Unified approach to decomposing commutative n-ary superalgebras with skew-symmetric forms.
problem Decomposing commutative n-ary superalgebras with skew-symmetric invariant forms.
method Unified approach using derived bracket formalism and inductive orthogonal sums and generalized double extensions.
result Any commutative n-ary superalgebra with a skew-symmetric invariant form can be obtained by inductive orthogonal sums and generalized double extensions.
We study n-ary commutative superalgebras and L∞-algebras that possess a skew-symmetric invariant form, using the derived bracket formalism. This class of superalgebras includes for instance Lie algebras and their n-ary generalizations, commutative associative and Jordan algebras with an invariant form. We…
A notion of n-Lie algebra introduced by V.T. Filippov can be viewed as a generalization of a concept of binary Lie algebra to the algebras with n-ary multiplication law. A notion of Lie algebra can be extended to Z_2-graded structures giving a notion of Lie superalgebra. Analogously a notion of n-Lie algebra can be ext…
Study of n-ary differential operators on weighted densities with canonical symbol and quantization maps.
problem Analysis of n-ary differential operators acting on weighted densities. method Existence and uniqueness of conformally equivariant symbol maps and quantization maps.
result Existence and explicit expression of conformally equivariant symbol and quantization maps.
The paper introduces transposed Poisson superalgebras and their properties.
problem Developing algebraic structures for quantum physics.
method Defining transposed Poisson superalgebras using derivations and proving their identities.
result Six identities are crucial for understanding transposed Poisson superalgebras.
We show that the first-order symmetry operators of twistor spinors can be constructed from conformal Killing-Yano forms in conformally-flat backgrounds. We express the conditions on conformal Killing-Yano forms to obtain mutually commuting symmetry operators of twistor spinors. Conformal superalgebras which consist of …
Extends Nambu-Poisson bracket to superspace R^{n|m}.
problem No new problem introduced.
method Constructs Nambu-Poisson algebras of even degree functions using superdeterminant.
result Proves the n-ary bracket satisfies conditions for Nambu-Poisson bracket in R^{n|1} and R^{n|2}.
Study examines Lie algebroids with homological sections, generalizing Q-manifolds and Lie superalgebras.
problem Exploring Lie algebroids with homological sections.
method Derived bracket formalism to define an odd Loday-Leibniz bracket on sections.
result Sections of inner Q-algebroids come equipped with an odd Loday-Leibniz bracket.
Three methods combine one-class classifiers with MST-CD and N-ary Trees for binary classification.
problem Binary classification with overlapping and imbalanced classes.
method Combining one-class classifiers with MST-CD and N-ary Trees to handle inconsistencies and spurious connections.
result The proposed methods are feasible and comparable to state-of-the-art algorithms.
Study cohomology spaces of sl(2) acting on n-ary differential operators.
problem Computing cohomology spaces for sl(2) action on n-ary differential operators.
method Analyzes polynomial μ-densities as sl(2) modules and computes cohomological spaces H^2.
result Computed cohomological spaces H^2 of sl(2) on n-ary differential operators.
These notes are devoted to the multiple generalization of a Lie algebra introduced by A.M.Vinogradov and M.M.Vinogradov. We compare definitions of such algebras in the usual and invariant case. Furthermore, we show that there are no simple n-ary Lie algebras of type (n−1,l) for l>0.
We show that one can skip the skew-symmetry assumption in the definition of Nambu-Poisson brackets. In other words, a n-ary bracket on the algebra of smooth functions which satisfies the Leibniz rule and a n-ary version of the Jacobi identity must be skew-symmetric. A similar result holds for a non-antisymmetric versio…
A C-infinity ring is a set equipped with n-ary operations corresponding to smooth n-ary functions on the real line (satisfying natural axioms). We prove that the cosimplicial abelian group associated to the de Rham complex of Euclidean space has the structure of a cosimplicial C-infinity ring. We also analyse the notio…
Paper proposes a novel model to improve n-ary cross-sentence relation extraction by addressing noisy data and non-consecutive sentences.
problem Noisy labeled data and non-consecutive sentences in n-ary cross-sentence relation extraction.
method Two-level agent reinforcement learning model and hybrid attention mechanism/PCNN approach.
result The model reduces the impact of noisy data and achieves better performance.
The study examines conditions for completeness and simplicity in hom-Lie superalgebras.
problem Conditions for completeness and simplicity in hom-Lie superalgebras.
method Equivalent conditions and derivations analysis.
result Conditions for completeness and simplicity in hom-Lie superalgebras.
Extends graph theory to hypergraphs with manifold-valued nodes.
problem Representing complex N-ary relationships on manifolds.
method Defined function spaces and symmetric products for manifold-valued nodes and edges.
result Generalized hypergraph Laplacians to manifold-valued hypergraphs.
The paper defines Z-graded hom-Lie superalgebras and explores their properties.
problem Understanding the structure and properties of Z-graded hom-Lie superalgebras.
method Definition and exploration of Z-graded hom-Lie superalgebras, invariant bilinear forms, and simplicity conditions.
result Maximal and minimal Z-graded hom-Lie superalgebras for local hom-Lie superalgebras are identified, and conditions for simplicity are checked.
New invariant for 3-manifolds from topological Hopf superalgebra.
problem Constructing an invariant for 3-manifolds.
method Using a ribbon superalgebra from a topological Hopf superalgebra, we construct an invariant of 3-manifolds of Hennings type.
result Constructs a new invariant for 3-manifolds from a topological Hopf superalgebra.
Killing-Yano and conformal Killing-Yano superalgebras are rigid in constant curvature manifolds.
problem Understanding the rigidity of Killing-Yano and conformal Killing-Yano superalgebras in constant curvature manifolds.
method Defining Z-gradations and filtrations, showing trivial second cohomology groups, and proving non-deformability. result Killing-Yano and conformal Killing-Yano superalgebras are rigid and correspond to geometric invariants of constant curvature manifolds.
Starting from the four normed division algebras - the real numbers, complex numbers, quaternions and octonions - a systematic procedure gives a 3-cocycle on the Poincare Lie superalgebra in dimensions 3, 4, 6 and 10. A related procedure gives a 4-cocycle on the Poincare Lie superalgebra in dimensions 4, 5, 7 and 11. In…
We present supersymmetric, curved space, quantum mechanical models based on deformations of a parabolic subalgebra of osp(2p+2|Q). The dynamics are governed by a spinning particle action whose internal coordinates are Lorentz vectors labeled by the fundamental representation of osp(2p|Q). The states of the theory are t…
Study uses superalgebra homology to classify 4D Engel-like Lie algebras.
problem Classifying 4-dimensional Engel-like Lie algebras.
method Applied homology groups of Lie superalgebras.
result Distinguished and classified 4D Engel-like Lie algebras.
Recent work applying higher gauge theory to the superstring has indicated the presence of `higher symmetry'. Infinitesimally, this is realized by a `Lie 2-superalgebra' extending the Poincare superalgebra in precisely the dimensions where the classical supersymmetric string makes sense: 3, 4, 6 and 10. In the previous …
Study pseudo-Riemannian metrics on Jordan superalgebras.
problem No specific problem stated; focus on metrics.
method Coadjoint orbit-like construction for pseudo-Euclidean Jordan superalgebras.
result Investigated canonical pseudo-Riemannian metrics.
Study Euler and Betti numbers of homology groups for a specific type of superalgebra.
problem Calculating Euler and Betti numbers for homology groups of pre Lie superalgebras.
method Introduced double weighted chain spaces to analyze pre Lie superalgebras of multi-vector fields with polynomial coefficients. Calculated Euler and Betti numbers for these homology groups.
result Derived formulas for Euler and Betti numbers of homology groups of pre Lie superalgebras.
Researchers study Killing superalgebras in 2D manifolds.
problem Characterizing Killing superalgebras in 2D pseudo-Riemannian manifolds.
method Computing Spencer cohomology and filtered deformations of superalgebras.
result Killing superalgebras arise as solutions for geometric and skew-Killing spinors.
This paper investigates higher order generalizations of well known results for Lie algebroids and bialgebroids. It is proved that n-Lie algebroid structures correspond to n-ary generalization of Gerstenhaber algebras and are implied by n-ary generalization of linear Poisson structures on the dual bundle. A Nambu-…
This research classifies deformations of Yang-Baxter operators using cohomology of n-Lie algebras.
problem Classifying deformations of Yang-Baxter operators via cohomology of n-Lie algebras. method Introducing a cohomology theory for n-ary self-distributive objects, showing natural injections and isomorphisms, and constructing deformation theories. result The self-distributive deformations classify the Yang-Baxter operator deformations, with nontrivial examples provided.
Two geometric realizations of F(4) Lie superalgebra found.
problem Geometric realization of F(4) Lie superalgebra. method Explicit geometric realizations as supersymmetries of super-PDE systems.
result Two orders of super-PDE systems found for F(4). From the four normed division algebras--the real numbers, complex numbers, quaternions and octonions, of dimension k=1, 2, 4 and 8, respectively--a systematic procedure gives a 3-cocycle on the Poincare superalgebra in dimensions k+2=3, 4, 6 and 10, and a 4-cocycle on the Poincare superalgebra in dimensions k+3=4, 5, 7…
We determine the structure of the ∗-Lie superalgebra generated by a set of carefully chosen natural operators of an orientable WSD manifold of rank three. This Lie superalgebra is formed by global sections of a natural Lie superalgebra bundle, and turns out to be a product of $\mathbf{sl}(4,\C)$ with the full special…
New cohomology groups generalize Euler number for Lie superalgebras.
problem Generalizing cohomology groups for Lie superalgebras.
method Abstracted Poisson cohomology groups to Poisson-like cohomology groups for general Lie superalgebras.
result De Rham cohomology groups match Poisson-like cohomology groups for differential forms.
New q-series found for osp(2|2n) Lie superalgebra.
problem Finding a q-series invariant for osp(2|2n).
method Invariant series for 3-manifolds associated with Lie superalgebras.
result A q-series found for osp(2|2n).
Let M be a manifold and g a Lie algebra acting on M. Differential forms Omega(M) carry a natural action of Lie derivatives L(x) and contractions I(x) of fundamental vector fields for x \in g. Contractions (anti-) commute with each other, [I(x), I(y)]=0. Together with the de Rham differential, they satisfy the Cartan's …
We study the algebraic structure of the Killing superalgebra of a supersymmetric background of 11-dimensional supergravity and show that it is isomorphic to a filtered deformation of a Z-graded subalgebra of the Poincaré superalgebra. We are able to map the classification problem for highly supersymmetric b…
We determine the Killing superalgebras underpinning field theories with rigid unextended supersymmetry on Lorentzian four-manifolds by re-interpreting them as filtered deformations of Z-graded subalgebras with maximum odd dimension of the N=1 Poincaré superalgebra in four dimensions. Part of this calcula…
New algebraic structures extend Courant algebroids to higher multi-Courant algebroids.
problem Extending Courant algebroid structures to higher multi-Courant algebroids.
method Constructing higher geometric versions of algebraic structures defined by Keller and Waldmann.
result Higher multi-Courant algebroids form a Poisson algebra.
Extends Killing superalgebras to higher dimensions and signatures.
problem Generalizing Killing superalgebras to higher dimensions and signatures.
method Definition of Killing superalgebras for connections on spinor bundles, sufficient conditions for existence, abstract study using Spencer cohomology.
result Existence of Killing superalgebras as filtered deformations of graded subalgebras of the Poincaré superalgebra.
Extends symmetry superalgebras to include all hidden symmetries of manifolds.
problem Tackles hidden symmetries of manifolds generated by Killing spinors.
method Defines generalized symmetry superalgebras, constructs Lie algebra structure, and defines symmetry operators.
result Constructs special Killing-Yano and conformal Killing-Yano forms from bilinears of Killing spinors.
We describe explicitly Lie superalgebra isomorphisms between the Lie superalgebras of first-order superdifferential operators on supermanifolds, showing in particular that any such isomorphism induces a diffeomorphism of the supermanifolds. We also prove that the group of automorphisms of such a Lie superalgebra is a s…
The paper realizes Lie superalgebras G(3) and F(4) as symmetries of supergeometries.
problem Addressing whether exceptional Lie superalgebras G(3) and F(4) are maximal symmetries of supergeometries.
method Considering negatively graded Lie superalgebras for every choice of parabolic subgroup, computing Tanaka-Weisfeiler prolongations, and reducing the structure group when required.
result Realization of 19 inequivalent G(3)-supergeometries and 55 inequivalent F(4)-supergeometries.
The study explores Lie superalgebras constructed from Lie algebras using Schouten-like brackets.
problem Investigate how the core Lie algebra controls the Lie superalgebra.
method Construct Lie superalgebras from abstract Lie algebras using Schouten-like brackets and analyze Betti numbers of super homology groups.
result For low dimensional non-abelian Lie algebras, the Betti numbers of super homology groups provide insights into the control of the core Lie algebra.
We discuss the correspondence between Nahm's equations, the Basu-Harvey-Terashima equations, and Lie superalgebras.
Bounds on knot polynomials for Lie superalgebras of type I.
problem Determining genus bounds for knot polynomials colored by Lie superalgebra representations.
method Proved bounds on the t-degree of knot polynomials, relating it to the number of odd roots and the genus of the knot. result Proved bounds on knot polynomials for Lie superalgebras of type I, showing equality for certain knots.
Defines Killing spinors and bosonic backgrounds in 5D supergravity.
problem Characterizing backgrounds in 5D supergravity.
method Calculates Spencer cohomology, defines Killing spinors, and imposes constraints on spinor connection curvature.
result Recover field equations of 5D supergravity and find new field equations for sp(1)-valued one-form. Classifies kinematical and aristotelian Lie superalgebras and their superspaces.
problem Classifying Lie superalgebras and their associated superspaces.
method Quaternionic formalism to simplify calculations, classification via isomorphism and central extensions, geometric limits to explore relationships.
result 27 homogeneous superspaces classified, some with parameters.
Novel symmetry found in colored HOMFLY polynomials from superalgebras.
problem Understanding symmetries in colored HOMFLY polynomials.
method Exploring the sl(N∣M) superalgebra to find a symmetry. result A symmetry relating polynomials colored by different representations.
Given a finite dimensional representation of a semisimple Lie algebra there are two ways of constructing link invariants: 1) quantum group invariants using the R-matrix, 2) the Kontsevich universal link invariant followed by the Lie algebra based weight system. Le and Murakami showed that these two link invariants are …