Tensor product of partial modules

Solution to a conjecture by Hofmeier–Wittstock

The tensor product is a fundamental mathematical concept with applications spanning linear algebra, graph theory, quantum computing, and representation theory. In graph theory, the tensor product provides a framework for analyzing structural relationships, particularly through the study of complete graphs, which yield complex networks from simple structures. Closely related is the Kronecker product of matrices, an essential tool for investigating tensor products via adjacency matrices. The Kronecker product preserves key algebraic properties, including linearity, distributivity, and associativity, and has played a central role in matrix analysis, systems theory, and signal processing. This work presents the definitions and core properties of the tensor product, supported by illustrative examples with complete graphs, and explores the Kronecker product along with its fundamental properties and theorems. By combining theoretical foundations with applications, the study offers both conceptual insights and practical perspectives on these algebraic constructions.

Fourier transform algorithms are described using tensor (Kronecker) products and an associated class of permutations. Algebraic properties of tensor products and the related permutations are used to derive variants of the Cooley-Tukey fast Fourier transform algorithm. These algorithms can be implemented by translating tensor products and permutations to programming constructs. An implementation can be matched to a specific computer architecture by selecting the appropriate variant. This methodology is carried out for the Cray X-MP and the AT&T DSP32.

In the context of operator-space modules over C ∗ C^* -algebras, we give a complete characterisation of those C ∗ C^* -correspondences whose associated Haagerup tensor product functors admit left adjoints. The characterisation, which builds on previous joint work with N. Higson, exhibits a close connection between the notions of adjoint operators and adjoint functors. As an application, we prove a Frobenius reciprocity theorem for representations of locally compact groups on operator spaces: the functor of unitary induction for a closed subgroup H H of a locally compact group G G admits a left adjoint in this setting if and only if H H is cocompact in G G . The adjoint functor is given by the Haagerup tensor product with the operator-theoretic adjoint of Rieffel’s induction bimodule.

Recall that a left module A (resp. right module B) is said to be singly injective (resp. singly flat) if $$Ext^{1}_{R}(F/K, A) = 0$$ (resp. $${\text {Tor}}_{1}^{R}(B, F/K)= 0$$ ) for any cyclic submodule K of any finitely generated free left R-module F. In this paper, we continue to study and investigate the homological objects related to singly flat and singly injective modules and module homomorphisms. Along the way, the right orthogonal class of singly flat right modules and the left orthogonal class of singly injective left modules are introduced and studied. These concepts are used to extend the some known results and to characterize pseudo-coherent rings and left singly injective rings. In terms of some derived functors, some homological dimensions are investigated. As applications, some new characterizations of von Neumann regular rings and left PP rings are given. Finally, we study the singly flatness and singly injectivity of homomorphism modules over a commutative ring.

Homomorphisms of progenerator modules

This chapter is devoted to tensor categories which axiomatize the properties of tensor products of vector spaces. Its importance became more evident when quantum groups produced rich examples of non commutative tensor categories and this notion is now used in many areas, mathematical physics, knot theory, computer sciences, etc. Tensor categories and their applications deserve at least a whole book, and we shall be extremely superficial and sketchy here. Among the vast literature on this subject, let us only quote [15, 40].We begin this chapter by introducing projectors in categories. Then we define and study tensor categories, dual pairs, braidings and the Yang-Baxter equations. We also introduce the notions of a ring in a tensor category and a module over this ring in a category on which the tensor category operates. As a particular case we treat monads, and finally we prove the Bar-Beck theorem.Most of the notions introduced in this Chapter (with the exception of §4.1) are not necessary for the understanding of the rest of the book, and this chapter may be skipped.KeywordsTensor ProductFull SubcategoryDual PairMonoidal CategoryTensor CategoryThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Bilinear transformations and bilinear forms are introduced and studied. Their matrix representation, and especially the representation of symmetric bilinear forms, is presented. Orthogonality relative to a bilinear form is considered. These notions are then used to define the tensor product of vector spaces, and the properties of the tensor product are considered in detail.

In this work, we introduce the notion of tensor product of spaces with a generalized 2-inner product (see Definition 5), and we establish several interesting properties (see Proposition5), thereby generalizing the classical properties of the tensor product of inner product spaces. Moreover, we equip this tensor product with a mapping that defines a generalized 2-inner product(see Theorem 3) and, consequently, endow it with a generalized 2-norm (see Theorem 1). In this context, we also define the tensor product of linear operators (see Definition 9) and prove a series of results for example, that the tensor product of two 2-bounded linear operators is again 2-bounded under the tensor product (see Proposition 10).

One of the main problems in the theory of quaternion quantum mechanics has been the construction of a tensor product of quaternion Hilbert modules. A solution to this problem is given by studying the tensor product of quaternion algebras (over the reals) and some of its quotient modules. Real, complex, and (covariant) quaternion scalar products are found in the tensor product spaces. Annihilationcreation operators are constructed, corresponding to the second quantization of the quaternion quantum theory with Bose-Einstein or Fermi-Dirac statistics. The gauge transformations of a tensor product vector and the gauge fields are studied.

In this paper, we have the principal goal is to study a topography property of important algebraic construction namely the quotient module. We use a new tool with a quotient module which is a tensor product of modules. Therefore all topography submodules in this notion are a tensor product. The meaning of the tensor module introduced in this notion and the important fact of this article is to explain the quotient module when all submodules are tensor. Finally, several results have been obtained about the direct sum of the finite quotient module.

The authors obtain in this paper a classification of projective tensor products of C(K) spaces, in terms of the behaviour of certain classes of multilinear operators on the product of the spaces, or the verification of certain Banach space properties of the corresponding tensor product. The main tool used is an improvement of some results of Emmanuele and Hensgen on the reciprocal Dunford-Pettis and Pełczy´nski’s (V) properties of the projective tensor product of Banach spaces. Finally, the paper ends with a study of the relationships between some classes of multilinear operators and their linearizations.

Casimir invariants and infinitesimal characters for semi-simple Lie algebras

In this paper, we define fractal bases and fractal frames of L2(I×J), where I and J are real compact intervals, in order to approximate two-dimensional square-integrable maps whose domain is a rectangle, using the identification of L2(I×J) with the tensor product space L2(I)⨂L2(J). First, we recall the procedure of constructing a fractal perturbation of a continuous or integrable function. Then, we define fractal frames and bases of L2(I×J) composed of product of such fractal functions. We also obtain weaker families as Bessel, Riesz and Schauder sequences for the same space. Additionally, we study some properties of the tensor product of the fractal operators associated with the maps corresponding to each variable.

In the present paper, we provide a decomposition of a k-tridiagonal Toeplitz matrix via tensor product. By the decomposition, the required memory of the matrix is reduced and the matrix is easily analyzed since we can use properties of tensor product.