Quadratic Identities Research Articles

Towards higher-spin holography in flat space

We study the chiral flat space higher-spin algebra, which is the global symmetry algebra of the chiral higher-spin theory in the 4d Minkowski space. We find that it can be constructed as the universal enveloping algebra of a certain chiral deformation of the Poincaré algebra quotiented by a set of quadratic identities. These identities allow us to identify a representation of the latter algebra, which by analogy with the AdS space higher-spin holography, we interpret as the flat space singleton representation. We provide two explicit realisations of this singleton representation — in terms of sl(2, ℂ) spinors and in terms of oscillator-like variables — as well as briefly discuss its properties.

Products of quadratic residues and related identities

We study products of quadratic residues modulo odd primes and prove some identities involving quadratic residues. For instance, let $p$ be an odd prime. We prove that if $p\equiv 5\,({\rm mod}\, 8)$, then $$\prod _{0 \lt x \lt p/2,\,(\frac {x}{p})=1}x\equ

Basic quadratic identities on quantum minors

This paper continues an earlier research of the authors on universal quadratic identities (QIs) on minors of quantum matrices. We demonstrate situations when the universal QIs are provided, in a sense, by the ones of four special types (Plűcker, co-Plűcker, Dodgson identities and quasi-commutation relations on flag and co-flag interval minors).

On ternary quasigroup quadratic identities of the small length

In this article, it has been proved that each quadratic identity of the lengths one, two, three is parastrophically primarily equivalent to at least one of the given identities. The identities of the length three have been analyzed in the class of universal loops, i.e., quasigroups in which every element is neutral. It has been proved that there are five non-equivalent identities. The first identity defines the class of all universal loops, the second one defines the variety of the boolean skeins and the other three identities define three parastrophic varieties whose operations are repetition-free compositions of binary commutative middle loops. Cite as: F. M. Sokhatsky, A. V. Tarasevych, “On ternary quasigroup quadratic identities of the small length,” Prykl. Probl. Mekh. Mat. , Issue 18, 150–161 (2020), https://doi.org/10.15407/apmm2020.18.150-161

A note on dual third-order Jacobsthal vectors

Dual third order Jacobsthal and dual third order Jacobsthal-Lucas numbers are defined. In this study, we work on these dual numbers and we obtain the properties e.g. some quadratic identities, summation, norm, negadual third order Jacobsthal identities, Binet formulas and relations of them. We also define new vectors which are called dual third order Jacobsthal vectors and dual third order Jacobsthal-Lucas vectors. We give properties of these vectors to exert in geometry of dual space.

Extensions of quadratic transformation identities for hypergeometric functions

Extensions of quadratic transformation identities for hypergeometric functions

Extensions of the classical transformations of the hypergeometric function [formula omitted

It is shown that the classical quadratic and cubic transformation identities satisfied by the hypergeometric function F23 can be extended to include additional parameter pairs, which differ by integers. In the extended identities, which involve hypergeometric functions of arbitrarily high order, the added parameters are nonlinearly constrained: in the quadratic case, they are the negated roots of certain orthogonal polynomials of a discrete argument (dual Hahn and Racah ones). Specializations and applications of the extended identities are given, including an extension of Whipple's identity relating very well poised F67(1) series and balanced F34(1) series, and extensions of other summation identities.

$${{\varvec{A}}}_{\varvec{\infty }}$$ A ∞ -algebras associated with elliptic curves and Eisenstein–Kronecker series

We compute the \(A_{\infty }\)-structure on the self-\({\text {Ext}}\) algebra of the vector bundle G over an elliptic curve of the form \(G=\bigoplus _{i=1}^r P_i\oplus \bigoplus _{j=1}^s L_j\), where \((P_i)\) and \((L_j)\) are line bundles of degrees 0 and 1, respectively. The answer is given in terms of Eisenstein–Kronecker numbers \((e^*_{a,b}(z,w))\). The \(A_\infty \)-constraints lead to quadratic polynomial identities between these numbers, allowing to express them in terms of few ones. Another byproduct of the calculation is the new representation for \(e^*_{a,b}(z,w)\) by rapidly converging series.

On universal quadratic identities for minors of quantum matrices

We give a complete combinatorial characterization of homogeneous quadratic relations of “universal character” valid for minors of quantum matrices (more precisely, for minors in the quantized coordinate ring Oq(Mm,n(K)) of m×n matrices over a field K, where q∈K⁎). This is obtained as a consequence of a study of quantized minors of matrices generated by paths in certain planar graphs, called SE-graphs, generalizing the ones associated with Cauchon diagrams. Our efficient method of verifying universal quadratic identities for minors of quantum matrices is illustrated with many appealing examples.

Isotopy invariant quasigroup identities

According to S. Krstic, there are only four quadratic varieties which are closed under isotopy. We give a simple procedure generating quadratic identities and deciding which of the four varieties they define. There are about 37000 such identities with up to five variables.

Quadratic Identities for a Class of Fibonacci-Like Polynomials

Quadratic Identities for a Class of Fibonacci-Like Polynomials

A generalization of Bakerʼs quadratic formulae for hyperelliptic ℘-functions

We present a generalization of a compact form, due to Baker, for quadratic identities satisfied by the three-index ℘-functions on curves of genus g=2, and a further generalization of a new result in genus g=3. The compact forms involve a bordered determinant containing 2(g−1)(g+1) free parameters.

Integrals of products of Bernoulli polynomials

Extending known results on integrals of products of two or three Bernoulli polynomials with limits of integration 0 and 1, we obtain identities for such integrals with limits of integration 0 and x, for a variable x. As applications we obtain certain quadratic and cubic identities for Bernoulli polynomials.

Remarks on the identities of gluon tree amplitudes

Recently, Bjerrum-Bohr, Damgaard, Feng, and Sondergaard derived a set of new interesting quadratic identities of the Yang-Mills (YM) tree scattering amplitudes, besides Bern-Carrasco-Johansson (BCJ) identities. Here we comment that these quadratic identities of YM amplitudes actually follow directly from the KLT (Kawai-Lewellen-Tye) relation for graviton-dilaton-axion scattering amplitudes (in four-dimensional spacetime). This clarifies their physical origin and also provides a simpler version of the new identities. We also comment that the recently discovered BCJ identities of YM helicity amplitudes, at least for the maximal helicity-violating case, can be verified by using (repeatedly) the Schouten identity. We also point out additional quadratic identities that can be written down from the KLT relations.

Quadratic Central Differential Identities on a Multilinear Polynomial

Let K be a commutative ring with unity, R a prime K-algebra, Z(R) the center of R, d and δ nonzero derivations of R, and f(x 1,…, x n ) a multilinear polynomial over K. If [d(f(r 1,…, r n )), δ (f(r 1,…, r n ))] ∈ Z(R), for all r 1,…, r n ∈ R, then either f(x 1,…, x n ) is central valued on R or {d, δ} are linearly dependent over C, the extended centroid of R, except when char(R) = 2 and dim C RC = 4.

Identities for hyperelliptic ℘-functions of genus one, two and three in covariant form

We give a covariant treatment of the quadratic differential identities satisfied by the ℘-functions on the Jacobian of smooth hyperelliptic curves of genus ⩽3.

A constructive theory of triple and quintuple product identities of the second degree

The groundwork for a theory of quadratic identities involving the classical triple and quintuple products is layed. The approach is through the study and use of affine maps that act on indexing lattices associated with the terms (double sums) in the given identity. The terms of the identity are found to be connected by the invariant of a ternary quadratic form.

Contiguous relations, continued fractions and orthogonality

We examine a special linear combination of balanced very-well-poised 10 ϕ 9 {_{10} \phi _{9}} basic hypergeometric series that is known to satisfy a transformation. We call this Φ \Phi and show that it satisfies certain three-term contiguous relations. From two of these contiguous relations for Φ \Phi we obtain fifty-six pairwise linearly independent solutions to a three-term recurrence that generalizes the recurrence for Askey-Wilson polynomials. The associated continued fraction is evaluated using Pincherle’s theorem. From this continued fraction we are able to derive a discrete system of biorthogonal rational functions. This ties together Wilson’s results for rational biorthogonality, Watson’s q q -analogue of Ramanujan’s Entry 40 continued fraction, and a conjecture of Askey concerning the latter. Some new q q -series identities are also obtained. One is an important three-term transformation for Φ \Phi ’s which generalizes all the known two- and three-term 8 ϕ 7 {_{8} \phi _{7}} transformations. Others are new and unexpected quadratic identities for these very-well-poised 8 ϕ 7 {_{8} \phi _{7}} ’s.

Polynomial Relations Among Characters coming from Quantum Affine Algebras

The Jacobi-Trudi formula implies some interesting quadratic identities for characters of representations of gln. Earlier work of Kirillov and Reshetikhin proposed a generalization of these identities to the other classical Lie algebras, and conjectured that the characters of certain finite-dimensional representations of Uq(ĝ) satisfy it. Here we use a positivity argument to show that the generalized identities have only one solution.

Quadratische Identitäten bei Polygonen

In a letter to Christian Goldbach (1690–1764) Leonhard Euler (1707–1783) communicates the following theorem on quadrilaterals: IfABCD is a quadrilateral andM, N are the mid-points of the diagonals, then |AB|2+|BC|2+|CD|2+|DA|2=|AC|2+|BD|2+4|MN|2. This theorem, a generalization of a classical theorem of Apollonius of Perge (262-190 B.C.) on parallelograms, was rediscovered recently by A. R. Amir-Moez and J. D. Hamilton (1976). A. J. Douglas (1981) carried the generalization a stage further and proved a theorem on polygons in a Euclidean space, which have an even number of points. On the basis of the Fourier analysis of polygons this paper establishes a wide class of quadratic identities, whose geometrical interpretation leads to polygon-theorems of the above type.