Elliptic Algebra Research Articles

Algorithms for representations of quiver Yangian algebras

In this note, we aim to review algorithms for constructing crystal representations of quiver Yangians in detail. Quiver Yangians are believed to describe an action of the BPS algebra on BPS states in systems of D-branes wrapping toric Calabi-Yau three-folds. Crystal modules of these algebras originate from molten crystal models for Donaldson-Thomas invariants of respective three-folds. Despite the fact that this subject was originally at the crossroads of algebraic geometry with effective supersymmetric field theories, equivariant toric action simplifies applied calculations drastically. So the sole pre-requisite for this algorithm’s implementation is linear algebra. It can be easily taught to a machine with the help of any symbolic calculation system. Moreover, these algorithms may be generalized to toroidal and elliptic algebras and exploited in various numerical experiments with those algebras. We illustrate the work of the algorithms in applications to simple cases of Ysl2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ extrm{Y}\\left({\\mathfrak{sl}}_2\\right) $$\\end{document}, Ygl̂1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ extrm{Y}\\left({\\hat{\\mathfrak{gl}}}_1\\right) $$\\end{document} and Ygl̂11\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ extrm{Y}\\left({\\hat{\\mathfrak{gl}}}_{\\left.1\\right|1}\\right) $$\\end{document}.

Symbolic studies of Maxwell’s equations in space-time algebra formalism

Different implementations of Clifford algebra: spinors, quaternions, and geometric algebra, are used to describe physical and technical systems. The geometric algebra formalism is a relatively new approach, destined to be used primarily by engineers and applied researchers. In a number of works, the authors examined the implementation of the geometric algebra formalism for computer algebra systems. In this article, the authors extend elliptic geometric algebra to hyperbolic space-time algebra. The results are illustrated by different representations of Maxwell’s equations. Using a computer algebra system, Maxwell’s vacuum equations in the space-time algebra representation are converted to Maxwell’s equations in vector formalism. In addition to practical application, the authors would like to draw attention to the didactic significance of these studies.

On the isomorphism between four-dimensional Sklyanin and elliptic algebras

Let [Formula: see text] be an elliptic curve, and let [Formula: see text] be a point on [Formula: see text]. In his groundbreaking paper [E. K. Sklyanin, Some algebraic structures connected with the Yang-Baxter equation. Representations of a quantum algebra, Funktsional. Anal. i Prilozhen. 17(4) (1983) 34–48], Sklyanin introduced a family of algebras [Formula: see text], consisting of four generators and six relations. These algebras hold profound connections with his work on the Quantum Scattering Method. Subsequently, Feigin and Odesskii defined algebras [Formula: see text] for any number of generators [Formula: see text] [Sklyanin’s algebras associated to elliptic curves, Inst. Theor. Phys., Akad. Nauk Ukrain. SSR, Kiev (1988)] and asserted that, for [Formula: see text], they coincide with Sklyanin’s algebras. In this paper, we provide an explicit isomorphism between the algebras [Formula: see text] and [Formula: see text]. As an application, we present a new proof of the Calabi–Yau property for [Formula: see text].

Separation of variables for the classical elliptic reflection equation algebra

In the present paper we construct separation of variables (SoV) for all Lax-integrable systems, two by two Lax matrix of which enjoys classical reflection equation algebra with the elliptic r−s matrices. We show that, similar to the cases of SoV for the classical XYZ and XXZ models [1,2], the constructed SoV admits two types of momenta, which are important in the quantum case [1,2]. We consider two examples of such the integrable hamiltonian systems governed by the simplest Poisson brackets connected with classical reflection equation algebra with the elliptic r−s matrices. They are the classical Sklyanin algebra and its four-dimensional extension which provide a quadratic structure for the classical Steklov top. We explicitly construct the variables of separation, Abel equations and reconstruction formulae for the corresponding integrable models.

The characteristic variety for Feigin and Odesskii's elliptic algebras

The characteristic variety for Feigin and Odesskii's elliptic algebras

More on affine Dynkin quiver Yangians

We consider the quiver Yangians associated to general affine Dynkin diagrams. Although the quivers are generically not toric, the algebras have some similar structures. The odd reflections of the affine Dynkin diagrams should correspond to Seiberg duality of the quivers, and we investigate the relations of the dual quiver Yangians. We also mention the construction of the twisted quiver Yangians. It is conjectured that the truncations of the (twisted) quiver Yangians can give rise to certain \U0001d4b2-algebras. Incidentally, we give the screening currents of the \U0001d4b2-algebras in terms of the free field realization in the case of generalized conifolds. Moreover, we discuss the toroidal and elliptic algebras for any general quivers.

Imaginary scators quadratic mapping in 1+2D dynamic space

The elliptic scator algebra quadratic iteration is evaluated in 1+2 dimensions in dynamic space. There exists a non divergent K set in the scator three dimensional space with a highly complex boundary J=∂K. Two and three dimensional renderings of the sets in dynamic space are published for the first time. The sets exhibit a rich fractal boundary in all three directions. Some of the salient features of the sets can be described in terms of square nilpotent iterations. The Julia and filled in Julia sets are identically reproduced at two perpendicular planes where only one non-vanishing hypercomplex director component is present. The fixed points of K in S1+2 with real constant c, can be obtained from the roots of a scator quadratic polynomial equation. In S1+2 there can be, in addition to the usual complex roots, hypercomplex roots that give rise to four additional fixed points. The inverse orbits of the hypercomplex roots reveal a rich complex structure, that allow for the evaluation of an infinite set of points in ∂K. The ∂K set of the origin is equal to the unit magnitude scator surface, named a cusphere. The J set exhibits self similar structures in 3D at different scales, typical of fractal phenomena. The ix set, is the three dimensional equivalent of the M-set in three dimensions. It is conjectured that the ix-set with some restrictions, is the set of parameters where the J set is connected.

Elliptic R-matrices and Feigin and Odesskii’s elliptic algebras

The algebras $$Q_{n,k}(E,\tau )$$ introduced by Feigin and Odesskii as generalizations of the 4-dimensional Sklyanin algebras form a family of quadratic algebras parametrized by coprime integers $$n>k\ge 1$$ , a complex elliptic curve E, and a point $$\tau \in E$$ . The main result in this paper is that $$Q_{n,k}(E,\tau )$$ has the same Hilbert series as the polynomial ring on n variables when $$\tau $$ is not a torsion point. We also show that $$Q_{n,k}(E,\tau )$$ is a Koszul algebra, hence of global dimension n when $$\tau $$ is not a torsion point, and, for all but countably many $$\tau $$ , $$Q_{n,k}(E,\tau )$$ is Artin–Schelter regular. The proofs use the fact that the space of quadratic relations defining $$Q_{n,k}(E,\tau )$$ is the image of an operator $$R_{\tau }(\tau )$$ that belongs to a family of operators $$R_{\tau }(z):{\mathbb {C}}^n\otimes {\mathbb {C}}^n\rightarrow {\mathbb {C}}^n\otimes {\mathbb {C}}^n$$ , $$z\in {\mathbb {C}}$$ , that (we will show) satisfy the quantum Yang–Baxter equation with spectral parameter.

Elliptic quantum toroidal algebra $$U_{q,t,p}({\mathfrak {gl}}_{1,tor})$$ and affine quiver gauge theories

Elliptic quantum toroidal algebra $$U_{q,t,p}({\mathfrak {gl}}_{1,tor})$$ and affine quiver gauge theories

English Intelligent Question Answering System Based on elliptic fitting equation

Abstract The elliptic equation intelligent question answering system based on the elliptic equation intelligent question answering system is an intelligent device that combines natural language and data retrieval technology. Knowledge domains used for different purposes are divided into open domains and restricted domains. In order to simplify the existing knowledge of English intelligence answer and elliptic algebra and reduce the efficiency of the questionnaire, a similar algorithm is designed and implemented in the process of English intelligence answer, and the method of distance ellipse switching is adopted. The experimental results show that the accuracy of the questionnaire answering the questions described in this paper is 17% higher than that of general research answers, which confirms the authenticity of the questionnaire algorithm mentioned in this paper. The system not only provides accurate answers to English questions for primary school students, but also plays an important role in the cultivation of English question and answer skills in primary and secondary schools.

4d higgsed network calculus and elliptic DIM algebra

Supersymmetric gauge theories of certain class possess a large hidden nonperturbative symmetry described by the Ding-Iohara-Miki (DIM) algebra which can be used to compute their partition functions and correlators very efficiently. We lift the DIM-algebraic approach developed to study holomorphic blocks of 3d linear quiver gauge theories one dimension higher. We employ an algebraic construction in which the underlying trigonometric DIM algebra is elliptically deformed, and an alternative geometric approach motivated by topological string theory. We demonstrate the equivalence of these two methods, and motivated by this, prove that elliptic DIM algebra is isomorphic to the direct sum of a trigonometric DIM algebra and an additional Heisenberg algebra.

Toroidal and elliptic quiver BPS algebras and beyond

The quiver Yangian, an infinite-dimensional algebra introduced recently in [1], is the algebra underlying BPS state counting problems for toric Calabi-Yau three-folds. We introduce trigonometric and elliptic analogues of quiver Yangians, which we call toroidal quiver algebras and elliptic quiver algebras, respectively. We construct the representations of the shifted toroidal and elliptic algebras in terms of the statistical model of crystal melting. We also derive the algebras and their representations from equivariant localization of three-dimensional mathcal{N} = 2 supersymmetric quiver gauge theories, and their dimensionally-reduced counterparts. The analysis of supersymmetric gauge theories suggests that there exist even richer classes of algebras associated with higher-genus Riemann surfaces and generalized cohomology theories.

Representations of the elliptic affine Hecke algebras

We prove that irreducible representations of the elliptic affine Hecke algebras of Ginzburg, Kapranov, and Vasserot are in one-to-one correspondence with certain nilpotent Higgs bundles on the elliptic curve. The main tool we use is the equivariant elliptic cohomology of the Steinberg variety of the Springer resolution. As a by-product, we study representations at roots of unity in type-A. As another by-product, we define a version of elliptic Demazure-Lusztig operators with dynamical parameters that satisfy the braid relations. We discuss speculative indications of this correspondence in 4d N=2 gauge theory.

Poisson geometry and representations of PI 4-dimensional Sklyanin algebras

Take S to be a 4-dimensional Sklyanin (elliptic) algebra that is module-finite over its center Z; thus, S is PI. Our first result is the construction of a Poisson Z-order structure on S such that the induced Poisson bracket on Z is non-vanishing. We also provide the explicit Jacobian structure of this bracket, leading to a description of the symplectic core decomposition of the maximal spectrum Y of Z. We then classify the irreducible representations of S by combining (1) the geometry of the Poisson order structures, with (2) algebro-geometric methods for the elliptic curve attached to S, along with (3) representation-theoretic methods using line and fat point modules of S. Along the way, we improve results of Smith and Tate obtaining a description the singular locus of Y for such S. The classification results for irreducible representations are in turn used to determine the zero sets of the discriminants ideals of these algebras S.

Powers of Elliptic Scator Numbers

Elliptic scator algebra is possible in 1+n dimensions, n∈N. It is isomorphic to complex algebra in 1 + 1 dimensions, when the real part and any one hypercomplex component are considered. It is endowed with two representations: an additive one, where the scator components are represented as a sum; and a polar representation, where the scator components are represented as products of exponentials. Within the scator framework, De Moivre’s formula is generalized to 1+n dimensions in the so called Victoria equation. This novel formula is then used to obtain compact expressions for the integer powers of scator elements. A scator in S1+n can be factored into a product of n scators that are geometrically represented as its projections onto n two dimensional planes. A geometric interpretation of scator multiplication in terms of rotations with respect to the scalar axis is expounded. The powers of scators, when the ratio of their director components is a rational number, lie on closed curves. For 1 + 2 dimensional scators, twisted curves in a three dimensional space are obtained. Collecting previous results, it is possible to evaluate the exponential of a scator element in 1 + 2 dimensions.

Numerical Algorithm for Solving General Linear Elliptic Quaternionic Matrix Equations

In this study, we develop a general method to solve the general linear elliptic quaternionic matrix equations by means of real representation of elliptic quaternion matrices. A pseudocode for our approach that provides the solution of the linear elliptic quaternionic matrix equations is expressed. Moreover, we apply this method to the well-known Slyvester matrix equations and Kalman Yakubovich matrix equations over the elliptic quaternion algebra.

PBWD bases and shuffle algebra realizations for $$U_{\varvec{v}}(L\mathfrak {sl}_n), U_{{\varvec{v}}_1,{\varvec{v}}_2}(L\mathfrak {sl}_n), U_{\varvec{v}}(L\mathfrak {sl}(m|n))$$ and their integral forms

We construct a family of PBWD (Poincaré–Birkhoff–Witt–Drinfeld) bases for the quantum loop algebras \(U_{\varvec{v}}(L\mathfrak {sl}_n),U_{{\varvec{v}}_1,{\varvec{v}}_2}(L\mathfrak {sl}_n),U_{\varvec{v}}(L\mathfrak {sl}(m|n))\) in the new Drinfeld realizations. In the 2-parameter case, this proves (Hu et al. in Commun Math Phys 278(2):453–486, 2008, Theorem 3.11) (stated in loc. cit. without a proof), while in the super case it proves a conjecture of Zhang (Math. Z. 278(3–4):663–703, 2014). The main ingredient in our proofs is the interplay between those quantum loop algebras and the corresponding shuffle algebras, which are trigonometric counterparts of the elliptic shuffle algebras of Feigin and Odesskii (Anal. i Prilozhen 23(3):45–54, 1989; Anal i Prilozhen 31(3):57–70, 1997; Integrable Structures of Exactly Solvable Two-Dimensional Models of Quantum Field Theory (Kiev, 2000). NATO Sci Ser II Math Phys Chem, vol 35, Kluwer Academic Publishers, Dordrecht, pp 123–137, 2001). Our approach is similar to that of Enriquez (J Lie Theory 13(1):21–64, 2003) in the formal setting, but the key novelty is an explicit shuffle algebra realization of the corresponding algebras, which is of independent interest. This also allows us to strengthen the above results by constructing a family of PBWD bases for the RTT forms of those quantum loop algebras as well as for the Lusztig form of \(U_{\varvec{v}}(L\mathfrak {sl}_n)\). The rational counterparts provide shuffle algebra realizations of type A (super) Yangians and their Drinfeld–Gavarini dual subalgebras.

Feigin and Odesskii's elliptic algebras

We study the elliptic algebras Qn,k(E,τ) introduced by Feigin and Odesskii as a generalization of Sklyanin algebras. They form a family of quadratic algebras parametrized by coprime integers n>k≥1, an elliptic curve E, and a point τ∈E. We consider and compare several different definitions of the algebras and provide proofs of various statements about them made by Feigin and Odesskii. For example, we show that Qn,k(E,0), and Qn,n−1(E,τ) are polynomial rings on n variables. We also show that Qn,k(E,τ+ζ) is a twist of Qn,k(E,τ) when ζ is an n-torsion point. This paper is the first of several we are writing about the algebras Qn,k(E,τ).

Maps from Feigin and Odesskii's elliptic algebras to twisted homogeneous coordinate rings

Abstract The elliptic algebras in the title are connected graded $\mathbb {C}$ -algebras, denoted $Q_{n,k}(E,\tau )$ , depending on a pair of relatively prime integers $n>k\ge 1$ , an elliptic curve E and a point $\tau \in E$ . This paper examines a canonical homomorphism from $Q_{n,k}(E,\tau )$ to the twisted homogeneous coordinate ring $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ on the characteristic variety $X_{n/k}$ for $Q_{n,k}(E,\tau )$ . When $X_{n/k}$ is isomorphic to $E^g$ or the symmetric power $S^gE$ , we show that the homomorphism $Q_{n,k}(E,\tau ) \to B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ is surjective, the relations for $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ are generated in degrees $\le 3$ and the noncommutative scheme $\mathrm {Proj}_{nc}(Q_{n,k}(E,\tau ))$ has a closed subvariety that is isomorphic to $E^g$ or $S^gE$ , respectively. When $X_{n/k}=E^g$ and $\tau =0$ , the results about $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ show that the morphism $\Phi _{|\mathcal {L}_{n/k}|}:E^g \to \mathbb {P}^{n-1}$ embeds $E^g$ as a projectively normal subvariety that is a scheme-theoretic intersection of quadric and cubic hypersurfaces.

On Abelianity Lines in Elliptic W-Algebras

We present a systematic derivation of the abelianity conditions for the $q$-deformed $W$-algebras constructed from the elliptic quantum algebra $\mathcal{A}_{q,p}\big(\widehat{\mathfrak{gl}}(N)_{c}\big)$. We identify two sets of conditions on a given critical surface yielding abelianity lines in the moduli space ($p, q, c$). Each line is identified as an intersection of a countable number of critical surfaces obeying diophantine consistency conditions. The corresponding Poisson brackets structures are then computed for which some universal features are described.