Related papers: R-matrices and Tensor Product Graph Method
We propose a method to compute the $R$-matrix $R$ on a tensor product of Fock modules from coproduct relations in a Hopf algebra. We apply this method to the quantum toroidal algebra $U_{q,t}(\overset{..}{gl}_1)$ for which $R$ is currently…
Tensor factorization is a powerful tool to analyse multi-way data. Compared with traditional multi-linear methods, nonlinear tensor factorization models are capable of capturing more complex relationships in the data. However, they are…
Pairings are particular bilinear maps, and as any bilinear maps they factor through the tensor product as group homomorphisms. Besides, nothing seems to prevent us to construct pairings on other abelian groups than elliptic curves or more…
Daily internet communication relies heavily on tree-structured graphs, embodied by popular data formats such as XML and JSON. However, many recent generative (probabilistic) models utilize neural networks to learn a probability distribution…
We propose a method to construct the initial tensor representation of partition functions and observables for the tensor renormalization group (TRG). The TRG is a numerical calculation technique that utilizes a tensor network…
Let $\mathfrak{g}$ be a symmetrizable Kac-Moody algebra, $U_q(\mathfrak{g})$ its quantum group, and $U_q(\mathfrak{k}) \subset U_q(\mathfrak{g})$ a quantum symmetric pair subalgebra determined by a Lie algebra automorphism $\theta$. We…
It is well-known that the tensor product of two bialgebras constitutes the binary product in the category of cocommutative bialgebras and morphisms of bialgebras between them. In this paper, we extend this result to triangular bialgebras…
In this paper we give a method to associate a graph with an arbitrary density matrix referred to a standard orthonormal basis in the Hilbert space of a finite dimensional quantum system. We study the related issues like classification of…
This work concerns the construction and characterization of product kernels for multivariate approximation from a finite set of discrete samples. To this end, we consider composing different component kernels, each acting on a…
For a quasitriangular C*-quantum group, we enrich the twisted tensor product constructed in the first part of this series to a monoidal structure on the category of its continuous coactions on C*-algebras. We define braided C*-quantum…
Motivated by Maulik-Okounkov stable maps associated to quiver varieties, we define and construct algebraic stable maps on tensor products of representations in the category O of the Borel subalgebra of an arbitrary untwisted quantum affine…
We survey the matrix product solutions of the Yang-Baxter equation obtained recently from the tetrahedron equation. They form a family of quantum $R$ matrices of generalized quantum groups interpolating the symmetric tensor representations…
Braided groups and braided matrices are novel algebraic structures living in braided or quasitensor categories. As such they are a generalization of super-groups and super-matrices to the case of braid statistics. Here we construct braided…
For arbitrary F-algebra, in which the operation of addition is defined, I explore biring of matrices of mappings. The sum of matrices is determined by the sum in F-algebra, and the product of matrices is determined by the product of…
The Norton product is defined on each eigenspace of a distance regular graph by the orthogonal projection of the entry-wise product. The resulting algebra, known as the Norton algebra, is a commutative nonassociative algebra that is useful…
We prove a variety results on tensor product factorizations of finite dimensional Hopf algebras (more generally Hopf algebras satisfying chain conditions in suitable braided categories). The results are analogs of well-known results on…
Quantum machine learning (QML) is a rapidly expanding field that merges the principles of quantum computing with the techniques of machine learning. One of the powerful mathematical frameworks in this domain is tensor networks. These…
The algebraic formulation of the quantum group gauge models in the framework of the $R$-matrix approach to the theory of quantum groups is given. We consider gauge groups taking values in the quantum groups and noncommutative gauge fields…
We compute the braided groups and braided matrices $B(R)$ for the solution $R$ of the Yang-Baxter equation associated to the quantum Heisenberg group. We also show that a particular extension of the quantum Heisenberg group is dual to the…
A classical t-tensor product expander is a natural way of formalising correlated walks of t particles on a regular expander graph. A quantum t-tensor product expander is a completely positive trace preserving map that is a straightforward…