Trilinear Operators Research Articles

On G-derivations of Lie–Yamaguti algebras

This paper primarily deals with the study of [Formula: see text]-derivations associated with Lie–Yamaguti algebras (LY-algebras). Taking [Formula: see text] as an automorphism group, the concept of [Formula: see text]-derivations, which is a derivation under both the bilinear and trilinear operations is defined for LY-algebras. Then some important properties of [Formula: see text]-derivations are studied along with their relationship with other generalized derivations of LY-algebras.

Extensions and tangent prolongations of differentiable loops

The tangent Akivis algebra and Sabinin algebra of degree 3 of a differentiable loop is the tangential object determined by the third-order Taylor polynomial of the multiplication function of the loop. It is endowed with a bilinear skew-symmetric and a trilinear operation defined by the infinitesimal commutator and associator of the loop. The aim of our work is to study tangent algebras of degree 3 of abelian extensions of differentiable loops, which are affine extensions of the tangent algebras of the loop by abelian algebras. This class of loop extensions has previously been studied in terms of computational complexity and in terms of universal algebra. We apply the obtained results to the determination of tangent algebras of degree 3 of tangent prolongation of differentiable loops.

Confinement on the moose lattice

In this work we present a new class of mathcal{N} = 1 supersymmetric confining gauge theories, with strikingly simple infrared theories that descend from intricate interconnected networks of product gauge groups. A diagram of the gauge groups and the charged matter content of the ultraviolet theory has the structure of a triangular lattice, with SU(N) or SU(3N) gauge groups at each of the vertices, connected by bifundamental chiral superfields. This structure admits a U(1)R conserving superpotential with marginal (trilinear) operators. With the introduction of this superpotential, the SU(3N) and SU(N) gauge groups confine: in the far infrared limit of the supersymmetric theory, the relevant degrees of freedom are gauge invariant “mesons” and “baryons.” In this paper we show how the properties of the infrared degrees of freedom depend on the topology and shape of the moose/quiver “lattice” of the original gauge theory. Between the local structure of the moose lattice at higher energies, and the boundary dependence that emerges in the infrared, the 4d moose lattice theory reproduces features of a 6d spacetime with two compact dimensions. We investigate various deformations of the theory, and propose some phenomenological applications for BSM models.

Poisson triple systems

We introduce Poisson triple systems, which are vector spaces with 3 trilinear operations satisfying 9 polynomial identities of degree 5. We show that every Poisson triple system has a universal enveloping Poisson algebra. Finally, we briefly discuss operadic aspects of Poisson triple systems.

Boundedness of some multi-parameter fiber-wise multiplier operators

We prove $L^p$ estimates for various multi-parameter bi- and trilinear operators with symbols acting on fibers of the two-dimensional functions. In particular, this yields estimates for the general bi-parameter form of the twisted paraproduct studied in arXiv:1011.6140.

Endpoint estimates for a trilinear pseudo-differential operator with flag symbols

Abstract In this paper, we establish the endpoint estimate ( 0 < p ≤ 1 {0<p\leq 1} ) for a trilinear pseudo-differential operator, where the symbol involved is given by the product of two standard symbols from the bilinear Hörmander class B ⁢ S 1 , 0 0 {BS^{0}_{1,0}} . The study of this operator is motivated from the L p {L^{p}} ( 1 < p < ∞ {1<p<\infty} ) estimates for the trilinear Fourier multiplier operator with flag singularities considered in [C. Muscalu, Paraproducts with flag singularities. I. A case study, Rev. Mat. Iberoam. 23 2007, 2, 705–742] and Hardy space estimates in [A. Miyachi and N. Tomita, Estimates for trilinear flag paraproducts on L ∞ L^{\infty} and Hardy spaces, Math. Z. 282 2016, 1–2, 577–613], and the L p {L^{p}} ( 1 < p < ∞ {1<p<\infty} ) estimates for the trilinear pseudo-differential operator with flag symbols in [G. Lu and L. Zhang, L p L^{p} -estimates for a trilinear pseudo-differential operator with flag symbols, Indiana Univ. Math. J. 66 2017, 3, 877–900]. More precisely, we will show that the trilinear pseudo-differential operator with flag symbols defined in (1.3) maps from the product of local Hardy spaces to the Lebesgue space, i.e., h p 1 × h p 2 × h p 3 → L p {h^{p_{1}}\times h^{p_{2}}\times h^{p_{3}}\rightarrow L^{p}} with 1 p 1 + 1 p 2 + 1 p 3 = 1 p {\frac{1}{p_{1}}+\frac{1}{p_{2}}+\frac{1}{p_{3}}=\frac{1}{p}} with 0 < p < ∞ {0<p<\infty} (see Theorem 1.6 and Theorem 1.7).

A trilinear analysis for lump-type wave, breather wave and BK-type wave solutions of a (3+1)-dimensional [formula omitted]-gKP equation

This paper investigates some novel exact analytical solutions, including lump-type wave, breather wave as well as BK-type wave solutions for a (3+1)-dimensional p¯-gKP equation. Firstly, the trilinear p¯-gKP equation is well established by the multivariate trilinear operators. Secondly, three types of wave solutions for the p¯-gKP equation are obtained by utilizing the trilinear method. A kind of new interaction solutions among the breather wave and the kink-type waves are also generated, i.e. the so called BK-type waves, which enriches the types of solutions of PDEs. Through the way of resetting different space constants, we adjust the coordinates of kink-type waves in order for them colliding with the breather wave, after that, the transformed kink-type waves gradually swallow the breather wave. Finally, the solitary wave solutions for the (3+1)-dimensional gKP equation are successfully constructed through using a perturbation method.

Cohomology of the vector fields Lie algebras on acting on trilinear differential operators, vanishing on

The main topic of this paper is to compute the first relative cohomology group of the Lie algebra of smooth vector fields with coefficients in the space of trilinear differential operators that acts on tensor densities, vanishing on the Lie algebra where and

Lump-type solution and breather lump–kink interaction phenomena to a ($$\mathbf{{3{\varvec{+}}1}}$$)-dimensional GBK equation based on trilinear form

In this paper, the multivariate trilinear operators in the ($$3+1$$)-dimensional space are applied to a ($$3+1$$)-dimensional GBK equation. The resulting trilinear form is used to study its wave dynamics. Particularly, we generate a type of new interaction solutions between breather lump-type solitons and other multi-kink solitons, thereby formulating a kind of breather lump–kink solitons. By setting time constants, we change the coordinates of kink solitons to make them collide with the breather lump-type soliton, during which breather lump-type soliton is swallowed eventually by those kink solitons. The evolution behaviours of the breather lump–kink solitons are depicted by plotting 3-D and density graphs from the perspective of wave characteristics.

Representations of the comtrans algebra of n × n matrices

We show that the universal enveloping algebra of the comtrans algebra of all n × n (n > 2) matrices over a field of characteristic 0 with respect to the trilinear operations: (the special commutator) and (the special translator) is finite-dimensional. We explicitly determine the decomposition of the universal envelope into matrix algebras. We also show that the universal enveloping algebra of a comtrans algebra is not necessary be finite-dimensional.

Tr(F3) supersymmetric form factors and maximal transcendentality. Part II. 0 < mathcal{N} < 4 super Yang-Mills

The study of form factors has many phenomenologically interesting applications, one of which is Higgs plus gluon amplitudes in QCD. Through effective field theory techniques these are related to form factors of various operators of increasing classical dimension. In this paper we extend our analysis of the first finite top-mass correction, arising from the operator Tr(F3), from mathcal{N} = 4 super Yang-Mills to theories with mathcal{N} < 4, for the case of three gluons and up to two loops. We confirm our earlier result that the maximally transcendental part of the associated Catani remainder is universal and equal to that of the form factor of a protected trilinear operator in the maximally supersymmetric theory. The terms with lower transcendentality deviate from the mathcal{N} = 4 answer by a surprisingly small set of terms involving for example ζ2, ζ3 and simple powers of logarithms, for which we provide explicit expressions.

Special identities for comtrans algebras

ABSTRACT Comtrans algebras, arising in web geometry, have two trilinear operations, commutator and translator. We determine a Gröbner basis for the comtrans operad, and state a conjecture on its dimension formula. We study multilinear polynomial identities for the special commutator and special translator in associative triple systems. In degree 3, the defining identities for comtrans algebras generate all identities. In degree 5, we simplify known identities for each operation and determine new identities relating the operations. In degree 7, we use representation theory of the symmetric group to show that each operation satisfies identities which do not follow from those of lower degree but there are no new identities relating the operations. We use noncommutative Gröbner bases to construct the universal associative envelope for the special comtrans algebra of matrices.

Trilinear Lp estimates with applications to the Cauchy problem for the Hartree-type equation

In this paper, Lp estimates for a trilinear operator associated with the Hartree type nonlinearity are proved. Moreover, as application of these estimates, it is proved that after a linear transformation, the Cauchy problem for the Hartree-type equation becomes locally well posed in the Bessel potential and homogeneous Besov spaces under certain regularity assumptions on the initial data. This notion of well-posedness and the functional framework to solve the equation were firstly proposed by Y. Zhou.

Quasi Pieces of the Bilinear Hilbert Transform Incorporated into a Paraproduct

We prove the boundedness of a class of tri-linear operators consisting of a quasi piece of bilinear Hilbert transform whose scale equals to or dominates the scale of its linear counter part. Such type of operators is motivated by the tri-linear Hilbert transform and its curved versions.

A weak Galerkin finite element method for the Navier–Stokes equations

In this paper, we propose and analyze a weak Galerkin finite element method for the Navier–Stokes equations. The new formulation hinges upon the introduction of weak gradient, weak divergence and weak trilinear operators. Moreover, by choosing the matching finite element triples, this new method not only obtains stability and optimal error estimates but also has a lot of attractive computational features: general finite element partitions of arbitrary polygons or polyhedra with certain shape regularity and parameter free. Finally, several numerical experiments assess the convergence properties of the new method and show its computational advantages.

Higgs Amplitudes from N=4 Supersymmetric Yang-Mills Theory.

Higgs plus multigluon amplitudes in QCD can be computed in an effective Lagrangian description. In the infinite top-mass limit, an amplitude with a Higgs boson and n gluons is computed by the form factor of the operator TrF^{2}. Up to two loops and for three gluons, its maximally transcendental part is captured entirely by the form factor of the protected stress tensor multiplet operator T_{2} in N=4 supersymmetric Yang-Mills theory. The next order correction involves the calculation of the form factor of the higher-dimensional, trilinear operator TrF^{3}. We present explicit results at two loops for three gluons, including the subleading transcendental terms derived from a particular descendant of the Konishi operator that contains TrF^{3}. These are expressed in terms of a few universal building blocks already identified in earlier calculations. We show that the maximally transcendental part of this quantity, computed in nonsupersymmetric Yang-Mills theory, is identical to the form factor of another protected operator, T_{3}, in the maximally supersymmetric theory. Our results suggest that the maximally transcendental part of Higgs amplitudes in QCD can be entirely computed through N=4 super Yang-Mills theory.

Cohomology of the vector fields Lie algebras on ℝ acting on trilinear differential operators, vanishing on 𝔞𝔣𝔣(1)

The main topic of this paper is to compute the first relative cohomology group of the Lie algebra of smooth vector fields [Formula: see text], with coefficients in the space of trilinear differential operators that act on tensor densities, [Formula: see text], vanishing on the Lie algebra [Formula: see text].

Iterated trilinear Fourier integrals with arbitrary symbols

We prove Lp estimates for trilinear multiplier operators with singular symbols. These operators arise in the study of iterated trilinear Fourier integrals, which are trilinear variants of the bilinear Hilbert transform. Specifically, we consider trilinear operators determined by multipliers that are products of two functions m1(ξ1,ξ2) and m2(ξ2,ξ3), such that the singular set of m1 lies in the hyperplane ξ1=ξ2 and that of m2 lies in the hyperplane ξ2=ξ3. While previous work [8] requires that the multipliers satisfy χξ1<ξ2⋅χξ2<ξ3, our results allow for the case of the arbitrary multipliers, which have common singularities.

On a hybrid of bilinear Hilbert transform and paraproduct

We introduce a class of tri-linear operators that combine features of the bilinear Hilbert transform and paraproduct. For two instances of these operators, we prove boundedness property in a large range \(D = \left\{ {\left( {{p_1},{p_2},{p_3}} \right) \in {\mathbb{R}^3}:1 < {p_1},{p_2} < \infty ,\frac{1}{{{p_1}}} + \frac{1}{{{p_2}}} < \frac{3}{2},1 < {p_3} < \infty } \right\}\).

L^p estimates for a trilinear pseudo-differential operator with flag symbols

L^p estimates for a trilinear pseudo-differential operator with flag symbols