On some sequence of graded Lie algebras associated to manifolds

A simple construction associates to any linear mapping a short exact sequence of graded Lie algebras. The sequence associated to the de Rham differential of an arbitrary smooth manifold is never split. Combined with a sort of algebraic Chern-Weil homomorphism adapted from [1] to the graded case, this leads to a family of cohomology classes of the Nijenhuis-Richardson algebra of the space of functions of the manifold. Some of these characteristic classes of degree 2 are computed. They are the classes constructed by hand in [2] and used in the theory of star-products.     

On the real zeroes of the Hurwitz zeta-function and Bernoulli polynomials

The behaviour of real zeroes of the Hurwitz zeta-function

     

Automorphic equivalence in the varieties of representations of Lie algebras

Abstract

In this paper, we consider the wide class of subvarieties of the variety of all representation of Lie algebras over a field k of characteristic 0. We study the relation between the geometric equivalence and automorphic equivalence of the representations from these subvarieties.     

Lie algebraic characterization of manifolds

Results on characterization of manifolds in terms of certain Lie algebras growing on them, especially Lie algebras of differential operators, are reviewed and extended. In particular, we prove that a smooth (real-analytic, Stein) manifold is characterized by the corresponding Lie algebra of linear differential operators, i.e. isomorphisms of such Lie algebras are induced by the appropriate class of diffeomorphisms of the underlying manifolds. The research of Janusz Grabowski supported by the Polish Ministry of Scientific Research and Information Technology under the grant No. 2 P03A 020 24, that of Norbert Poncin by grant C.U.L./02/010.     

Representations of Yangians and multiplicities of occurrence of the irreducible components of the tensor product of representations of simple Lie algebras

New combinatorial formulas are obtained for the multiplicities in the decomposition of the tensor product of the representations of simple Lie algebras into irreducible components. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 160, pp. 211–221, 1987. The authors are grateful to L. D. Faddeev, N. A. Liskov, A. M. Vershik, and M. A. Semenov-Tyan-Shanskii for interesting discussions and for support.     

Construction and classification of complex simple Lie algebras via projective geometry

We construct the complex simple Lie algebras using elementary algebraic geometry. We use our construction to obtain a new proof of the classification of complex simple Lie algebras that does not appeal to the classification of root systems.     

Representation of Hilbert algebras and implicative semilattices

In this paper we shall give a topological representation for Hilbert algebras that extend the topological representation given by A. Diego in [4]. For implicative semilattices this representation gives a full duality. We shall also consider the representation for Boolean ring.     

Nondegenerate cohomology pairing for transitive Lie algebroids,characterization

The Evens-Lu-Weinstein representation (Q _A , D) for a Lie algebroid A on a manifold M is studied in the transitive case. To consider at the same time non-oriented manifolds as well, this representation is slightly modified to (Q _A ^or , D^or) by tensoring by orientation flat line bundle, Q _A ^or =Q_A⊗or (M) and D ^or=D⊗∂ _A ^or . It is shown that the induced cohomology pairing is nondegenerate and that the representation (Q _A ^or , D^or) is the unique (up to isomorphy) line representation for which the top group of compactly supported cohomology is nontrivial. In the case of trivial Lie algebroid A=TM the theorem reduce to the following: the orientation flat bundle (or (M), ∂ _A ^or ) is the unique (up to isomorphy) flat line bundle (ξ, ∇) for which the twisted de Rham complex of compactly supported differential forms on M with values in ξ possesses the nontrivial cohomology group in the top dimension. Finally it is obtained the characterization of transitive Lie algebroids for which the Lie algebroid cohomology with trivial coefficients (or with coefficients in the orientation flat line bundle) gives Poincaré duality. In proofs of these theorems for Lie algebroids it is used the Hochschild-Serre spectral sequence and it is shown the general fact concerning pairings between graded filtered differential ℝ-vector spaces: assuming that the second terms live in the finite rectangular, nondegeneration of the pairing for the second terms (which can be infinite dimensional) implies the same for cohomology spaces.     

Isotropy of non-nilpotent Riemannian solvable Lie groups

We describe a family of non-nilpotent Riemannian solvable Lie groups whose isotropy group has a prescribed compact Lie algebra.     

Flat orbits and kernels of irreducible representations of the group algebra of a completely solvable Lie group

We show that the kernel of an irreducible unitary representation π of the group algebra L¹(G) of a completely solvable Lie group G is given by the functions, whose abelian Fourier transform vanish on the Kirillov orbit Oπ of π if and only if this orbit Oπ is flat. This is a generalization of a result obtained before for nilpotent Lie groups.     

Cartan matrices of selfinjective algebras of tubular type

The Cartan matrix of a finite dimensional algebra A is an important combinatorial invariant reflecting frequently structural properties of the algebra and its module category. For example, one of the important features of the modular representation theory of finite groups is the nonsingularity of Cartan matrices of the associated group algebras (Brauer’s theorem). Recently, the class of all tame selfinjective algebras having simply connected Galois coverings and the stable Auslander-Reiten quiver consisting only of stable tubes has been shown to be the class of selfinjective algebras of tubular type, that is, the orbit algebras /G of the repetitive algebras of tubular algebras B with respect to the actions of admissible groups G of automorphisms of . The aim of the paper is to describe the determinants of the Cartan matrices of selfinjective algebras of tubular type and derive some consequences.     

Truncated Lie groups and almost Klein models

We consider a real analytic dynamical system G×M→M with nonempty fixed point subset M ^G . Using symmetries of G×M→M, we give some conditions which imply the existence of transitive Lie transformation group with G as isotropy subgroup.     

The Frattini Subalgebra of Restricted Lie Superalgebras

In the present paper, we study the Frattini subalgebra of a restricted Lie superalgebra （L, [p]）. We show first that if L = A1 ＋ A2 ＋… ＋An, then Фp（L） = Фp（A1） ＋Фp（A2） ＋…＋Фp（An）, where each Ai is a p-ideal of L. We then obtain two results： F（L） = Ф（L） = J（L） = L if and only if L is nilpotent; Fp（L） and F（L） are nilpotent ideals of L if L is solvable. In addition, necessary and sufficient conditions are found for Фp-free restricted Lie superalgebras. Finally, we discuss the relationships of E-p-restricted Lie superalgebras and E-restricted Lie superalgebras.     

Lie algebras and Lie super algebra for the integrable couplings of NLS–MKdV hierarchy

Lie algebras and Lie super algebra are constructed and integrable couplings of NLS–MKdV hierarchy are obtained. Furthermore, its Hamiltonian and Super-Hamiltonian are presented by using of quadric-form identity and super-trace identity. The method can be used to produce the Hamiltonian structures of the other integrable and super-integrable systems.     

The closure diagram for nilpotent orbits of the split real form of E8

Let and be adjoint nilpotent orbits in a real semisimple Lie algebra. Write ≥ if is contained in the closure of . This defines a partial order on the set of such orbits, known as the closure ordering. We determine this order for the split real form of the simple complex Lie algebra, E ₈. The proof is based on the fact that the Kostant-Sekiguchi correspondence preserves the closure ordering. We also present a comprehensive list of simple representatives of these orbits, and list the irreeducible components of the boundaries and of the intersections .     

On the duality of generalized Lie and Hopf algebras

We show how, under certain conditions, an adjoint pair of braided monoidal functors can be lifted to an adjoint pair between categories of Hopf algebras. This leads us to an abstract version of Michaelis' theorem, stating that given a Hopf algebra H  , there is a natural isomorphism of Lie algebras Q(H)^?≅P(H^°)$Q{(H)}^{?}\cong P(H^{°})$, where Q(H)^?$Q{(H)}^{?}$ is the dual Lie algebra of the Lie coalgebra of indecomposables of H  , and P(H^°)$P(H^{°})$ is the Lie algebra of primitive elements of the Sweedler dual of H. We apply our theory to Turaev's Hopf group-(co)algebras.     

Representation Rings of Lie Superalgebras

Given a Lie superalgebra , we introduce several variants of the representation ring, built as subrings and quotients of the ring of virtual -supermodules, up to (even) isomorphisms. In particular, we consider the ideal of virtual -supermodules isomorphic to their own parity reversals, as well as an equivariant K-theoretic super representation ring on which the parity reversal operator takes the class of a virtual -supermodule to its negative. We also construct representation groups built from ungraded -modules, as well as degree-shifted representation groups using Clifford modules. The full super representation ring , including all degree shifts, is then a -graded ring in the complex case and a -graded ring in the real case. Our primary result is a six-term periodic exact sequence relating the rings , and . We first establish a version of it working over an arbitrary (not necessarily algebraically closed) field of characteristic 0. In the complex case, this six-term periodic long exact sequence splits into two three-term sequences, which gives us additional insight into the structure of the complex super representation ring . In the real case, we obtain the expected 24-term version, as well as a surprising six-term version, of this periodic exact sequence. (Received: October 2004)     

On the semigroup nilpotency and the Lie nilpotency of associative algebras

To each associative ringR we can assign the adjoint Lie ringR ⁽⁻⁾ (with the operation(a,b)=ab−ba) and two semigroups, the multiplicative semigroupM(R) and the associated semigroupA(R) (with the operationaob=ab+a+b). It is clear that a Lie ringR ⁽⁻⁾ is commutative if and only if the semigroupM(R) (orA(R)) is commutative. In the present paper we try to generalize this observation to the case in whichR ⁽⁻⁾ is a nilpotent Lie ring. It is proved that ifR is an associative algebra with identity element over an infinite fieldF, then the algebraR ⁽⁻⁾ is nilpotent of lengthc if and only if the semigroupM(R) (orA(R)) is nilpotent of lengthc (in the sense of A. I. Mal'tsev or B. Neumann and T. Taylor). For the case in whichR is an algebra without identity element overF, this assertion remains valid forA(R), but fails forM(R). Another similar results are obtained. Translated fromMatematicheskie Zametki, Vol. 62, No. 4, pp. 510–519, October, 1997. Translated by A. I. Shtern