General Algebra Research Articles

Tetrahedron equation and Schur functions

Abstract The tetrahedron equation introduced by Zamolodchikov is a three-dimensional generalization of the Yang–Baxter equation. Several types of solutions to the tetrahedron equation that have connections to quantum groups can be viewed as q-oscillator valued vertex models with matrix elements of the L-operators given by generators of the q-oscillator algebra acting on the Fock space. Using one of the q = 0-oscillator valued vertex models introduced by Bazhanov–Sergeev, we introduce a family of partition functions that admits an explicit algebraic presentation using Schur functions. Our construction is based on the three-dimensional realization of the Zamolodchikov–Faddeev algebra provided by Kuniba–Maruyama–Okado. Furthermore, we investigate an inhomogeneous generalization of the three-dimensional lattice model. We show that the inhomogeneous analog of (a certain subclass of) partition functions can be expressed as loop elementary symmetric functions.

Anti-Leibniz algebras: A non-commutative version of mock-Lie algebras

Leibniz algebras are non skew-symmetric generalization of Lie algebras. In this paper we introduce the notion of anti-Leibniz algebras as a “non commutative version” of mock-Lie algebras. Low dimensional classification of such algebras is given. Then we investigate the notion of averaging operators and more general embedding tensors to build some new algebraic structures, namely anti-associative dialgebras, anti-associative trialgebras and anti-Leibniz trialgebras.

Symmetric group gauge theories and simple gauge/string dualities

Abstract We study two-dimensional topological gauge theories with gauge group equal to the symmetric group Sn and their string theory duals. The simplest such theory is the topological quantum field theory of principal Sn fiber bundles. Its correlators are equal to Hurwitz numbers. The operator products in the gauge theory for each finite value of n are coded in one partial permutation algebra. We propose a generalization of the partial permutation algebra to the symmetric orbifold topological quantum field theory of any seed theory and show that the theory factorizes into marked partial permutation combinatorics and seed Frobenius algebra properties. Moreover, we exploit the established correspondence between Hurwitz theory and the stationary sector of Gromov–Witten theory on the sphere to prove an exact gauge/string duality. The relevant field theory is a grand canonical version of Hurwitz theory and its two-point functions are obtained by summing over all values of the instanton degree of the maps covering the sphere. We stress that one must look for a multiplicative basis on the boundary to match the bulk operator algebra of single string insertions. The relevant boundary observables are completed cycles.

Chiral Virasoro algebra from a single wavefunction

Chiral edges of 2+1D systems can have very robust emergent conformal symmetry. When the edge is purely chiral, the Hilbert space of low-energy edge excitations can form a representation of a single Virasoro algebra. We propose a method to systematically extract the generators of the Virasoro algebra from a single ground state wavefunction, using entanglement bootstrap and an input from the edge conformal field theory. We corroborate our construction by numerically verifying the commutation relations of the generators. We also study the unitary flows generated by these operators, whose properties (such as energy and state overlap) are shown numerically to agree with our analytical predictions.

Supergravity in the geometric approach and its hidden graded Lie algebra

In this contribution, we present the geometric approach to supergravity. In the first part, we discuss in some detail the peculiarities of the approach and apply the formalism to the case of pure supergravity in four space–time dimensions. In the second part, we extend the discussion to theories in higher dimensions, which include antisymmetric tensors of degree higher than one, focussing on the case of eleven dimensional space–time. Here, we report the formulation first introduced by R. D’Auria and P. Fré in 1981, corresponding to a generalization of a Chevalley-Eilenberg Lie algebra, together with some more recent results, pointing out the relation of the formalism with the mathematical framework of L∞ algebras.

A generic classification of locally free representations of affine GLS algebras

Throughout, let K be an algebraically closed field of characteristic 0. We provide a generic classification of locally free representations of Geiß-Leclerc-Schröer's algebras HK(C,D,Ω) associated to affine Cartan matrices C with minimal symmetrizer D and acyclic orientation Ω. Affine GLS algebras are “smooth” degenerations of tame hereditary algebras and as such their representation theory is presumably still tractable. Indeed, we observe several “tame” phenomena of affine GLS algebras even though they are in general representation wild. For the GLS algebras of type BC˜1 we achieve a classification of all stable representations. For general GLS algebras of affine type, we construct a 1-parameter family of representations stable with respect to the defect. Our construction is based on a generalized one-point extension technique. This confirms in particular τ-tilted versions of the second Brauer-Thrall Conjecture recently raised by Mousavand and Schroll-Treffinger-Valdivieso for the class of GLS algebras. Finally, we show that generically every locally free H-module is isomorphic to a direct sum of τ-rigid modules and modules from our 1-parameter family. This generalizes Kac's canonical decomposition from the symmetric to the symmetrizable case in affine types and we obtain such a decomposition by folding the canonical decomposition of dimension vectors over path algebras. As a corollary we obtain that affine GLS algebras are E-tame in the sense of Derksen-Fei and Asai-Iyama.

SparseAuto: An Auto-scheduler for Sparse Tensor Computations using Recursive Loop Nest Restructuring

Automated code generation and performance enhancements for sparse tensor algebra have become essential in many real-world applications, such as quantum computing, physical simulations, computational chemistry, and machine learning. General sparse tensor algebra compilers are not always versatile enough to generate asymptotically optimal code for sparse tensor contractions. This paper shows how to generate asymptotically better schedules for complex sparse tensor expressions using kernel fission and fusion. We present generalized loop restructuring transformations to reduce asymptotic time complexity and memory footprint. Furthermore, we present an auto-scheduler that uses a partially ordered set (poset)-based cost model that uses both time and auxiliary memory complexities to prune the search space of schedules. In addition, we highlight the use of Satisfiability Module Theory (SMT) solvers in sparse auto-schedulers to approximate the Pareto frontier of better schedules to the smallest number of possible schedules, with user-defined constraints available at compile-time. Finally, we show that our auto-scheduler can select better-performing schedules and generate code for them. Our results show that the auto-scheduler provided schedules achieve orders-of-magnitude speedup compared to the code generated by the Tensor Algebra Compiler (TACO) for several computations on different real-world tensors.

Exceptional Periodicity and Magic Star algebras: Generalized roots, gradings, Hermitian Vinberg T-algebras and their derivations

We introduce countably infinite series of finite dimensional generalizations of the exceptional Lie algebras: in fact, each exceptional Lie algebra (but g2) is the first element of an infinite series of finite dimensional algebras, which we name Magic Star algebras. All these algebras (but the first elements of the infinite series) are not Lie algebras, but nevertheless they have remarkable similarities with many characterizing features of the exceptional Lie algebras; they also enjoy a kind of periodicity (inherited by Bott periodicity), which we name Exceptional Periodicity. We analyze the graded algebraic structures arising in a certain projection (named Magic Star projection) of the generalized root systems pertaining to Magic Star algebras, and we highlight the occurrence of a class of rank-3, Hermitian matrix (special Vinberg T)-algebras (which we call H algebras) on each vertex of such a projection. We then focus on the Magic Star algebra f4(n), which generalizes the non-simply laced exceptional Lie algebra f4, and deserves a treatment apart. Finally, we compute the Lie algebra of the inner derivations of the H algebras, pointing out the enhancements occurring for each first element of the series of Magic Star algebras, thus retrieving the result known for the derivations of cubic simple Jordan algebras.

CFD investigation of supersonic two parallel expansions nozzles

This paper aims to validate the design contours and result parameters of the newly developed Dual Expansion Nozzle (DEN) using the Fastran software when it enters the non-adaptation regime with the increase in altitude and variation of the Nozzle Pressure Ratio (NPR). DEN design and Computational Fluid Dynamic (CFD) application utilize the High Temperature (HT) model defined by a calorically imperfect and thermally perfect gas, providing good accuracy compared to PG model. The computational domain is decomposed into subdivisions of structured grids using the algebraic grid generator software CFD-GEOM. The mesh convergence was analyzed using three mesh resolutions: coarse, medium, and fine. A comparison was made between the two conventional nozzles, MLN and the newly developed BPN, as both are currently used in aerospace propulsion to enhance aerodynamic performance. If a rocket engine operates under strongly over-expanded conditions with ambient pressure considerably higher than the nozzle exit pressure, and by increasing NPR value, the flow separates from the wall and can lead to high side loads. These loads are reduced in DEN. The flow separation point is observed close to the exit section for the DEN in comparison with Minimum Length Nozzle (MLN) and Best Performance Nozzle (BPN) for the same NPR, resulting in low flow separation and low side load effects for DEN. This result demonstrates a small loss in the exit Mach number, leading to less loss in the thrust coefficient and a small effect of boundary layer friction for DEN. Enhanced over-expanded aerodynamics is observed in DEN compared to MLN and BPN. The application is made for air with an exit Mach number equal to 3.00.

Commutative families in DIM algebra, integrable many-body systems and q, t matrix models

We extend our consideration of commutative subalgebras (rays) in different representations of the W1+∞ algebra to the elliptic Hall algebra (or, equivalently, to the Ding-Iohara-Miki (DIM) algebra Uq,tgl̂̂1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {U}_{q,t}\\left({\\hat{\\hat{\\mathfrak{gl}}}}_1\\right) $$\\end{document}). Its advantage is that it possesses the Miki automorphism, which makes all commutative rays equivalent. Integrable systems associated with these rays become finite-difference and, apart from the trigonometric Ruijsenaars system not too much familiar. We concentrate on the simplest many-body and Fock representations, and derive explicit formulas for all generators of the elliptic Hall algebra en,m. In the one-body representation, they differ just by normalization from znqmD̂\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {z}^n{q}^{m\\hat{D}} $$\\end{document} of the W1+∞ Lie algebra, and, in the N -body case, they are non-trivially generalized to monomials of the Cherednik operators with action restricted to symmetric polynomials. In the Fock representation, the resulting operators are expressed through auxiliary polynomials of n variables, which define weights in the residues formulas. We also discuss q, t-deformation of matrix models associated with constructed commutative subalgebras.

The Racah Algebra of Rank 2: Properties, Symmetries and Representation

The goals of this paper are threefold. First, we provide a new ''universal'' definition for the Racah algebra of rank 2 as an extension of the rank-1 Racah algebra where the generators are indexed by subsets and any three disjoint indexing sets define a subalgebra isomorphic to the rank-1 case. With this definition, we explore some of the properties of the algebra including verifying that these natural assumptions are equivalent to other defining relations in the literature. Second, we look at the symmetries of the generators of the rank-2 Racah algebra. Those symmetries allows us to partially make abstraction of the choice of the generators and write relations and properties in a different format. Last, we provide a novel representation of the Racah algebra. This new representation requires only one generator to be diagonal and is based on an expansion of the split basis representation from the rank-1 Racah algebra.

On the interaction of the Coxeter transformation and the rowmotion bijection

Let P be a finite poset and L the associated distributive lattice of order ideals of P . Let \rho denote the rowmotion bijection of the order ideals of P viewed as a permutation matrix and C the Coxeter matrix for the incidence algebra kL of L . Then, we show the identity (\rho^{-1}C)^{2}=\mathrm{id} , as was originally conjectured by Sam Hopkins. Recently, it was noted that the rowmotion bijection is a special case of the much more general grade bijection R that exists for any Auslander regular algebra. This motivates to study the interaction of the grade bijection and the Coxeter matrix for general Auslander regular algebras. For the class of higher Auslander algebras coming from n -representation finite algebras, we show that (R^{-1}C)^{2}=\mathrm{id} if n is even and (R^{-1}C+\mathrm{id})^{2}=0 when n is odd.

Algebraic generators of the skein algebra of a surface

Algebraic generators of the skein algebra of a surface

Subinvariance and ascendancy in Leibniz Algebras

Leibniz algebras are certain generalizations of Lie algebras. Motivated by the concept of subinvariance and ascendancy in group theory, Schenkman studied properties of subinvariant subalgebras of a Lie algebra. Kawamoto studied ascendency in Lie algebras. In this paper we define subinvariance and asendency in Leibniz algebras and study their properties. It is shown that the signature results on subinvariance and ascendency in Lie algebras have analogs for Leibniz algebras.

The representation theory of Brauer categories II: Curried algebra

A representation of \mathfrak{gl}(V)=V \otimes V^{\ast} is a linear map \mu \colon \mathfrak{gl}(V) \otimes M \rightarrow M satisfying a certain identity. By currying, giving a linear map \mu is equivalent to giving a linear map a \colon V \otimes M \rightarrow V \otimes M , and one can translate the condition for \mu to be a representation into a condition on a . This alternate formulation does not use the dual of V and makes sense for any object V in a tensor category \mathcal{C} . We call such objects representations of the curried general linear algebra on V . The currying process can be carried out for many algebras built out of a vector space and its dual, and we examine several cases in detail. We show that many well-known combinatorial categories are equivalent to the curried forms of familiar Lie algebras in the tensor category of linear species; for example, the titular Brauer category is the curried form of the symplectic Lie algebra. This perspective puts these categories in a new light, has some technical applications, and suggests new directions to explore.

Generalized double affine Hecke algebra for double torus

We propose a generalization of the double affine Hecke algebra of type-C∨C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^\\vee C_1$$\\end{document} at specific parameters by introducing a “Heegaard dual” of the Hecke operators. Shown is a relationship with the skein algebra on double torus. We give automorphisms of the algebra associated with the Dehn twists on the double torus.

A hidden 2d CFT for self-dual Yang-Mills on the celestial sphere

Self-dual Yang-Mills theory admits an underlying infinite dimensional symmetry algebra, which has been obtained from mode expansion of Mellin transformed 4d scattering amplitudes and separately, Koszul duality on twistor space. In this paper, we propose to derive an explicit 2d realization of the algebra by performing a particular gauge transformation on the twistor action for self-dual Yang-Mills. The gauge parameter used in the transformation generates pure gauge connections corresponding to large gauge transformations on 4d Minkowski space, which localises part of the twistor action to a ℂℙ1 on ℂ* reduction of twistor space. Under a projection, it can be mapped to the celestial sphere at the light-cone cut of the origin on I\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{I} $$\\end{document}. Geometrically, this is the common boundary celestial sphere shared by Euclidean AdS3 or Lorentzian dS3 slices of Minkowski space. We comment on the geometric meaning of the derivation from the perspective of minitwistor spaces of the 3d slices embedded in 4d Minkowski space. Using the action functional of this 2d CFT, we compute its stress-energy tensor and central charge. By a further marginal deformation, we calculate correlation functions of current algebra generators purely from the 2d side which reproduce full 4d tree-level form factors.

On General Wide Graph Algebras and Orbit Equivalence

On General Wide Graph Algebras and Orbit Equivalence

Weak Hopf symmetry and tube algebra of the generalized multifusion string-net model

We investigate the multifusion generalization of string-net ground states and lattice Hamiltonians, delving into their associated weak Hopf symmetries. For the multifusion string-net, the gauge symmetry manifests as a general weak Hopf algebra, leading to a reducible vacuum string label; the charge symmetry, serving as a quantum double of gauge symmetry, constitutes a connected weak Hopf algebra. This implies that the associated topological phase retains its characterization by a unitary modular tensor category (UMTC). The bulk charge symmetry can also be captured by a weak Hopf tube algebra. We offer an explicit construction of the weak Hopf tube algebra structure and thoroughly discuss its properties. The gapped boundary and domain wall models are extensively discussed, with these 1d phases characterized by unitary multifusion categories (UMFCs). We delve into the gauge and charge symmetries of these 1d phases, as well as the construction of the boundary and domain wall tube algebras. Additionally, we illustrate that the domain wall tube algebra can be regarded as a cross product of two boundary tube algebras. As an application of our model, we elucidate how to interpret the defective string-net as a restricted multifusion string-net.

Automorphism groups of some non-nilpotent Leibniz algebras

Let $L$ be an algebra over a field $F$ with the binary operations $+$ and $[,]$. Then $L$ is called a left Leibniz algebra if it satisfies the left Leibniz identity: $[a,[b,c]]=[[a,b],c]+[b,[a,c]]$ for all $a,b,c\in L$. A linear transformation $f$ of $L$ is called an endomorphism of $L$, if $f([a,b])=[f(a),f(b)]$ for all elements $a,b\in L$. A bijective endomorphism of $L$ is called an automorphism of $L$. It is easy to show that the set of all automorphisms of Leibniz algebra is a group with respect to the operation of multiplication of automorphisms. The description of the structure of the automorphism groups of Leibniz algebras is one of the natural and important problems of the general Leibniz algebra theory. The main goal of this article is to describe the structure of the automorphism group of a certain type of non-nilpotent three-dimensional Leibniz algebras.