Tensor product of partial modules

We study tensor product decompositions of representations of the General Linear Supergroup Gl(m|n). We show that the quotient of Rep(Gl(m|n),\epsilon)$ by the tensor ideal of negligible representations is the representation category of a pro-reductive supergroup G red. In the Gl(m|1)-case we show G red = Gl(m-1) \times Gl(1) \times Gl(1). In the general case we study the image of the canonical tensor functor Fmn from Deligne's interpolating category Rep (Gl m-n) to Rep(Gl(m|n),\epsilon). We determine the image of indecomposable elements under Fmn. This implies tensor product decompositions between projective modules and between certain irreducible modules, including all irreducible representations in the Gl(m|1)-case. Using techniques from Deligne's category we derive a closed formula for the tensor product of two maximally atypical irreducible Gl(2|2)-representations. We study cohomological tensor functors DS : Rep(Gl(m|m), epsilon) -> Rep(Gl(m-1|m-1)) and describe the image of an irreducible element under DS. At the end we explain how these results can be used to determine the pro-reductive group G L \hookrightarrow Gl(m|m) red corresponding to the subcategory Rep(G L, epsilon) generated by the image of an irreducible element L in Rep(Gl(m|m) red, epsilon).

In this paper we introduce a new device, which enables us to treat the quaternions as though they were a commutative field. This is of interest both for its own sake, and because it can be applied to develop an algebraic geometry of noncompact hypercomplex manifolds. The basic building blocks of the theory are AH modules, which should be thought of spaces over the quaternions. An AH-module is a left module over the quaternions H, together with a real vector subspace. There are natural concepts of linear map and tensor product of AH-modules, which have many of the properties of linear maps and tensor products of vector spaces. However, the definition of tensor product of AH-modules is strange and has some unexpected properties. Let M be a hypercomplex manifold. Then there is a natural class of H-valued on M, satisfying a quaternionic analogue of the Cauchy-Riemann equations, which are analogues of holomorphic functions on complex manifolds. The vector space of q-holomorphic functions A on M is an AH-module. Now some pairs of q-holomorphic functions can be multiplied together to get another q-holomorphic function, but other pairs cannot. So A has a kind of partial algebra structure. It turns out that this structure can be very neatly described using the quaternionic tensor product and AH-morphisms, and that A has the structure of an H-algebra, a quaternionic analogue of commutative algebra.

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.

Let R be a commutative ring. We investigate the relationship between (pre)covers ((pre)envelopes) of an R-module and the counterparts of the corresponding homomorphism module or tensor product module. Some applications are also given.

Let R be a commutative ring. We write Hom( A; B) for the set of all fuzzy R-morphisms from A to B, where A and B are two fuzzy R-modules. We make Hom( A; B) into fuzzy R-module by redening a func- tion : Hom( A; B) ! (0; 1). We study the properties of the functor Hom( A; ) : FR-Mod! FR-Mod and get some unexpected results. In ad- dition, we prove that Hom( p; ) is exact if and only if P is a fuzzy projective R-module, when R is a commutative semiperfect ring. Finally, we investigate tensor product of two fuzzy R-modules and get some related properties. Also, we study the relationships between Hom functor and tensor functor.

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.

It is proved that each Schurian algebra is tensor-simple. Let K be a field, for any algebra A denote by  the set of isoclasses of simple A-modules. A module means a left module. If M and N are A- and B-modules respectively, then their tensor product M⊗ N (over K,⊗ means ⊗ k) is the module over the tensor product A⊗ B of algebras A and B.

Mathematical structures often start from simple ones and are extended by various constructions to structures of increasing complexity. This process is to be controlled, and the guiding lines should include, among other things, applicability to problems of interest. The present chapter is devoted to a circle of ideas from abstract linear algebra that covers several topics of interest to us because of their applicability to the study of linear systems. These topics include bilinear forms defined on vector spaces, module homomorphisms over various rings, and the analysis of classes of special structured matrices such as Bezoutian, Hankel, and Toeplitz matrices.

This paper discusses the generalization of the fundamental theorems of -module homomorphisms to the structure of -bimodules, where and are rings with identity. The study begins with a review of the definitions, properties, and types of -bimodule homomorphisms. Subsequently, three fundamental theorems of -module homomorphisms are generalized to the -bimodule setting. The results show that the fundamental structures and relationships in module theory can be naturally extended to bimodules by considering the actions of two rings that are compatible with the bimodule operations. This generalization provides a broader framework for studying algebraic structures involving two interacting ring actions.This paper discusses the generalization of the fundamental theorems of -module homomorphisms to the structure of -bimodules, where and are rings with identity. The study begins with a review of the definitions, properties, and types of -bimodule homomorphisms. Subsequently, three fundamental theorems of -module homomorphisms are generalized to the -bimodule setting. The results show that the fundamental structures and relationships in module theory can be naturally extended to bimodules by considering the actions of two rings that are compatible with the bimodule operations. This generalization provides a broader framework for studying algebraic structures involving two interacting ring actions.

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.

Although the categoryCLC of complete lattices and complete homomorphisms does not possess arbitrary coproducts, we show that the tensor product introduced by Wille has the universal property of coproducts for so-called distributing families of morphisms (and only for these). As every family of morphisms into a completely distributive lattice is distributing, this includes the known fact that in the category of completely distributive lattices, arbitrary coproducts exist and coincide with the tensor products. Since the definition of tensor products is based on the notion of contexts and their concept lattices, many results on tensor products extend from complete lattices to contexts. Thus we introduce two kinds of tensor products for arbitrary families of contexts, a “partial” and a “complete” one, and establish universal properties of these tensor products.

II.8 - CROSS PRODUCT IN FOUR DIMENSIONS AND BEYOND

We first prove the resonance theorem, closed graph theorem, inverse operator theorem, and open mapping theorem for module homomorphisms between random normed modules by simultaneously considering the two kinds of topologies—the -topology and the locally -convex topology for random normed modules. Then, for the future development of the theory of module homomorphisms on complete random inner product modules, we give a proof with better readability of the known orthogonal decomposition theorem and Riesz representation theorem in complete random inner product modules under two kinds of topologies. Finally, to connect module homomorphism between random normed modules with linear operators between ordinary normed spaces, we give a proof with better readability of the known result connecting random conjugate spaces with classical conjugate spaces, namely, , where and are a pair of Hölder conjugate numbers with a random normed module, the random conjugate space of the corresponding (resp., ) space derived from (resp., ), and the ordinary conjugate space of