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Nilpotent slices, Hilbert schemes, and the Jones polynomial  [PDF]
Ciprian Manolescu
Mathematics , 2004,
Abstract: Seidel and Smith have constructed an invariant of links as the Floer cohomology for two Lagrangians inside a complex affine variety Y. This variety is the intersection of a semisimple orbit with a transverse slice at a nilpotent in the Lie algebra $sl_{2m}.$ We exhibit bijections between a set of generators for the Seidel-Smith cochain complex, the generators in Bigelow's picture of the Jones polynomial, and the generators of the Heegaard Floer cochain complex for the double branched cover. This is done by presenting Y as an open subset of the Hilbert scheme of a Milnor fiber.
Strictly nilpotent elements and bispectral operators in the Weyl algebra  [PDF]
Emil Horozov
Mathematics , 2002,
Abstract: In this paper we give another characterization of the strictly nilpotent elements in the Weyl algebra, which (apart from the polynomials) turn out to be all bispectral operators with polynomial coefficients. This also allows to reformulate in terms of bispectral operators the famous conjecture, that all the endomorphisms of the Weyl algebra are automorphisms (Dixmier, Kirillov, etc).
Rigidity of the álvarez classes of Riemannian foliations with nilpotent structure Lie algebras  [PDF]
Hiraku Nozawa
Mathematics , 2008,
Abstract: We show that if the structure algebra of a Riemannian foliation F on a closed manifold M is nilpotent, then the integral of the \'Alvarez class of (M,F) along every closed path is the exponential of an algebraic number. By this result and the continuity of the \'Alvarez class under deformations shown in arXiv:1009.1098v2, we prove that the \'Alvarez class and the geometrically tautness of Riemannian foliations on a closed manifold M are invariant under deformation, if the fundamental group of M has polynomial growth.
Geometric Quantization of Real Minimal Nilpotent Orbits  [PDF]
Ranee Brylinski
Mathematics , 1998,
Abstract: In this paper, we begin a quantization program for nilpotent orbits of a real semisimple Lie group. These orbits and their covers generalize the symplectic vector space. A complex structure polarizing the orbit and invariant under a maximal compact subgroup is provided by the Kronheimer-Vergne Kaehler structure. We outline a geometric program for quantizing the orbit with respect to this polarization. We work out this program in detail for minimal nilpotent orbits in the non-Hermitian case. The Hilbert space of quantization consists of holomorphic half-forms on the orbit. We construct the reproducing kernel. The Lie algebra acts by explicit pseudo-differential operators on half-forms where the energy operator quantizing the Hamiltonian is inverted. The Lie algebra representation exponentiates to give a minimal unitary ladder representation. Jordan algebras play a key role in the geometry and the quantization.
Hidden algebra of the $N$-body Calogero problem  [PDF]
Alexander Turbiner
Physics , 1993, DOI: 10.1016/0370-2693(94)90657-2
Abstract: A certain generalization of the algebra $gl(N,{\bf R})$ of first-order differential operators acting on a space of inhomogeneous polynomials in ${\bf R}^{N-1}$ is constructed. The generators of this (non)Lie algebra depend on permutation operators. It is shown that the Hamiltonian of the $N$-body Calogero model can be represented as a second-order polynomial in the generators of this algebra. Given representation implies that the Calogero Hamiltonian possesses infinitely-many, finite-dimensional invariant subspaces with explicit bases, which are closely related to the finite-dimensional representations of above algebra. This representation is an alternative to the standard representation of the Bargmann-Fock type in terms of creation and annihilation operators.
Nilpotent integrability, reduction of dynamical systems and a third-order Calogero-Moser system  [PDF]
A. Ibort,G. Marmo,M. A. Rodriguez,P. Tempesta
Physics , 2015,
Abstract: We present an algebraic formulation of the notion of integrability of dynamical systems, based on a nilpotency property of its flow: it can be explicitly described as a polynomial on its evolution parameter. Such a property is established in a purely geometric--algebraic language, in terms both of the algebra of all higher-order constants of the motion (named the nilpotent algebra of the dynamics), and of a maximal Abelian algebra of symmetries (called a Cartan subalgebra of the dynamics). It is shown that this notion of integrability amounts to the annihilator of the nilpotent algebra being contained in a Cartan subalgebra of the dynamics. Systems exhibiting this property will be said to be nilpotent integrable. Our notion of nilpotent integrability offers a new insight into the intrinsic dynamical properties of a system, which is independent of any auxiliary geometric structure defined on its phase space. At the same time, it extends in a natural way the classical concept of complete integrability for Hamiltonian systems. An algebraic reduction procedure valid for nilpotent integrable systems, generalizing the well-known reduction procedures for symplectic and/or Poisson systems on appropriate quotient spaces, is also discussed. In particular, it is shown that a large class of nilpotent integrable systems can be obtained by reduction of higher-order free systems. The case of the third-order free system is analyzed and a nontrivial set of third-order Calogero-Moser-like nilpotent integrable equations is obtained.
The global nilpotent variety is Lagrangian  [PDF]
Victor Ginzburg
Mathematics , 1997,
Abstract: The purpose of this note is to present a short elementary proof of a theorem due to Faltings and Laumon, saying that the global nilpotent cone is a Lagrangian substack in the cotangent bundle of the moduli space of G-bundles on a complex compact curve. This result plays a crucial role in the Geometric Langlands program, due to Beilinson-Drinfeld, since it insures that the D-modules on the moduli space of G-bundles whose characteristic variety is contained in the global nilpotent cone are automatically holonomic, hence, e.g. have finite length.
The $BC_{1}$ Elliptic model: algebraic forms, hidden algebra $sl(2)$, polynomial eigenfunctions  [PDF]
Alexander V. Turbiner
Physics , 2014, DOI: 10.1088/1751-8113/48/9/095205
Abstract: The potential of the $BC_1$ quantum elliptic model is a superposition of two Weierstrass functions with doubling of both periods (two coupling constants). The $BC_1$ elliptic model degenerates to $A_1$ elliptic model characterized by the Lam\'e Hamiltonian. It is shown that in the space of $BC_1$ elliptic invariant, the potential becomes a rational function, while the flat space metric becomes a polynomial. The model possesses the hidden $sl(2)$ algebra for arbitrary coupling constants: it is equivalent to $sl(2)$-quantum top in three different magnetic fields. It is shown that there exist three one-parametric families of coupling constants for which a finite number of polynomial eigenfunctions (up to a factor) occur.
Enveloping algebras of the nilpotent Malcev algebra of dimension five  [PDF]
Murray R. Bremner,Hamid Usefi
Mathematics , 2010,
Abstract: Perez-Izquierdo and Shestakov recently extended the PBW theorem to Malcev algebras. It follows from their construction that for any Malcev algebra $M$ over a field of characteristic $\ne 2, 3$ there is a representation of the universal nonassociative enveloping algebra $U(M)$ by linear operators on the polynomial algebra $P(M)$. For the nilpotent non-Lie Malcev algebra $\mathbb{M}$ of dimension 5, we use this representation to determine explicit structure constants for $U(\mathbb{M})$; from this it follows that $U(\mathbb{M})$ is not power-associative. We obtain a finite set of generators for the alternator ideal $I(\mathbb{M}) \subset U(\mathbb{M})$ and derive structure constants for the universal alternative enveloping algebra $A(\mathbb{M}) = U(\mathbb{M})/I(\mathbb{M})$, a new infinite dimensional alternative algebra. We verify that the map $\iota\colon \mathbb{M} \to A(\mathbb{M})$ is injective, and so $\mathbb{M}$ is special.
Nilpotent commuting varieties of the Witt algebra  [PDF]
Yu-Feng Yao,Hao Chang
Mathematics , 2013, DOI: 10.1016/j.jpaa.2014.02.004
Abstract: Let $\mathfrak{g}$ be the $p$-dimensional Witt algebra over an algebraically closed field $k$ of characteristic $p>3$. Let $\mathscr{N}={x\in\ggg\mid x^{[p]}=0}$ be the nilpotent variety of $\mathfrak{g}$, and $\mathscr{C}(\mathscr{N}):=\{(x,y)\in \mathscr{N}\times\mathscr{N}\mid [x,y]=0\}$ the nilpotent commuting variety of $\mathfrak{g}$. As an analogue of Premet's result in the case of classical Lie algebras [A. Premet, Nilpotent commuting varieties of reductive Lie algebras. Invent. Math., 154, 653-683, 2003.], we show that the variety $\mathscr{C}(\mathscr{N})$ is reducible and equidimensional. Irreducible components of $\mathscr{C}(\mathscr{N})$ and their dimension are precisely given. Furthermore, the nilpotent commuting varieties of Borel subalgebras are also determined.
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