Verma Howe duality connects tensor products of Verma modules to LKB representations.
problem Understanding the relationship between tensor products of Verma modules and LKB representations.
method Established a quantized version of Verma Howe duality and used it to prove the simplicity of LKB representations.
result LKB representations arise from the quantized Verma Howe duality and are shown to be simple modules.
The paper studies algebraic structures related to quantum groups.
problem Understanding centralisers of tensor representations of Uq(glN). method Introducing fused permutations and braids, proving Schur--Weyl duality, and describing centralisers.
result A conjecture about a generating element of centralisers is proven in some cases.
Extends Lawrence's representations to integral Uqsl(2) Verma-modules and braid groups.
problem Integrating Lawrence's representations into Uqsl(2) Verma-modules and braid groups. method Defining homological operators and showing they provide a representation for Uqsl(2), establishing isomorphisms and preserving key properties. result Recovering an integral version of Kohno's theorem for Verma-modules and braid group representations.
New braided Frobenius algebras created from specific Hopf algebras.
problem Creating new algebraic structures from Hopf algebras.
method Heap operation and Yang-Baxter operator on tensor product.
result Heap operation induces a braiding compatible with Frobenius operations.
We study the structure of the virtual braid group. It is shown that the virtual braid group is a semi--direct product of the virtual pure braid group and the symmetric group. Also, it is shown that the virtual pure braid group is a semi--direct product of free groups. From these results we obtain a normal form of words…
The paper explores how the unit inclusion affects topological quantum field theories in non-semisimple categories.
problem Understanding the effects of unit inclusion in non-semisimple braided tensor categories on topological quantum field theories.
method Analyzes the dualizability of the unit inclusion morphism in Morita 4-category of braided tensor categories and applies the Cobordism Hypothesis.
result Shows that the unit inclusion in non-semisimple modular categories leads to non-compact relative 3D topological quantum field theories.
The paper studies algebraic and geometric properties of stated skein algebras of surfaces.
problem Understanding the algebraic and geometric properties of stated skein algebras of surfaces.
method Analyzes the skein algebra of surfaces, proving isomorphisms and lifting properties, and interpreting topologically.
result The skein algebra of a surface with n boundary components is an algebra-comodule over Oq2(SL(2))⊗n. Virtual singular braids embed in a group with normal form.
problem Embedding virtual singular braids into algebraic structures.
method Presented a semi-direct product structure and provided a normal form.
result Virtual singular braid group VSGn is a semi-direct product of VSPGn and Sn. This study introduces a unified cohomology theory for braided algebras.
problem Classifying infinitesimal deformations of braided algebras.
method Developed a cohomology theory unifying Hochschild and Yang-Baxter cohomology.
result The second cohomology group classifies infinitesimal deformations of braided algebras.
An analysis is made of reality conditions within the context of noncommutative geometry. We show that if a covariant derivative satisfies a given left Leibniz rule then a right Leibniz rule is equivalent to the reality condition. We show also that the matrix which determines the reality condition must satisfy the Yang-…
We categorify the notion of an infinitesimal braiding in a linear strict symmetric monoidal category, leading to the notion of a (strict) infinitesimal 2-braiding in a linear symmetric strict monoidal 2-category. We describe the associated categorification of the 4-term relation, leading to six categorified relations. …
We point out that the Homfly polynomial (that is to say, Ocneanu's trace functional) contains two polynomial-valued inner products on the Hecke algebra representation of Artin's braid group. These bear a close connection to the Morton-Franks-Williams inequality. In these structures, the sets of positive, respectively n…
We first show that the braid group over a graph topologically containing no Θ-shape subgraph has a presentation related only by commutators. Then using discrete Morse theory and triple Massey products, we prove that a graph topologically contains none of four prescribed graphs if and only if its 4-braid groups is a r…
Study on virtual singular braid groups with algebraic properties and homomorphisms.
problem Algebraic properties and homomorphisms of virtual singular braid groups.
method Numerical invariants, homomorphisms, semi-direct product decompositions, presentations, and quotients.
result Determined all group homomorphisms from VSGn to Sn and obtained corresponding semi-direct product decompositions. New braid group action defined on projective quantum sl(2) modules.
problem Defining a new braid group action on quantum sl(2) modules.
method Action via R-matrix on tensor powers of simple projective modules.
result The action is faithful for the extended representation.
The tensor-tensor product (t-product) [M. E. Kilmer and C. D. Martin, 2011] is a natural generalization of matrix multiplication. Based on t-product, many operations on matrix can be extended to tensor cases, including tensor SVD, tensor spectral norm, tensor nuclear norm [C. Lu, et al., 2018] and many others. The line…
The study finds a subgroup of graph braid groups that is a direct product of non-abelian free groups.
problem Finding subgroups of graph braid groups with specific properties.
method Observing the structure of graph braid groups and their subgroups.
result A subgroup of the same cohomological dimension is a direct product of non-abelian free groups.
The paper defines and analyzes curvature tensors on super twisted product spaces.
problem Investigating curvature tensors on super twisted product spaces.
method Defined W2-curvature tensor, computed curvature tensors and Ricci tensors, and studied curvature flatness. result Mixed Ricci-flat super twisted product semi-Riemannian manifolds can be expressed as super warped product manifolds.
This paper presents a decomposition of the 3-strand Singular Braid Group.
problem Decomposing the 3-strand Singular Braid Group.
method Investigation of the group ST3 through presentations and isomorphisms. result Proves ST3 is isomorphic to the singular pure braid group SP3 and is a semi-direct product. We prove that the n th pure braid group of a nonorientable surface (closed or with boundary, but different from RP2) is residually 2-finite. Consequently, this group is residually nilpotent. The key ingredient in the closed case is the notion of p-almost direct product, which is a generalization of the notion of almost…
Graphical notation simplifies tensor operations and decompositions.
problem Complex tensor operations are difficult to understand and represent.
method Introduces graphical notation to represent tensor operations.
result Simplified representation of tensor operations and decompositions.
The paper defines the OU matrix for braid diagrams and finds determinant relationships.
problem Understanding the layeredness of braid diagrams.
method Defining the OU matrix and analyzing its determinant for layered braid diagrams.
result The determinant of the OU matrix for layered braid diagrams is the product of the determinants of the layers.
We give a computational algorithm which decides if a braid is quasipositive or not. A braid is quasipositive if it's a product of conjuguates of generators. For this, we use the theory of Garside and the combinatorials properties of the Artin monoid.
Killing tensors on reducible spaces are reducible, except for special cases.
problem Characterizing Killing tensors on reducible spaces.
method Analyzing Killing tensors on product manifolds and their lifts.
result Killing tensors on product manifolds are reducible, except for specific cases.
Paper uses Long-Moody construction for new braid group representations.
problem Constructing new representations of braid groups.
method Applies Fox derivation to matrix presentation of Long-Moody construction.
result Shows relation to twisted Alexander invariants.
Study on embedding tree products into groups, distinguishing them.
problem Quasi-isometric embedding of tree products into various groups.
method Using coarse embeddings of products of bushy trees into hierarchically hyperbolic spaces.
result Quasi-isometrically distinguish and rule out embeddings between groups.
The paper analyzes orbits of integer tuples using braid diagrams.
problem Determining orbits of integer tuples under braid diagram actions.
method Monoid action of braid diagrams on integer tuples.
result Orbits of integer tuples under up-down action of braid diagrams.
We study the Morton-Franks-Williams inequality for closures of simple braids (also known as positive permutation braids). This allows to prove, in a simple way, that the set of simple braids is a orthonormal basis for the inner product of the Hecke algebra of the braid group defined by Kálmán, who first obtained this r…
Classifies orbits of Hurwitz actions on dihedral quandles.
problem Classifying orbits of Hurwitz actions on dihedral quandles.
method Introduced three computable invariants to classify orbits.
result Complete classification of orbits under Hurwitz action.
It has been conjectured that every (2+1)-TQFT is a Chern-Simons-Witten (CSW) theory labelled by a pair (G,λ), where G is a compact Lie group, and λ∈H4(BG;Z) a cohomology class. We study two TQFTs constructed from Jones' subfactor theory which are believed to be counterexamples to this conjecture: one is the…
We show that the sutured Khovanov homology of a balanced tangle in the product sutured manifold D x I has rank 1 if and only if the tangle is isotopic to a braid.
Two formulas for Chern classes of tensor products of vector bundles are presented.
problem Calculating Chern classes for tensor products of vector bundles.
method Two formulas using matrices and polynomials to compute Chern classes.
result Determinantal formulas for Chern classes of tensor products.
Paper studies pseudo-projective tensors on warped products.
problem Characterizing pseudo-projectively flat warped products.
method Analyzes sequential warped products and pseudo-projective tensors.
result Necessary and sufficient conditions for pseudo-projectively flat sequential warped products.
Explores tensor products in hyperdimensional computing.
problem Understanding tensor products in hyperdimensional computing.
method Generalized results from graph embeddings to vector symbolic architectures and hyperdimensional computing.
result Tensor product is the most general and expressive representation with errorless unbinding and detection.
In this paper we study the kernel of the homomorphism Bg,n→Bn of the braid group Bg,n in the handlebody Hg to the braid group Bn. We prove that this kernel is a semi-direct product of free groups. Also, we introduce an algebra Hg,n(q), which is some analog of the Hecke algebra $H_n(q…
Introduces tensor product for quiver representations and applies to stable bundles and character varieties.
problem Stability and classification of quiver bundles and their subvarieties.
method Definition of tensor product for quiver representations and application to stability and character varieties.
result Tensor products of polystable quiver bundles are polystable and provide insights into character varieties.
Some invariant tensors in two Naveira classes of Riemannian product manifolds are considered. These tensors are related with natural connections, i.e. linear connections preserving the Riemannian metric and the product structure.
Properties of pairs of product conjugate connections are stated with a special view towards the integrability of the given almost product structure. We define the analogous in product geometry of the structural and the virtual tensors from the Hermitian geometry and express the product conjugate connections in terms of…
Defines tensor products for A-infinity structures using diagonals.
problem No specific problem stated; focuses on new definitions.
method Uses diagonals of associahedra and multiplihedra to define tensor products.
result Defines tensor products for various A-infinity structures.
Study of wild mapping class groups and their cabled braids.
problem Understanding the structure of wild mapping class groups and their cabled versions.
method Define and study generalizations of pure g-braid groups, establish product decompositions, and introduce fission trees. result Obtain cabled versions of braid groups, related to braid operads.
The study extends Jacobi-orthogonality to indefinite scalar product spaces.
problem Generalizing Jacobi-orthogonality to indefinite scalar product spaces.
method Comparing principles, investigating tensor relations, proving properties.
result Every quasi-Clifford tensor is Jacobi-orthogonal; certain tensors are Jacobi-dual or Osserman.
The mixed braid groups are the subgroups of Artin braid groups whose elements preserve a given partition of the base points. We prove that the centralizer of any braid can be expressed in terms of semidirect and direct products of mixed braid groups. Then we construct a generating set of the centralizer of any braid on…
Formulates quantum jet bundles over noncommutative algebras with connections and braiding.
problem Defining jet bundles over noncommutative algebras with connections and braiding.
method Formalizes jet bundles over noncommutative algebras with flat connections and braiding tensor obeying Yang-Baxter equation.
result Examples include permutation groups, matrix algebras, and quantum spacetime models.
Defines semi-symmetric metric connection on super warped products.
problem Computing curvature and Ricci tensors on super warped products.
method Introduced semi-symmetric metric connection and conditions for Einstein spaces.
result Conditions for super warped product spaces to be Einstein with semi-symmetric metric connection.
New braid group representations into mapping class groups are derived from disk coverings.
problem Tackling the representation of braid groups into mapping class groups.
method Explicit expressions of braid group representations into automorphism groups of free groups using covering groupoids.
result Each generator of braid group inside mapping class group is a product of Dehn twists.
New concept of boundary braids defined for disk configurations.
problem Defining and studying braids with points fixed on the boundary.
method Using configuration spaces and fundamental groups, defining boundary braids and analyzing their geometric properties.
result Boundary braids form a subcomplex that metrically splits into a Euclidean polyhedron and a smaller rank dual braid complex.
The paper studies special warped products with a specific connection on super Riemannian manifolds.
problem Investigating curvature and Ricci tensors on super warped product spaces with a semi-symmetric non-metric connection.
method Defined a semi-symmetric non-metric connection, computed curvature and Ricci tensors, and introduced and analyzed two types of super warped product spaces.
result Conditions for two super warped product spaces with a semi-symmetric non-metric connection to be Einstein spaces are provided.
We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show that these invariants coincide with previous invariants defined by Khovanov for sl(2) and sl(3) and by Mazorchuk-Stroppel and Sussan for sl(n). Our technique uses categorifications of the tensor produ…