Full Algebra Research Articles

Unconventional conformal invariance of maximal depth partially massless fields on dS4 and its relation to complex partially massless SUSY

Deser and Waldron have shown that maximal depth partially massless theories of higher (integer) spin on four-dimensional de Sitter spacetime (dS4) possess infinitesimal symmetries generated by the conformal Killing vectors of dS4. However, it was later shown by Barnich, Bekaert, and Grigoriev that these theories are not invariant under the conformal algebra so(2, 4). To get some insight into these seemingly contradicting results we write down the full set of infinitesimal transformations of the fields generated by the fifteen conformal Killing vectors of dS4. In particular, although the infinitesimal transformations generated by the ten dS Killing vectors are well-known (these correspond to the conventional Lie derivatives), the transformations generated by the five non-Killing conformal Killing vectors were absent from the literature, and we show that they have an ‘unconventional’ form. In the spin-2 case (partially massless graviton), we show that the field equations and the action are invariant under the unconventional conformal transformations. For spin s > 2, the invariance is demonstrated only at the level of the field equations. For all spins s ≥ 2, we reproduce the result that the symmetry algebra does not close on the conformal algebra, so(2, 4). This is due to the appearance of new higher-derivative symmetry transformations in the commutator of two unconventional conformal transformations. Our results concerning the closure of the full symmetry algebra are inconclusive. Then we shift focus to the question of supersymmetry (SUSY) on dS4 and our objective is twofold. First, we uncover a non-interacting supermultiplet that consists of a complex partially massless spin-2 field and a complex spin-3/2 field on dS4. Second, we showcase the appearance of the unconventional conformal symmetries in the commutator of two SUSY transformations. Thus, this commutator closes on an algebra that is neither so(1, 4) nor so(2, 4), while its full structure is an open question. More open questions arising from our findings are also discussed.

T-duality and flavor symmetries in Little String Theories

We explore the T-duality web of 6D Heterotic Little String Theories, focusing on flavor algebra reducing deformations. A careful analysis of the full flavor algebra, including Abelian factors, shows that the flavor rank is preserved under T-duality. This suggests a new T-duality invariant in addition to the Coulomb branch dimension and the two-group structure constants. We also engineer Little String Theories with non-simply laced flavor algebras, whose appearance we attribute to certain discrete 3-form fluxes in M-theory. Geometrically, these theories are engineered in F-theory with non-Kähler favorable K3 fibers. This geometric origin leads us to propose that freezing fluxes are preserved across T-duality. Along the way, we discuss various exotic models, including two inequivalent Spin(32)/ℤ2 models that are dual to the same E8 × E8 theory, and a family of self-T-dual models.

Representations of toroidal and full toroidal Lie algebras over polynomial algebras

Toroidal Lie algebras are n variable generalizations of affine Kac-Moody Lie algebras. Full toroidal Lie algebra is the semidirect product of derived Lie algebra of toroidal Lie algebra and Witt algebra, also it can be thought of n-variable generalization of Affine-Virasoro algebras. Let h̃ be a Cartan subalgebra of a toroidal Lie algebra as well as full toroidal Lie algebra without containing the zero-degree central elements. In this paper, we classify the module structure on U(h̃) for all toroidal Lie algebras as well as full toroidal Lie algebras which are free U(h̃)-modules of rank 1. These modules exist only for type Al(l ≥ 1), Cl(l ≥ 2) toroidal Lie algebras and the same is true for full toroidal Lie algebras. Also, we determined the irreducibility condition for these classes of modules for both the Lie algebras.

Local properties of Schrödinger-Virasoro type Lie algebras and a full diamond thin Lie algebra

Let S V ( z ) be a class of Schrödinger-Virasoro type Lie algebras, where z ∈ Z . In this paper, we show that S V ( z ) is 2-local complete (namely, every 2-local derivation is a derivation) for almost all integers z’s. We also introduce the notion of full diamond thin Lie algebras, and construct an example F from S V ( − 1 ) . We completely determine the derivations of F by combinatoric techniques, and show that F is not 2-local complete.

Simple modules for twisted Hamiltonian extended affine Lie algebras

In this paper we consider the twisted Hamiltonian extended affine Lie algebra (THEALA). We classify the irreducible integrable modules for these Lie algebras with finite dimensional weight spaces when the finite dimensional center acts non-trivially. This Lie algebra has a triangular decomposition, which is different from the natural triangular decomposition of twisted full toroidal Lie algebra. Any irreducible integrable module of it is a highest weight module with respect to the given triangular decomposition. In this paper we describe the highest weight space in detail.

C⁎-extreme entanglement breaking maps on operator systems

Let E denote the set of all unital entanglement breaking (UEB) linear maps defined on an operator system S⊂Md and, mapping into Mn. As it turns out, the set E is not only convex in the classical sense but also in a quantum sense, namely it is C⁎-convex. The main objective of this article is to describe the C⁎-extreme points of this set E. By observing that every EB map defined on the operator system S dilates to a positive map with commutative range and also extends to an EB map on Md, we show that the C⁎-extreme points of the set E are precisely the UEB maps that are maximal in the sense of Arveson ([1] and [2]) and that they are also exactly the linear extreme points of the set E with commutative range. We also determine their explicit structure, thereby obtaining operator system generalizations of the analogous structure theorem and the Krein-Milman type theorem given in [8]. As a consequence, we show that C⁎-extreme (UEB) maps in E extend to C⁎-extreme UEB maps on the full algebra. Finally, we obtain an improved version of the main result in [8], which contains various characterizations of C⁎-extreme UEB maps between the algebras Md and Mn.

Derivations, extensions, and rigidity of subalgebras of the Witt algebra

Let k be an algebraically closed field of characteristic 0. We study some cohomological properties of Lie subalgebras of the Witt algebra W=Der(k[t,t−1]) and the one-sided Witt algebra W≥−1=Der(k[t]). In the first part of the paper, we consider finite codimension subalgebras of W≥−1. We compute derivations and one-dimensional extensions of such subalgebras. These correspond to ExtU(L)1(M,L), where L is a subalgebra of W≥−1 and M is a one-dimensional representation of L. We find that these subalgebras exhibit a kind of rigidity: their derivations and extensions are controlled by the full one-sided Witt algebra. As an application of these computations, we prove that any isomorphism between finite codimension subalgebras of W≥−1 extends to an automorphism of W≥−1.The second part of the paper is devoted to explaining the observed rigidity. We define a notion of “completely non-split extension” and prove that W≥−1 is the universal completely non-split extension of any of its subalgebras of finite codimension. In some sense, this means that even when studying subalgebras of W≥−1 as abstract Lie algebras, they remember that they are contained in W≥−1. We also consider subalgebras of infinite codimension, explaining the similarities and differences between the finite and infinite codimension situations.Almost all of the results above are also true for subalgebras of the Witt algebra. We summarise results for W at the end of the paper.

Transposed Poisson structures on the Lie algebra of upper triangular matrices

We describe transposed Poisson structures on the upper triangular matrix Lie algebra T_{n}(F) , n>1 , over a field F of characteristic zero. We prove that, for n>2 , any such structure is either of Poisson type or the orthogonal sum of a fixed non-Poisson structure with a structure of Poisson type, and for n=2 , there is one more class of transposed Poisson structures on T_{n}(F) . We also show that, up to isomorphism, the full matrix Lie algebra M_{n}(F) admits only one non-trivial transposed Poisson structure, and it is of Poisson type.

Renormalized electric and magnetic charges for O(rn) large gauge symmetries

In this work we present the construction of a renormalized symplectic form on an extended phases space where the higher order large gauge transformations (LGT) act canonically. The expressions of the subn-leading electric charges associated with each O(rn) LGT are then obtained, in agreement with the expressions previously proposed in [1] by means of the tree-level subn-leading formulas. We also present the duality extension of the extended phase space, computing the full electromagnetic charge algebra, showing a tower of central extensions.

Limits of traces of Temperley-Lieb algebras

We review the classification of positive extremal traces on the generic infinite Temperley-Lieb algebra, and then extend the classification to the non-semisimple root of unity case. As a result, we obtain Hilbert space structures on the full infinite Temperley-Lieb algebra at roots of unity.

Graphical methods and rings of invariants on the symmetric algebra

Abstract Let G be a complex classical group, and let V be its defining representation (possibly plus a copy of the dual). A foundational problem in classical invariant theory is to write down generators and relations for the ring of G-invariant polynomial functions on the space $\mathcal P^m(V)$ of degree-m homogeneous polynomial functions on V. In this paper, we replace $\mathcal P^m(V)$ with the full polynomial algebra $\mathcal P(V)$ . As a result, the invariant ring is no longer finitely generated. Hence, instead of seeking generators, we aim to write down linear bases for bigraded components. Indeed, when G is of sufficiently high rank, we realize these bases as sets of graphs with prescribed number of vertices and edges. When the rank of G is small, there arise complicated linear dependencies among the graphs, but we remedy this setback via representation theory: in particular, we determine the dimension of an arbitrary component in terms of branching multiplicities from the general linear group to the symmetric group. We thereby obtain an expression for the bigraded Hilbert series of the ring of invariants on $\mathcal P(V)$ . We conclude with examples using our graphical notation, several of which recover classical results.

The valuation pairing on an upper cluster algebra

Abstract It is known that many (upper) cluster algebras are not unique factorization domains. We exhibit the local factorization properties with respect to any given seed t: any non-zero element in a full rank upper cluster algebra can be uniquely written as the product of a cluster monomial in t and another element not divisible by the cluster variables in t. Our approach is based on introducing the valuation pairing on an upper cluster algebra: it counts the maximal multiplicity of a cluster variable among the factorizations of any given element. We apply the valuation pairing to obtain many results concerning factoriality, d-vectors, F-polynomials and the combinatorics of cluster Poisson variables. In particular, we obtain that full rank and primitive upper cluster algebras are factorial; an explanation of d-vectors using valuation pairing; a cluster monomial in non-initial cluster variables is determined by its F-polynomial; the F-polynomials of non-initial cluster variables are irreducible; and the cluster Poisson variables parametrize the exchange pairs of the corresponding upper cluster algebra.

On minimal Gorenstein Hilbert functions

We conjecture that a class of Artinian Gorenstein Hilbert algebras called full Perazzo algebras always have minimal Hilbert function, fixing codimension and length. We prove the conjecture in length four and five, in low codimension. We also prove the conjecture for a particular subclass of algebras that occurs in every length and certain codimensions. As a consequence of our methods we give a new proof of part of a known result about the asymptotic behavior of the minimum entry of a Gorenstein Hilbert function.

Primal quantization postulate for quantum fields

We formulate a theory of quantum fields underpinned solely by the full geometric algebra structure generated by the vector space of spacetime. A modified dynamical equation of a field with associated background field potentials, distinct from Dirac’s equation, is developed and corrects the inertial potential that acts on the field. A discrete wavenumber spectrum with a possible correlation to the mass spectrum of elementary particles is obtained, and explicit expressions are derived. The formulation is mathematically driven and constitutes what we refer to as the primal field quantization postulate. While manifestly a relativistic quantum field theory, our formulation may provide a natural account of mass hierarchy and family generation. A fourth family of elementary particles is predicted, indicating that lepton and quark oscillations across generation members, as well as quantum decay processes and fluctuations, may need modifying.

Super-spin chains for 6D SCFTs

Nearly all 6D superconformal field theories (SCFTs) have a partial tensor branch description in terms of a generalized quiver gauge theory consisting of a long one-dimensional spine of quiver nodes with links given by conformal matter; a strongly coupled generalization of a bifundamental hypermultiplet. For theories obtained from M5-branes probing an ADE singularity, this was recently leveraged to extract a protected large R-charge subsector of operators, with operator mixing controlled at leading order in an inverse large R-charge expansion by an integrable spin s Heisenberg spin chain, where s is determined by the su(2)R R-symmetry representation of the conformal matter operator. In this work, we show that this same structure extends to the full superconformal algebra osp(6,2|1). In particular, we determine the corresponding Bethe ansatz equations which govern this super-spin chain, as well as distinguished subsectors which close under operator mixing. Similar considerations extend to 6D little string theories (LSTs) and 4D N=2 SCFTs with the same generalized quiver structures.

Two-dimensional conformal field theory, full vertex algebra and current-current deformation

The main purpose of this paper is a mathematical construction of non-perturbative deformations of two-dimensional conformal field theories.We introduce the notion of a full vertex algebra which formulates a compact two-dimensional conformal field theory. Then, we construct a deformation family of full vertex algebras which serves as a current-current deformation of conformal field theory in physics. The parameter space of the deformations is expressed as a quotient of an orthogonal Grassmannian. This parameter space is a part of the CFT moduli space expected in physics. We demonstrate that nontrivial quotients of Grassmannians are obtained from holomorphic vertex operator algebras of central charge 24.

DBSP: Automatic Incremental View Maintenance for Rich Query Languages

Incremental view maintenance (IVM) has long been a central problem in database theory. Many solutions have been proposed for restricted classes of database languages, such as the relational algebra, or Datalog. These techniques do not naturally generalize to richer languages. In this paper we give a general, heuristic-free solution to this problem in 3 steps: (1) we describe a simple but expressive language called DBSP for describing computations over data streams; (2) we give a new mathematical definition of IVM and a general algorithm for solving IVM for arbitrary DBSP programs, and (3) we show how to model many rich database query languages using DBSP (including the full relational algebra, queries over sets and multisets, arbitrarily nested relations, aggregation, flatmap (unnest), monotonic and non-monotonic recursion, streaming aggregation, and arbitrary compositions of all of these). SQL and Datalog can both be implemented in DBSP. As a consequence, we obtain efficient incremental view maintenance algorithms for queries written in all these languages.

Differential polynomial identities of upper triangular matrices of size three

We determine the differential polynomial identities of 3×3 upper triangular matrices over a base field of characteristic zero, under the action of its full Lie algebra of derivations. We compute the exact differential codimension sequence of the multilinear ones and describe their Sn-structure by means of an explicit decomposition of the Sn-cocharacter of their proper part.

Universal corner symmetry and the orbit method for gravity

A universal symmetry algebra organizing the gravitational phase space has been recently found. It corresponds to the subset of diffeomorphisms that become physical at corners – codimension-2 surfaces supporting Noether charges. It applies to both finite distance and asymptotic corners. In this paper, we study this algebra and its representations, via the coadjoint orbit method. We show that generic orbits of the universal algebra split into sub-orbits spanned by finite distance and asymptotic corner symmetries, such that the full universal symmetry algebra gives rise to a unified treatment of corners in a manifold. We then identify the geometric structure that captures these algebraic properties on corners, which is the Atiyah Lie algebroid associated to a principal GL(2,R)⋉R2-bundle. This structure is suggestive of the existence of a novel quantum gravitational theory which would unitarily glue such geometric structures, with spacetime geometries appearing as semi-classical configurations.

Learning rotations

Many problems in computer vision today are solved via deep learning. Tasks like pose estimation from images, pose estimation from point clouds or structure from motion can all be formulated as a regression on rotations. However, there is no unique way of parametrizing rotations mathematically: matrices, quaternions, axis‐angle representation or Euler angles are all commonly used in the field. Some of them, however, present intrinsic limitations, including discontinuities, gimbal lock or antipodal symmetry. These limitations may make the learning of rotations via neural networks a challenging problem, potentially introducing large errors. Following recent literature, we propose three case studies: a sanity check, a pose estimation from 3D point clouds and an inverse kinematic problem. We do so by employing a full geometric algebra (GA) description of rotations. We compare the GA formulation with a 6D continuous representation previously presented in the literature in terms of regression error and reconstruction accuracy. We empirically demonstrate that parametrizing rotations as bivectors outperforms the 6D representation. The GA approach overcomes the continuity issue of representations as the 6D representation does, but it also needs fewer parameters to be learned and offers an enhanced robustness to noise. GA hence provides a broader framework for describing rotations in a simple and compact way that is suitable for regression tasks via deep learning, showing high regression accuracy and good generalizability in realistic high‐noise scenarios.