Trivial Coefficients Research Articles

Equivariant Fredholm modules for the full quantum flag manifold of $\mathrm{SU}_q(3)$

We introduce C^* -algebras associated to the foliation structure of a quantum flag manifold. We use these to construct {SL}_q(3,{C}) -equivariant Fredholm modules for the full quantum flag manifold X_q = {SU}_q(3)/T of {SU}_q(3) , based on an analytical version of the Bernstein-Gelfand-Gelfand complex. As a consequence we deduce that the flag manifold X_q satisfies Poincaré duality in equivariant KK -theory. Moreover, we show that the Baum-Connes conjecture with trivial coefficients holds for the discrete quantum group dual to {SU}_q(3) .

Cohomology of the Lie algebra ℌ2: Experimental results and conjectures

The cohomology with trivial coefficients of the Lie algebra ℌ of Hamiltonian vector fields in the plane and of its maximal nilpotent subalgebra L 1ℌ is considered. The cohomology H 2(L 1ℌ) is calculated, and some far-reaching conjectures concerning the cohomology of the Lie algebras mentioned above and based on an extensive experimental material are formulated.

On a p-adic invariant cycles theorem

Abstract For a proper semistable curve X over a DVR of mixed characteristics and perfect residue field we prove the “invariant cycles theorem” with trivial coefficients, i.e. that the group of elements of the first de Rham cohomology group of the generic fiber of X annihilated by the monodromy operator coincides with the first rigid cohomology group of its special fiber. This is done using an explicit description of the monodromy operator on the de Rham cohomology of the generic fiber of X with coefficients a convergent F-isocrystal (see [Représentations p-adiques de groupes p-adiques III: Méthodes globales et géométriques, Astérisque 331, 179–254]) and the proof exhibits an interesting interplay between this cohomology group and the combinatorics of the graph of the reduction of X. The result was proved in a different way in the case the DVR has finite residue field in [Duke Math. J. 97 (1999), no. 1, 155–169]. We also study the case where the coefficients are unipotent convergent F-isocrystals on the special fiber of X (without log-structure): we show that the invariant cycles theorem does not hold in general in this setting and give a sufficient condition for non-exactness.

Efficient Minimum Error Bounded Particle Resampling L1 Tracker With Occlusion Detection

Recently, sparse representation has been applied to visual tracking to find the target with the minimum reconstruction error from a target template subspace. Though effective, these L1 trackers require high computational costs due to numerous calculations for l1 minimization. In addition, the inherent occlusion insensitivity of the l1 minimization has not been fully characterized. In this paper, we propose an efficient L1 tracker, named bounded particle resampling (BPR)-L1 tracker, with a minimum error bound and occlusion detection. First, the minimum error bound is calculated from a linear least squares equation and serves as a guide for particle resampling in a particle filter (PF) framework. Most of the insignificant samples are removed before solving the computationally expensive l1 minimization in a two-step testing. The first step, named τ testing, compares the sample observation likelihood to an ordered set of thresholds to remove insignificant samples without loss of resampling precision. The second step, named max testing, identifies the largest sample probability relative to the target to further remove insignificant samples without altering the tracking result of the current frame. Though sacrificing minimal precision during resampling, max testing achieves significant speed up on top of τ testing. The BPR-L1 technique can also be beneficial to other trackers that have minimum error bounds in a PF framework, especially for trackers based on sparse representations. After the error-bound calculation, BPR-L1 performs occlusion detection by investigating the trivial coefficients in the l1 minimization. These coefficients, by design, contain rich information about image corruptions, including occlusion. Detected occlusions are then used to enhance the template updating. For evaluation, we conduct experiments on three video applications: biometrics (head movement, hand holding object, singers on stage), pedestrians (urban travel, hallway monitoring), and cars in traffic (wide area motion imagery, ground-mounted perspectives). The proposed BPR-L1 method demonstrates an excellent performance as compared with nine state-of-the-art trackers on eleven challenging benchmark sequences.

Trivial module for ortho-symplectic Lie superalgebras and Littlewood’s formula

We show that the denominator identity for ortho-symplectic Lie superalgebras osp(k|2n) is equivalent to the Littlewood’s formula. Such an equivalence also implies the relation between the trivial module and generalized Verma modules for osp(k|2n). Furthermore, we discuss the harmonic representative elements of the Kostant’s u-cohomology with trivial coefficients.

Homotopy transfer and self-dual Schur modules

We consider the free 2-nilpotent graded Lie algebra $$\mathfrak{g}$$ generated in degree one by a finite dimensional vector space V. We recall the beautiful result that the cohomology $$H^ \cdot \left( {\mathfrak{g},\mathbb{K}} \right)$$ of $$\mathfrak{g}$$ with trivial coefficients carries a GL(V)-representation having only the Schur modules V with self-dual Young diagrams {λ: λ = λ′} in its decomposition into GL(V)-irreducibles (each with multiplicity one). The homotopy transfer theorem due to Tornike Kadeishvili allows to equip the cohomology of the Lie algebra g with a structure of homotopy commutative algebra.

Hecke Operators and the Stable Homology of GL(n)

Let R be a field of any characteristic and A a principal ideal domain. We make a conjecture that asserts that any Hecke operator T acts punctually on any Hecke eigenclass in the stable homology with trivial coefficients R of a principal congruence subgroup Γ in GL(n, A), i.e., as multiplication by the number of single cosets contained in T. In the case where A = ℤ, this conjecture implies that the Galois representation consisting of the sum of the 0th, 1st, ⋅, n − 1st powers of the cyclotomic character is attached to any stable Hecke eigenclass. When A = ℤ and Γ =GL(n, ℤ), this conjecture was already made by Calegari and Venkatesh. We obtain partial results giving evidence for the conjecture. These results imply that some part of the Hecke algebra acts punctually. If the characteristic of R is 0, they show that the entire Hecke algebra acts punctually, which was already known in a completely different way using a result of A. Borel.

Decompositions of binomial ideals

The package Binomials contains implementations of specialized algorithms for binomial ideals, including primary decomposition into binomial ideals. The current implementation works in characteristic zero. Primary decomposition is restricted to binomial ideals with trivial coefficients to avoid computations over the algebraic numbers. The basic ideas of the algorithms go back to Eisenbud and Sturmfels' seminal paper on the subject. Two recent improvements of the algorithms are discussed and examples are presented.

Mass Transfer Resistance Analysis and Semipermeable Membrane Optimization for Pressure Retarded Osmosis

Salinity power is a promising renewable energy with a large amount, which can be tapped by pressure retarded osmosis(PRO), however, efficient membranes are in lack. It was reported subtrates of asymmetrical reverse osmosis(RO) membranes had severe concentration polarizations in PRO process, degrading performance greatly and limiting PRO application. Based on solute transportation equations, the paper presented a common mass transfer resistance model of osmosis process, and disclosed essential differences between PRO and RO by the comparison of resistances. In PRO water permeates through membranes against solute diffusion, which facilitates solute to accumulate in porous subtrate, as a result, subtrate mass transfer resistance is big, water flux and power density is small, especially in the case of trivial subtrate mass transfer coefficient. Commercial RO membrane CA-3000 was studied for PRO application, whose subtrate resistance was found much bigger in PRO, taking the majority of total resistance. Subtrate was optimized for PRO process and membrane performance was projected. When subtrate mass transfer coefficient was improved to 4×10-6 m/s, power density of 4.75 W/m2 was obtained with average sea water, which approached the threshold of commercial salinity power exploitation.

Iterated bar complexes ofE–infinity algebras and homology theories

We proved in a previous article that the bar complex of an E-infinity algebra inherits a natural E-infinity algebra structure. As a consequence, a well-defined iterated bar construction B^n(A) can be associated to any algebra over an E-infinity operad. In the case of a commutative algebra A, our iterated bar construction reduces to the standard iterated bar complex of A. The first purpose of this paper is to give a direct effective definition of the iterated bar complexes of E-infinity algebras. We use this effective definition to prove that the n-fold bar complex B^n(A) admits an extension to categories of algebras over E_n-operads. Then we prove that the n-fold bar complex determines the homology theory associated to a category of algebras over E_n-operads. For n infinite, we obtain an isomorphism between the homology of an infinite bar construction and the usual Gamma-homology with trivial coefficients.

Filtering bases and cohomology of nilpotent subalgebras of the Witt algebra and the algebra of loops in sl 2

We study the cohomology with trivial coefficients of the Lie algebras Lk, k ≥ 1, of polynomial vector fields with zero k-jet on the circle and the cohomology of similar subalgebras ℒk of the algebra of polynomial loops with values in sl2 The main result is a construction of special bases in the exterior complexes of these algebras. Using this construction, we obtain the following results. We calculate the cohomology of Lk and ℒL. We obtain formulas in terms of Schur polynomials for cycles representing the homology of these algebras. We introduce “stable” filtrations of the exterior complexes of Lk and ℒk, thus generalizing Goncharova’s notion of stable cycles for Lk, and give a polynomial description of these filtrations. We find the spectral resolutions of the Laplace operators for L1 and ℒ1.

Knots with a Trivial Coefficient Polynomial

By using a tangle decomposition of a knot, we give a method for the construc- tion of a knot with the lowest trivial HOMFLY coefficient polynomial. Applying this, we show that there exist infinitely many 2-bridge knots with such a coefficient polynomial.

The $K(\pi,1)$ problem for the affine Artin group of type $\widetilde{B}_n$ and its cohomology

We prove that the complement to the affine complex arrangement of type \widetilde B_n is a K(π,1) space. We also compute the cohomology of the affine Artin group G_{\widetilde B_n} (of type \widetilde B_n ) with coefficients in interesting local systems. In particular, we consider the module ℚ[q^{±1} ,t^{±1}] , where the first n standard generators of G_{\widetilde B_n} act by (−q) -multiplication while the last generator acts by (−t) multiplication. Such a representation generalizes the analogous 1-parameter representation related to the bundle structure over the complement to the discriminant hypersurface, endowed with the monodromy action of the associated Milnor fibre. The cohomology of G_{\widetilde B_n} with trivial coefficients is derived from the previous one.

Higher Hopf formula for homology of Leibniz [formula omitted]-algebras

We fit the homology with trivial coefficients of Leibniz n -algebras into the context of Quillen homology and provide the Hopf type formula for the higher homology.

Weight reduction for mod ℓ Bianchi modular forms

Let K be an imaginary quadratic field with class number one and ℓ be a rational prime that splits in K. We prove that mod ℓ, a system of Hecke eigenvalues occurring in the first cohomology group of some congruence subgroup Γ of SL 2 ( O K ) can be realized, up to twist, in the first cohomology with trivial coefficients after increasing the level of Γ by ( ℓ).

Abelian ideals and cohomology of symplectic type

Let b be a Borel subalgebra of the symplectic Lie algebra sp (2n, C) and let n be the corresponding maximal nilpotent subalgebra. We find a connection between the abelian ideals of b and the cohomology of n with trivial coefficients. Using this connection, we are able to enumerate the number of abelian ideals of b with given dimension via the Poincare polynomials of Weyl groups of types A n- 1 and C n .

Homology with Trivial Coefficients and Universal Central Extensions of Algebras with Bracket

A 5-term exact sequence and the interpretation of low dimensional groups of homology with trivial coefficients of algebras with bracket are obtained. The endofunctor in the category of algebras with bracket which assigns to a perfect algebra with bracket its universal central extension is constructed and the characterization of the universal central extension by means of the low-dimensional homology groups is done. The conditions required to lift an automorphism or a derivation of A to A′ in a covering f:A′ ↠ A are analyzed.

Braids, mapping class groups, and categorical delooping

Dehn twists around simple closed curves in oriented surfaces satisfy the braid relations. This gives rise to a group theoretic map $$\phi : \beta_{2g} \to \Gamma _{g,1}$$ from the braid group to the mapping class group. We prove here that this map is trivial in homology with any trivial coefficients in degrees less than g/2. In particular this proves an old conjecture of J. Harer. The main tool is categorical delooping in the spirit of (Tillmann in Invent Math 130:257–175, 1997). By extending the homomorphism to a functor of monoidal 2-categories, $$\phi$$ is seen to induce a map of double loop spaces on the plus construction of the classifying spaces. Any such map is null-homotopic. In an appendix we show that geometrically defined homomorphisms from the braid group to the mapping class group behave similarly in stable homology.

Cohomology of 3-dimensional color Lie algebras

We develop the cohomology theory of color Lie algebras due to Scheunert–Zhang in a framework of non-homogeneous quadratic Koszul algebras. In this approach, the Chevalley–Eilenberg complex of a color Lie algebra becomes a standard Koszul complex for its universal enveloping algebra, providing a constructive method for computation of cohomology. As an application, we compute cohomologies with trivial coefficients of Z 2 n -graded 3-dimensional color Lie algebras.

On Solvability and Nilpotency of Leibniz n-Algebras

The concepts of solvable and nilpotent Leibniz n-algebra are introduced, and classical results of solvable and nilpotent Lie algebras theory are extended to Leibniz n-algebras category. A homological criterion similar to Stallings Theorem for Lie algebras is obtained in Leibniz n-algebras category by means of the homology with trivial coefficients of Leibniz n-algebras.