Howe duality and combinatorial character formula for orthosymplectic Lie superalgebras

We study the Howe dualities involving the reductive dual pairs (O(d),spo(2m|2n)) and (Sp(d),osp(2m|2n)) on the (super)symmetric tensor of . We obtain complete decompositions of this space with respect to their respective joint actions. We also use these dualities to derive a character formula for these irreducible representations of spo(2m|2n) and osp(2m|2n) that appear in these decompositions.     

Analysis and algorithms for the computation of the excited states of helium

In this paper, we study a dimensionally scaled helium atom model for excited states of helium. The mathematical analysis of the corresponding effective energy potential is presented. Two simple numerical algorithms are developed for the computation of the excited states of helium. Comparison between our numerical results and those in the existing literature is given to indicate the accuracy and efficiency of the proposed algorithms.     

Representations of centrally-extended Lie algebras over differential operators and vertex algebras

We construct irreducible modules of centrally-extended classical Lie algebras over left ideals of the algebra of differential operators on the circle, through certain irreducible modules of centrally-extended classical Lie algebras of infinite matrices with finite number of nonzero entries. The structures of vertex algebras associated with the vacuum representations of these algebras are determined. Moreover, we prove that under certain conditions, the highest-weight irreducible modules of centrally-extended classical Lie algebras of infinite matrices with finite number of nonzero entries naturally give rise to the irreducible modules of the simple quotients of these vertex algebras. From vertex algebra and its representation point of view, our results with positive integral central charge are high-order differential operator analogues of the well-known WZW models in conformal field theory associated with affine Kac-Moody algebras. Indeed, when the left ideals are the algebra of differential operators, our Lie algebras do contain affine Kac-Moody algebras as subalgebras and our results restricted on them are exactly the representation contents in WZW models. Similar results with negative central charge are also obtained.     

Current superalgebras and unitary representations

In this paper we determine the projective unitary representations of finite dimensional Lie supergroups whose underlying Lie superalgebra is $\mathfrak{g}=A?\mathfrak{k}$, where $\mathfrak{k}$ is a compact simple Lie superalgebra and A is a supercommutative associative (super)algebra; the crucial case is when $A={\mathrm{\Lambda }}_s(\mathbb{R})$ is a Graßmann algebra. Since we are interested in projective representations, the first step consists in determining the cocycles defining the corresponding central extensions. Our second main result asserts that, if $\mathfrak{k}$ is a simple compact Lie superalgebra with ${\mathfrak{k}}_1\ne \{0\}$, then each (projective) unitary representation of ${\mathrm{\Lambda }}_s(\mathbb{R})?\mathfrak{k}$ factors through a (projective) unitary representation of $\mathfrak{k}$ itself, and these are known by Jakobsen's classification. If ${\mathfrak{k}}_1=\{0\}$, then we likewise reduce the classification problem to semidirect products of compact Lie groups K with a Clifford–Lie supergroup which has been studied by Carmeli, Cassinelli, Toigo and Varadarajan.     

Diagram automorphisms of quiver varieties

We show that the fixed-point subvariety of a Nakajima quiver variety under a diagram automorphism is a disconnected union of quiver varieties for the ‘split-quotient quiver’ introduced by Reiten and Riedtmann. As a special case, quiver varieties of type D arise as the connected components of fixed-point subvarieties of diagram involutions of quiver varieties of type A. In the case where the quiver varieties of type A correspond to small self-dual representations, we show that the diagram involutions coincide with classical involutions of two-row Slodowy varieties. It follows that certain quiver varieties of type D are isomorphic to Slodowy varieties for orthogonal or symplectic Lie algebras.     

Conformal nets associated with lattices and their orbifolds

In this paper, we study representations of conformal nets associated with positive definite even lattices and their orbifolds with respect to isometries of the lattices. Using previous general results on orbifolds, we give a list of all irreducible representations of the orbifolds, which generate a unitary modular tensor category.     

Weak approximation on del Pezzo surfaces of degree 1

We study del Pezzo surfaces of degree 1 of the form

w²=z³+Ax⁶+By⁶     

Enclosing all zeros of an analytic function — A rigorous approach

We present a method to find all zeros of an analytic function in a rectangular domain. The approach is based on finding guaranteed enclosures rather than approximations of the zeros. Well-isolated simple zeros are determined fast and with high accuracy. Clusters of zeros can in many cases be distinguished from multiple zeros by applying the argument principle to sufficiently high-order derivatives of the function. We illustrate the proposed method through five examples of varying levels of complexity.     

On transitive Lie bialgebroids and Poisson groupoids

We prove that, for any transitive Lie bialgebroid (A, A^∗), the differential associated to the Lie algebroid structure on A^∗ has the form d_∗=A[Λ,⋅]+Ω, where Λ is a section of ^∧2A and Ω is a Lie algebroid 1-cocycle for the adjoint representation of A. Globally, for any transitive Poisson groupoid (Γ,Π), the Poisson structure has the form , where ΠF is a bivector field on Γ associated to a Lie groupoid 1-cocycle.     

Algebraic identity for the Schouten tensor and bi-Hamiltonian systems

A (0,3)-tensor Tijk is introduced in an invariant form. Algebraic identities are derived that connect the Schouten (2,1)-tensor and tensor Tijk with the Nijenhuis tensor . Applications to the bi-Hamiltonian dynamical systems are presented.     

Differential Galois theory and non-integrability of planar polynomial vector fields

We study a necessary condition for the integrability of the polynomials vector fields in the plane by means of the differential Galois Theory. More concretely, by means of the variational equations around a particular solution it is obtained a necessary condition for the existence of a rational first integral. The method is systematic starting with the first order variational equation. We illustrate this result with several families of examples. A key point is to check whether a suitable primitive is elementary or not. Using a theorem by Liouville, the problem is equivalent to the existence of a rational solution of a certain first order linear equation, the Risch equation. This is a classical problem studied by Risch in 1969, and the solution is given by the “Risch algorithm”. In this way we point out the connection of the non integrability with some higher transcendent functions, like the error function.     

Commutative quotients of finite W-algebras

In this paper we reduce the problem of 1-dimensional representations for the finite W-algebras and Humphreys' conjecture on small representations of reduced enveloping algebras to the case of rigid nilpotent elements in exceptional Lie algebras. We use Katsylo's results on sections of sheets to determine the Krull dimension of the largest commutative quotient of the finite W-algebra U(g,e).     

A k-tableau characterization of k-Schur functions

We study k-Schur functions characterized by k-tableaux, proving combinatorial properties such as a k-Pieri rule and a k-conjugation. This new approach relies on developing the theory of k-tableaux, and includes the introduction of a weight-permuting involution on these tableaux that generalizes the Bender-Knuth involution. This work lays the groundwork needed to prove that the set of k-Schur Littlewood-Richardson coefficients contains the 3-point Gromov-Witten invariants; structure constants for the quantum cohomology ring.     

Infinite-dimensional modular special odd contact superalgebras

The infinite-dimensional special odd contact Lie superalgebras SKO(n,n+1;c) over a field of positive characteristic are studied. In particular, we prove that Lie superalgebras SKO(n,n+1;c) are simple. Besides, the Weisfeiler filtration of SKO(n,n+1;c) is proved to be invariant under automorphisms by characterizing -nilpotent elements. Thereby, we obtain some properties of these Lie superalgebras.     

Algebraic identities for the Nijenhuis tensors

The general algebraic identities are discovered for the Nijenhuis and Haantjes tensors on an arbitrary manifold Mn. For n=3, the special algebraic identities involving the symmetric bilinear form H(u,v) are derived.     

Formality for the nilpotent cone and a derived Springer correspondence

Recall that the Springer correspondence relates representations of the Weyl group to perverse sheaves on the nilpotent cone. We explain how to extend this to an equivalence between the triangulated category generated by the Springer perverse sheaf and the derived category of differential graded modules over a dg-ring related to the Weyl group.     

Evolution of conformal maps with quasiconformal extensions

We study one-parameter curves on the universal Teichmüller space T and on the homogeneous space M=DiffS¹/RotS¹ embedded into T. As a result, we deduce evolution equations for conformal maps that admit quasiconformal extensions and, in particular, such that the associated quasidisks are bounded by smooth Jordan curves. This approach allows us to understand the Laplacian growth (Hele-Shaw problem) as a flow in the Teichmüller space.     

Discrete Tomography and plane partitions

A plane partition   is a p×q$p\times q$ matrix A=(_aij)$A=(a_{ij})$, where 1?i?p$1?i?p$ and 1?j?q$1?j?q$, with non-negative integer entries, and whose rows and columns are weakly decreasing. From a geometric point of view plane partitions are equivalent to pyramids  , subsets of the integer lattice Z³${\mathbb{Z}}^3$ which play an important role in Discrete Tomography. As a consequence, some typical problems concerning the tomography of discrete lattice sets can be rephrased and considered via plane partitions. In this paper we focus on some of them. In particular, we get a necessary and sufficient condition for additivity, a canonical procedure for checking the existence of (weakly) bad configurations, and an algorithm which constructs minimal pyramids (with respect to the number of levels) with assigned projection of a bad configurations.     

Representations of twisted current algebras

We use evaluation representations to give a complete classification of the finite-dimensional simple modules of twisted current algebras. This generalizes and unifies recent work on multiloop algebras, current algebras, equivariant map algebras, and twisted forms.