [ABSTRACT FROM AUTHOR]

We offer a new approach to large N limits using the Batalin-Vilkovisky formalism, both commutative and noncommutative, and we exhibit how the Loday-Quillen-Tsygan Theorem admits BV quantizations in that setting. Matrix integrals offer a key example: we demonstrate how this formalism leads to a recurrence relation that in principle allows us to compute all multi-trace correlation functions. We also explain how the Harer-Zagier relations may be expressed in terms of this noncommutative geometry derived from the BV formalism. As another application, we consider the problem of quantization in the large N limit and demonstrate how the Loday-Quillen-Tsygan Theorem leads us to a solution in terms of noncommutative geometry. These constructions are relevant to open topological field theories and string field theory, providing a mechanism that relates moduli of categories of branes to moduli of brane gauge theories. [ABSTRACT FROM AUTHOR]

The strong Macdonald theorems state that, for L reductive and s an odd variable, the cohomology algebras H ⁎ ( L [ z ] / z N ) and H ⁎ ( L [ z , s ] ) are freely generated, and describe the cohomological, s -, and z -degrees of the generators. The resulting identity for the z -weighted Euler characteristic is equivalent to Macdonald's constant term identity for a finite root system. We calculate H ⁎ ( p / z N p ) and H ⁎ ( p [ s ] ) for p a standard parahoric in a twisted loop algebra, giving strong Macdonald theorems that take into account both a parabolic component and a possible diagram automorphism twist. In particular we show that H ⁎ ( p / z N p ) contains a parabolic subalgebra of the coinvariant algebra of the fixed-point subgroup of the Weyl group of L , and thus is no longer free. We also prove a strong Macdonald theorem for H ⁎ ( b ; S ⁎ n ⁎ ) and H ⁎ ( b / z N n ) when b and n are Iwahori and nilpotent subalgebras respectively of a twisted loop algebra. For each strong Macdonald theorem proved, taking z -weighted Euler characteristics gives an identity equivalent to Macdonald's constant term identity for the corresponding affine root system. As part of the proof, we study the regular adjoint orbits for the adjoint action of the twisted arc group associated to L , proving an analogue of the Kostant slice theorem. [ABSTRACT FROM AUTHOR]

We construct a seven-term exact sequence involving low degree cohomology spaces of a Lie algebra 𝔤, an ideal 𝔥 of 𝔤, and the quotient 𝔤/𝔥 with coefficients in a 𝔤-module. The existence of such a sequence follows from the Hochschild–Serre spectral sequence associated to the Lie algebra extension. However, some of the maps occurring in this induced sequence are not always explicitly known or easy to describe. In this article, we give alternative maps that yield an exact sequence of the same form, making use of the interpretations of the low-dimensional cohomology spaces in terms of derivations, extensions, etc. The maps are constructed using elementary methods. This alternative approach to the seven term exact sequence can certainly be useful, especially since we include straightforward cocycle descriptions of the constructed maps. [ABSTRACT FROM AUTHOR]

Homogeneous generalized holomorphic structures in the context of homogeneous principal fiber bundles are investigated. They are characterized in terms of Lie algebra data, and the generalized Dolbeault cohomology groups associated to a homogeneous generalized holomorphic vector bundle are identified with certain relative Lie algebra cohomology groups. We also provide some examples, using generalized flag manifolds. [ABSTRACT FROM AUTHOR]

We propose a unified framework in which the different constructions of cohomology groups for topological and Lie groups can all be treated on an equal footing. In particular, we show that the cohomology of "locally continuous" cochains (respectively "locally smooth" in the case of Lie groups) fits into this framework, which provides an easily accessible cocycle model for topological and Lie group cohomology. We illustrate the use of this unified framework and the relation between the different models in various applications. This includes the construction of cohomology classes characterizing the string group and a direct connection to Lie algebra cohomology. [ABSTRACT FROM AUTHOR]

Given a nilpotent Lie algebra 픫 we construct a spectral sequence which is derived from a filtration of its Chevalley-Eilenberg differential complex and converges to the Lie algebra cohomology of 픫. The limit of this spectral sequence gives a grading for the Lie algebra cohomology, except for the cohomology groups of degree 0, 1, dim 픫 - 1 and dim 픫 as we shall prove. We describe the spectral sequence associated to a nilpotent Lie algebra which is a direct sum of two ideals, one of them of dimension one, in terms of the spectral sequence of the co-dimension one ideal. Also, we compute the spectral sequence corresponding to each real nilpotent Lie algebra of dimension less than or equal to six. [ABSTRACT FROM AUTHOR]

Abstract: We describe the structure of the cohomology of the filiform Lie algebras and as a module over their (2-dimensional) torus of derivations. Our approach relies on the fact that both filiform algebras have an ideal of codimension 1 for which the structure of its cohomology under the action of the Levi factor of the algebra of derivations of is known. [Copyright &y& Elsevier]

By elementary and direct calculations the vanishing of the (algebraic) second Lie algebra cohomology of the Witt and the Virasoro algebra with values in the adjoint module is shown. This yields infinitesimal and formal rigidity or these algebras. The first (and up to now only) proof of this important result was given 1989 by Fialowski in an unpublished note. It is based on cumbersome calculations. Compared to the original proof the presented one is quite elegant and considerably simpler. [ABSTRACT FROM AUTHOR]

Abstract: Using the machinery of the Batalin–Vilkovisky formalism, we construct cohomology classes on compactifications of the moduli space of Riemann surfaces from the data of a contractible differential graded Frobenius algebra. We describe how evaluating these cohomology classes upon a well-known construction producing homology classes in the moduli space can be expressed in terms of the Feynman diagram expansion of some functional integral. By computing these integrals for specific examples, we are able to demonstrate that this construction produces families of nontrivial classes. [Copyright &y& Elsevier]

For a pair (¶, σ) of cuspidal automorphic representations of GLn and GLn-i, both of non-vanishing cohomology, we show algebraicity properties of critical values of the associated Rankin-Selberg //-function twisted by finite-order characters. A certain non-vanishing assumption about the associated archimedean Rankin-Selberg pairings on the cohomology is established for n = 3. [ABSTRACT FROM AUTHOR]

For a pair (π, σ) of cuspidal automorphic representations of n and n-1, both of non-vanishing cohomology, we show algebraicity properties of critical values of the associated Rankin-Selberg L-function twisted by finite-order characters. A certain non-vanishing assumption about the associated archimedean Rankin-Selberg pairings on the cohomology is established for n = 3. [ABSTRACT FROM AUTHOR]

We show that the Dirac cohomology coincides with nilpotent Lie algebra (co)homology up to a twist of a one-dimensional character for simple highest weight modules. As a consequence, we determine the Dirac cohomology of simple highest weight modules explicitly. [ABSTRACT FROM AUTHOR]

Given a singular Schubert variety X in a compact Hermitian symmetric space X, it is a long-standing question to determine when X is homologous to a smooth variety Y. We identify those Schubert varieties for which there exist first-order obstructions to the existence of Y. This extends (independent) work of M. Walters, R. Bryant and J. Hong. Key tools include (i) a new characterization of Schubert varieties that generalizes the well-known description of the smooth Schubert varieties by connected sub-diagrams of a Dynkin diagram and (ii) an algebraic Laplacian (à la Kostant), which is used to analyze the Lie algebra cohomology group associated with the problem. [ABSTRACT FROM AUTHOR]

Abstract: The worldsheet sigma model of the superstring in has a one-parameter family of flat connections parametrized by the spectral parameter. The corresponding Wilson line is not BRST invariant for an open contour, because the BRST transformation leads to boundary terms. These boundary terms define a cohomological complex associated with the endpoint of the contour. We study the cohomology of this complex for Wilson lines in some infinite-dimensional representations. We find that for these representations the cohomology is nontrivial at the ghost number 2. This implies that it is possible to define a BRST invariant open Wilson line. The central point in the construction is the existence of massless vertex operators transforming exactly covariantly under the action of the global symmetry group. In flat space, massless vertices transform covariantly only up to adding BRST-exact terms. But in AdS we show that it is possible to define vertices so that they transform exactly covariantly. [Copyright &y& Elsevier]

Given a Lie group action G we show, using the method of equivariant moving frames, that the local cohomology of the invariant Euler-Lagrange complex is isomorphic to the Lie algebra cohomology of G. [ABSTRACT FROM AUTHOR]

Abstract: Let G be a simple simply connected algebraic group scheme defined over an algebraically closed field of characteristic . Let T be a maximal split torus in G, be a Borel subgroup of G and U its unipotent radical. Let be the Frobenius morphism. For define the Frobenius kernel, , to be the kernel of F iterated with itself r times. Define (respectively ) to be the kernel of the Frobenius map restricted to U (respectively B). Let be the integral weight lattice and be the dominant integral weights. The computations of particular importance are , for , for , and for . The above cohomology groups for the case when the field has characteristic 2 are computed in this paper. These computations complete the picture started by Bendel, Nakano, and Pillen (2007) for . Furthermore, the computations show has a good filtration. [Copyright &y& Elsevier]

Abstract: We compute the second Hochschild cohomology space of Connes–Moscovici''s Hopf algebra , giving the infinitesimal deformations (up to equivalence) of the associative structure. The space is shown to be one-dimensional. [Copyright &y& Elsevier]

These are expository notes from the 2008 Srní Winter School. They have two purposes: (1) to give a quick introduction to exterior differential systems (EDS), which is a collection of techniques for determining local existence to systems of partial differential equations, and (2) to give an exposition of recent work (joint with C. Robles) on the study of the Fubini-Griffiths-Harris rigidity of rational homogeneous varieties, which also involves an advance in the EDS technology. [ABSTRACT FROM AUTHOR]

Abstract: Fialowski and Schlichenmaier constructed global deformations of the infinitesimally and formally rigid Lie algebra of polynomial vector fields on the circle from families of projective curves with marked points. In the present article, we show how to obtain these examples in a conceptual way. For this, we define a stack of deformations of Lie algebras and study the morphism from the moduli stack to this deformation stack which associates to a family of marked curves the Lie algebra of vertical vector fields on the punctured family. [Copyright &y& Elsevier]

On any manifold, any nondegenerate symmetric 2-form (metric) and any nondegenerate skew-symmetric differential form ω can be reduced to a canonical form at any point but not in any neighborhood: the corresponding obstructions are the Riemannian tensor and dω. The obstructions to flatness (to reducibility to a canonical form) are well known for any G-structure, not only for Riemannian or almost symplectic structures. For a manifold with a nonholonomic structure (nonintegrable distribution), the general notions of flatness and obstructions to it, although of huge interest (e.g., in supergravity) were not known until recently, although particular cases have been known for more than a century (e.g., any contact structure is nonholonomically “flat”: it can always be reduced locally to a canonical form). We give a general definition of the nonholonomic analogues of the Riemann tensor and its conformally invariant analogue, the Weyl tensor, in terms of Lie algebra cohomology and quote Premet’s theorems describing these cohomologies. Using Premet’s theorems and the SuperLie package, we calculate the tensors for flag manifolds associated with each maximal parabolic subalgebra of each simple Lie algebra (and in several more cases) and also compute the obstructions to flatness of the G(2)-structure and its nonholonomic superanalogue. [ABSTRACT FROM AUTHOR]

Abstract: Let G be a simple simply connected affine algebraic group over an algebraically closed field k of characteristic p for an odd prime p. Let B be a Borel subgroup of G and U be its unipotent radical. In this paper, we determine the second cohomology groups of B and its Frobenius kernels for all simple B-modules. We also consider the standard induced modules obtained by inducing a simple B-module to G and compute all second cohomology groups of the Frobenius kernels of G for these induced modules. Also included is a calculation of the second ordinary Lie algebra cohomology group of Lie(U) with coefficients in k. [Copyright &y& Elsevier]

Abstract: We study Novikov algebras and Novikov structures on finite-dimensional Lie algebras. We show that a Lie algebra admitting a Novikov structure must be solvable. Conversely we present an example of a nilpotent two-step solvable Lie algebra without any Novikov structure. We construct Novikov structures on certain Lie algebras via classical r-matrices and via extensions. In the latter case we lift Novikov structures on an abelian Lie algebra and a Lie algebra to certain extensions of by . We apply this to prove the existence of affine and Novikov structures on several classes of two-step solvable Lie algebras. In particular we generalize a well known result of Scheuneman concerning affine structures on three-step nilpotent Lie algebras. [Copyright &y& Elsevier]

Let g be a simple Lie algebra, V an irreducible g-module, W the Weyl group and b the Borel subalgebra of g, it [b, b], h the Cartan subalgebra of g. The Borel-Weil-Bott theorem states that the dimension of Hi (n; V) is equal to the cardinality of the set of elements of length i from W. Here a more detailed description of Hi (n; V) as an (g-module is given in terms of generating functions. Results of Leger and Luks and Williams who described Hi(n;n) for i ≤ 2 are generalized: dim H*(n;Λ*(n)) and dim Hi(n; it) for i ≤ 3 are calculated and dim Hi (n; n) as function of i and rank g is described for the classical series. [ABSTRACT FROM AUTHOR]

By considering non-trivial global deformations of the Witt (and the Virasoro) algebra given by geometric constructions it is shown that, despite their infinitesimal and formal rigidity, they are globally not rigid. This shows the need of a clear indication of the type of deformations considered. The families appearing are constructed as families of algebras of Krichever–Novikov type. They show up in a natural way in the global operator approach to the quantization of two-dimensional conformal field theory. In addition, a proof of the infinitesimal and formal rigidity of the Witt algebra is presented. [ABSTRACT FROM AUTHOR]