Homology of Lie algebra of supersymmetries and of super Poincaré Lie algebra

We study the homology and cohomology groups of super Lie algebras of supersymmetries and of super Poincaré Lie algebras in various dimensions. We give complete answers for (non-extended) supersymmetry in all dimensions ?11. For dimensions D=10,11 we describe also the cohomology of reduction of supersymmetry Lie algebra to lower dimensions. Our methods can be applied to extended supersymmetry Lie algebras.     

Lie 2-Bialgebras

In this paper, we study Lie 2-bialgebras, paying special attention to coboundary ones, with the help of the cohomology theory of L _∞-algebras with coefficients in L _∞-modules. We construct examples of strict Lie 2-bialgebras from left-symmetric algebras (also known as pre-Lie algebras) and symplectic Lie algebras (also called quasi-Frobenius Lie algebras).     

Symplectic reduction,BRS cohomology,and infinite-dimensional Clifford algebras

This paper gives the mathematical foundations for the BRS quantization procedure. We first discuss the classical finite dimensional BRS procedure and relate it to Marsden-Weinstein reduction. This leads to interesting relations between Lie algebras and Clifford algebras and a novel way of computing Lie algebra cohomology in terms of the spin representation. We then discuss infinite-dimensional Clifford algebras and their spin representations. We find that in the infinite-dimensional case, the analog of the finite-dimensional construction of Lie algebra cohomology breaks down, the obstruction (anomaly) being the Kac-Peterson class which is the cohomology class associated to the representation of the Lie algebra on spinors which is now only a projective representation. Tensoring by a projective representation of opposite class kills the obstruction and gives rise to a cohomology theory and a quantization procedure. We discuss the gradings and Hermitian structures on the absolute and relative complexes.     

Pre-symplectic algebroids and their applications

In this paper, we introduce the notion of a pre-symplectic algebroid and show that there is a one-to-one correspondence between pre-symplectic algebroids and symplectic Lie algebroids. This result is the geometric generalization of the relation between left-symmetric algebras and symplectic (Frobenius) Lie algebras. Although pre-symplectic algebroids are not left-symmetric algebroids, they still can be viewed as the underlying structures of symplectic Lie algebroids. Then we study exact pre-symplectic algebroids and show that they are classified by the third cohomology group of a left-symmetric algebroid. Finally, we study para-complex pre-symplectic algebroids. Associated with a para-complex pre-symplectic algebroid, there is a pseudo-Riemannian Lie algebroid. The multiplication in a para-complex pre-symplectic algebroid characterizes the restriction to the Lagrangian subalgebroids of the Levi–Civita connection in the corresponding pseudo-Riemannian Lie algebroid.     

Algebraic approach to the geometry of the (super-) string model

We discuss the extension of constraint algebras to include subsidiary constraints within a larger algebra. The interplay between various mathematical aspects of this procedure is described. Tools from Lie algebra cohomology and differential geometry are used to gain new insights into BRS techniques for nonabelian constrained systems. We show that cohomology considerations restrict our formalism to non-semisimple constraint algebras, such as the (super-) string model; we illustrate the ideas by presenting concrete results for this case.     

Analytic continuation of group representations II

The paper deals with the general background connecting the ideas of analytic continuation and contraction of Lie algebras and their representations. A connection is also established between the Kodaira-Spencer deformation theory and the theory of cohomology of Lie algebras.Supported by the Office of Naval Research, NONR 3656(09).     

A New Cohomology Theory Associated to Deformations of Lie Algebra Morphisms

We introduce a new cohomology theory related to deformations of Lie algebra morphisms. This notion involves simultaneous deformations of two Lie algebras and a homomorphism between them.This revised version was published online in March 2005 with corrections to the cover date.     

Formality and Deformations of Universal Enveloping Algebras

We describe enveloping algebras of finite-dimensional Lie algebras which are formal in the sense that their Hochschild complex as a differential graded Lie algebra is quasi-isomorphic to its Hochschild cohomology. For Abelian Lie algebras this is true thanks to the Kontsevich formality theorem. We are using his formality map twisted by the group-like element generated by the linear Poisson structure to simplify the problem, and then study examples. For instance, the universal enveloping algebras of the Lie algebras are formal. We also recover our rigidity results for enveloping algebras from this new angle and present some explicit deformations of linear Poisson structure in low dimensions.     

BRS cohomology and Casimir operators

Casimir operators play a central role in the study of cohomology problems for semisimple Lie algebras. An attempt is made to generalize this to strings. This generalization may be useful for studying small oscillations around nontrivial backgrounds.     

SAYD Modules over Lie-Hopf Algebras

In this paper a general van Est type isomorphism is proved. The isomorphism is between the Lie algebra cohomology of a bicrossed sum Lie algebra and the Hopf cyclic cohomology of its Hopf algebra. We first prove a one to one correspondence between stable-anti-Yetter-Drinfeld (SAYD) modules over the total Lie algebra and those modules over the associated Hopf algebra. In contrast to the non-general case done in our previous work, here the van Est isomorphism is proved at the first level of a natural spectral sequence, rather than at the level of complexes. It is proved that the Connes-Moscovici Hopf algebras do not admit any finite dimensional SAYD modules except the unique one-dimensional one found by Connes-Moscovici in 1998. This is done by extending our techniques to work with the infinite dimensional Lie algebra of formal vector fields. At the end, the one to one correspondence is applied to construct a highly nontrivial four dimensional SAYD module over the Schwarzian Hopf algebra. We then illustrate the whole theory on this example. Finally explicit representative cocycles of the cohomology classes for this example are calculated.     

Lie-admissible structures on Witt type algebras

In this paper we study Lie-admissible structures on Witt type algebras. Witt type algebras are Γ$\Gamma$-graded Lie algebras (where Γ$\Gamma$ is an abelian group) which generalize the Witt algebra. We give all third power-associative and flexible Lie-admissible structures on these algebras. In particular we generalize some results on the Witt algebra. After describing the second scalar cohomology group of Witt type algebras, we investigate third power-associative and flexible Lie-admissible structures on the central extension of some Witt type algebras. Finally we study a left-symmetric structure induced by a symplectic form for some Witt type algebras.     

Semiinfinite Cohomology of Quantum Groups

We define a new cohomology theory of associative algebras called semiinfinite cohomology in the derived categories' setting. We investigate the case of a small quantum group u, calculate semiinfinite cohomology spaces of the trivial u-module and express them in terms of local cohomology of the nilpotent cone for the corresponding semisimple Lie algebra. We discuss the connection between the semiinfinite homology of u and the conformal blocks' spaces. Received: 14 October 1996 / Accepted: 25 February 1997     

Quantization and representation theory of finiteW algebras

In this paper we study the finitely generated algebras underlyingW algebras. These so called finiteW algebras are constructed as Poisson reductions of Kirillov Poisson structures on simple Lie algebras. The inequivalent reductions are labeled by the inequivalent embeddings ofsl ₂ into the simple Lie algebra in question. For arbitrary embeddings a coordinate free formula for the reduced Poisson structure is derived. We also prove that any finiteW algebra can be embedded into the Kirillov Poisson algebra of a (semi)simple Lie algebra (generalized Miura map). Furthermore it is shown that generalized finite Toda systems are reductions of a system describing a free particle moving on a group manifold and that they have finiteW symmetry. In the second part we BRST quantize the finiteW algebras. The BRST cohomology is calculated using a spectral sequence (which is different from the one used by Feigin and Frenkel). This allows us to quantize all finiteW algebras in one stroke. Examples are given. In the last part of the paper we study the representation theory of finiteW algebras. It is shown, using a quantum version of the generalized Miura transformation, that the representations of finiteW algebras can be constructed from the representations of a certain Lie subalgebra of the original simple Lie algebra. As a byproduct of this we are able to construct the Fock realizations of arbitrary finiteW algebras.     

Differential cohomology of C (a)

Let be a finite dimensional real Lie algebra and ^* its dual. ^* is a Poisson manifold. Thus the space C^∞( ^*) of C^∞ functions on ^* has an associative and a Lie algebra structure. The problem of formal deformations of such a structure needs the determination of some cohomology groups of C^∞( ^*), considered as a module on itself for left multiplication or adjoint representation. We determine here these groups. The result is very similar to the case of C^∞(W), where W is a symplectic manifold except for the Lie algebras h_r × ^m, direct products of Heisenberg and abelian Lie algebras.     

On the structure of DeWitt supermanifolds

We show that for a wide and most natural class of (possibly infinite-dimensional) Grassmannian algebras of coefficients, the structure sheaf of every smooth DeWitt supermanifold is acyclic (i.e. its cohomology vanishes in positive degree). This result was previously known for finite-dimensional ground algebras and is new even for the original DeWitt algebra of supernumbers /GL∞. From here we deduce that (equivalence classes of) smooth DeWitt supermanifolds over a fixed ground algebra and of graded smooth manifolds are in a natural bijection with each other. However, contrary to what was stated previously by some authors, this correspondence fails to be functorial; so it happens, for instance, for Rogers' ground algebra B∞. Finally, we observe that every DeWitt super Lie group is a deformation of a graded Lie group over the spectrum Spec /GL of the ground algebra.     

On the 1-cohomology of Lie groups

Given a continous representation of a Lie group in a Banach space we study its 1-cohomology. We prove that the computation of the 1-cocycles can be reduced to that of the 1-cocycles of the differential of the representation. When the group is semi-simple and the representation is K-finite, we prove that the cohomology is equivalent to the cohomology of the Lie algebra representation on K-finite vectors. We prove, using Casimir operators, that there exist only a finite number of irreducible representation of a semi-simple Lie group with a non-trivial cohomology. Exemples of such representations are given.     

Hypersymplectic Lie algebras

We characterize real Lie algebras carrying a hypersymplectic structure as bicrossproducts of two symplectic Lie algebras endowed with a compatible flat torsion-free connection. In particular, we obtain the classification of all hypersymplectic structures on 4-dimensional Lie algebras, and we describe the associated metrics on the corresponding Lie groups.