Narrow Results

In this paper, we construct a class of non-weight modules $\mathrm{\Omega }(\lambda ,\alpha ,h)\otimes \mathrm{\text{Ind}}(V)$ over the $W$-algebra $W(2,2)$ by taking tensor products of modules $\mathrm{\Omega }(\lambda ,\alpha ,h)$ with restricted modules $\mathrm{\text{Ind}}(V)$. Then, we give the necessary and sufficient conditions for the irreducibility of such tensor product modules and determine the conditions under which two irreducible tensor product modules are isomorphic. These non-weight modules are different from known non-weight modules. Finally, we transform some tensor product modules into induced modules from modules of certain subalgebras over the $W$-algebra $W(2,2)$. And the conditions for these induced modules to be irreducible are determined by applying the established results.

In this paper, we prove that there exists an almost resolvable $k$-cycle decomposition of $(K_u\times K_g)(\lambda )$, (where × represents tensor product of graphs) for all even integers $k\ge 8$ with some possible exceptions.

In order to give numerical characterizations of the notion of "mutual orthogonality", we introduce two kinds of family of positive definite matrices for a unitary u in a finite von Neumann algebra M. They are arising from u naturally depending on the decompositions of M. One corresponds to the tensor product decomposition and the other does to the crossed product decomposition. By using the von Neumann entropy for these positive definite matrices, we characterize the notion of mutual orthogonality between subalgebras.

We put two C*-algebras together in a noncommutative tensor product using quantum group coactions on them and a bicharacter relating the two quantum groups that act. We describe this twisted tensor product in two equivalent ways, based on certain pairs of quantum group representations and based on covariant Hilbert space representations, respectively. We establish basic properties of the twisted tensor product and study some examples.

We prove explicit formulas for Chern classes of tensor products of virtual vector bundles, whose coefficients are given by certain universal polynomials in the ranks of the two bundles.

Kobayashi–Pevzner discovered in [Differential symmetry breaking operators: II. Rankin–Cohen operators for symmetric pairs, Selecta Math. (N.S.) 22(2) (2016) 847–911, MR 3477337] that the failure of the multiplicity-one property in the fusion rule of Verma modules of ${𝔰𝔩}_2$ occurs exactly when the Rankin–Cohen brackets vanish, and classified all the corresponding parameters. In this paper, we provide yet another characterization for these parameters, and give a precise description of indecomposable components of the tensor product. Furthermore, we discuss when the tensor products of two Verma modules are isomorphic to each other for semisimple Lie algebras $𝔤$.

Given a graph $G(V,E)$ and $S\subseteq V$, the Steiner distance $d_G(S)$ is the minimum size among all connected subgraphs of $G$ whose vertex sets contain $S$. The Steiner $k$-diameter ${\mathrm{\text{sdiam}}}_k(G)$ is the maximum value of $d_G(S)$ among all sets of $k$ vertices. In this short note, we study the Steiner $k$-diameters of the tensor product of complete graphs.

We propose a way to understand the three fermion generations by the algebraic structures of noncommutative geometry, which is a promising framework to unify the standard model and general relativity. We make the tensor product extension and the quaternion extension on the framework. Each of the two extensions alone keeps the action invariant, and we consider them as the almost trivial structures of the geometry. We combine the two extensions, and show the corresponding physical effects, i.e. the emergence of three fermion generations and the mass relationships among those generations. We define the coordinate fiber space of the bundle of the manifold as the space in which the classical noncommutative geometry is expressed, then the tensor product extension explicitly shows the contribution of structures in the non-coordinate base space of the bundle to the action. The quaternion extension plays an essential role to reveal the physical effect of the structure in the non-coordinate base space.

In this paper, we use semigroupoids to describe a notion of algebraic bundles, mostly motivated by Fell ($C^{\ast }$-algebraic) bundles, and the sectional algebras associated to them. As the main motivational example, Steinberg algebras may be regarded as the sectional algebras of trivial (direct product) bundles. Several theorems which relate geometric and algebraic constructions — via the construction of a sectional algebra — are widely generalized: Direct products bundles by semigroupoids correspond to tensor products of algebras; semidirect products of bundles correspond to “naïve” crossed products of algebras; skew products of graded bundles correspond to smash products of graded algebras; Quotient bundles correspond to quotient algebras. Moreover, most of the results hold in the non-Hausdorff setting. In the course of this work, we generalize the definition of smash products to groupoid graded algebras. As an application, we prove that whenever $𝜃$ is a ∧-preaction of a discrete inverse semigroupoid $S$ on an ample (possibly non-Hausdorff) groupoid ${𝒢}$, the Steinberg algebra of the associated groupoid of germs is naturally isomorphic to a crossed product of the Steinberg algebra of ${𝒢}$ by $S$. This is a far-reaching generalization of analogous results which had been proven in particular cases.

In this paper, we compute the Hausdorff dimension of a graph-directed set when the underlying multigraph is a Cartesian product or a tensor product of several multigraphs. We give explicit formulas in terms of the eigenvalues of the graph and the similarity ratios used with each graph.

A notion of "quasi-universal product" for algebraic probability spaces is introduced as a generalization of Speicher's "universal product". It is proved that there exist only five quasi-universal products, namely, tensor product, free product, Boolean product, monotone product and anti-monotone product. This result means that, in a sense, there exist only five independences which have nice properties of "associativity" and "(quasi-)universality".

Let be the class of all algebraic probability spaces. A "natural product" is, by definition, a map which is required to satisfy all the canonical axioms of Ben Ghorbal and Schürmann for "universal product" except for the commutativity axiom. We show that there exist only five natural products, namely tensor product, free product, Boolean product, monotone product and anti-monotone product. This means that, in a sense, there exist only five universal notions of stochastic independence in noncommutative probability theory.

Let be a Fock space and, for any non-negative integer k, let be the sum of all homogeneous chaos spaces of order at most k. For all non-negative integers m and n, the Wick product is a bounded bilinear operator from Γ (H_c)_m × Γ (H_c)_n into Γ (H_c)_{m +n} with norm greater than or equal to . In the monograph¹ S. Janson conjectured that this lower bound is exact. In this paper we prove this conjecture. In addition, we prove that a pair of nonzero vectors (ϕ, ψ)∈ Γ (H_c)_m × Γ (H_c)_n achieve this bound if and only if both vectors are multiples of the homogeneous products, i.e. ϕ =αu^⊗m, ψ =βu^⊗n, with u, v∈ H_c and α, β ∈ ℂ.

We find the best constant c(m, n, r), such that the inequality:

Motivated by an observation of Namioka and Phelps on an approximation property of order unit spaces, we introduce the $p$-tensor product and the $m$-tensor product of two compact matrix convex sets. We define a new approximation property for operator systems, and give a characterization using the $p$- and $m$-tensor products in the spirit of Grothendieck. Thus, an operator system has the operator system approximation property if and only if it is $(p,m)$-nuclear in a natural sense.

This paper is to establish a theory of regular representations for vertex operator algebras. In the paper, for a vertex operator algebra V and a V-module W, we construct, out of the dual space W*, a family of canonical weak V ⊗ V-modules with a nonzero complex number z as the parameter. We prove that for V-modules W, W₁ and W₂, a P(z)-intertwining map of type in the sense of Huang and Lepowsky exactly amounts to a V ⊗ V-homomorphism from W₁ ⊗ W₂ into . Combining this with Huang and Lepowsky's one-to-one linear correspondence between the space of intertwining operators and the space of P(z)-intertwining maps of the same type we obtain a canonical linear isomorphism fromthe space of intertwining operators of the indicated type to . Denote by R_P(z)(W) the sum of all (ordinary) V ⊗ V-submodules of . Assuming that V satisfies certain suitable conditions, we obtain a canonical decomposition of R_P(z)(W) into irreducible V ⊗ V-modules. In particular, we obtain a decomposition of Peter–Weyl type for R_P(z)(V). Denote by ℱ_P(z) the functor from the category of V-modules to the category of weak V ⊗ V-modules such that ℱ_P(z)(W)=R_P(z)(W'). We prove that for V-modules W₁, W₂, a P(z)-tensor product of W₁ and W₂ in the sense of Huang and Lepowsky exactly amounts to a universal from W₁ ⊗ W₂ to the functor ℱ_P(z). This implies that the functor ℱ_P(z) is essentially a right adjoint of the Huang–Lepowsky's P(z)-tensor product functor. It is also proved that R_P(z)(W) for are canonically isomorphic V ⊗ V-modules.

For a finite lattice L, the congruence lattice Con L of L can be easily computed from the partially ordered set J(L) of join-irreducible elements of L and the join-dependency relation D_L on J(L). We establish a similar version of this result for the dimension monoid Dim L of L, a natural precursor of Con L. For L join-semidistributive, this result takes the following form:

Theorem 1. Let L be a finite join-semidistributive lattice. Then Dim L is isomorphic to the commutative monoid defined by generators Δ(p), for p ∈ J(L), and relations

As a consequence of this, we obtain the following results:

Theorem 2. Let L be a finite join-semidistributive lattice. Then L is a lower bounded homomorphic image of a free lattice iff Dim L is strongly separative, iff it satisfies the axiom

Theorem 3. Let A and B be finite join-semidistributive lattices. Then the box product A □ B of A and B is join-semidistributive, and the following isomorphism holds:

A right R-module M_R over any ring R with 1 is called torsion-free if it satisfies the equality for every r∈R. An equivalent definition was used by Hattori [11]. We establish various properties of this concept, and investigate rings (called torsion-free rings) all of whose right ideals are torsion-free. In a torsion-free ring, the right annihilators of elements are always idempotent flat right ideals. The right p.p. rings are characterized as torsion-free rings in which the right annihilators of elements are finitely generated. An example shows that torsion-freeness ness is not a Morita invariant.

Several ring and module properties are proved, showing that, in several respects, torsion-freeness ness behaves like flatness. We exhibit examples to point out that the concept of torsion-freeness ness discussed here is different from other notions.

We solve the Clebsch–Gordan problem for rep_𝕂 Q (i.e. we describe the Krull–Schmidt decomposition of A ⊗ B for all A, B ∈ rep_𝕂 Q) in case 𝕂 is an algebraically closed field of characteristic 0 and Q is a quiver of type . The underlying tensor product is defined point-wise and arrow-wise.

Fix any compact ring R with identity. We associate to R the following categories of topological R-modules:

(i) _R𝔇 (𝔇_R) the category of all discrete topological left (right) R-modules;

(ii) _Rℭ (ℭ_R) the category of all compact left (right) R-modules.

We have introduced the following notions (analogous with classical notions of module theory):

(i) the tensor product of A ∈ ℭ_R and B ∈_Rℭ ( has a structure of a compact Abelian group);

(ii) a topologically semisimple module;

(iii) a compact topologically flat module.

We give a characterization of compact semisimple rings by using of flat modules.