Tensor Products Of Algebras Research Articles

A Mikhlin multiplier theory for free groups and amalgamated free products of von Neumann algebras

We obtain a Mikhlin multiplier theory for the nonabelian free groups. Let F∞ be a free group on infinite many generators {g1,g2,⋯}. Given d≥1 and a bounded symbol m on Zd satisfying the classical Mikhlin condition, the linear map Mm:C[F∞]→C[F∞] defined by λ(g)↦m(k1,⋯,kd)λ(g) for g=gi1k1⋯ginkn∈F∞ in reduced form (with kl=0 in m(k1,⋯,kd) for l>n), extends to a completely bounded map on Lp(Fˆ∞) for all 1<p<∞, where Fˆ∞ is the group von Neumann algebra of F∞. In the process, we establish a platform to transfer Lp-completely bounded maps on tensor products of von Neumann algebras to Lp-completely bounded maps on the corresponding amalgamated free products. A similar result holds for any free product of discrete groups.

Rings whose elements are sums of m-potents and nilpotents

For a fixed natural number an element a of a ring R is m-nil clean if where and b is nilpotent; if further eb = be, a is called strongly m-nil clean. The ring R is called m-nil clean (resp., strongly m-nil clean) if each of its elements is m-nil clean (resp., strongly m-nil clean). For a fixed natural number the strongly m-nil clean rings are a big subclass of the periodic rings which contains the class of strongly nil clean rings. The strongly m-nil cleanness of the tensor product of algebras, matrix rings, Morita contexts, and group rings is discussed in detail. Our results extend several existing results. Examples are provided to illustrate our results.

Hasse invariant for the tame Brauer group of a higher local field

We generalize the Hasse invariant of local class field theory to the tame Brauer group of a higher dimensional local field, and use it to study the arithmetic of central simple algebras, which are given a priori as tensor products of standard cyclic algebras. We also compute the tame Brauer dimension (or period-index bound) and the cyclic length of a general henselian-valued field of finite rank and finite residue field.

Hyperquaternionic Representations of Conic Sections

In this paper, by means of an isomorphism, we express the Clifford algebra Cl5,3 as hyperquaternion algebra H ⊗ H ⊗ H ⊗ H (a four-fold tensor product of quaternion algebras) and we provide the hyperquaternionic approach to the inner product null space (IPNS) representation of conic sections.

Arithmetic invariants of pseudoreflection groups and regular graded algebras

The main result of this paper is a generalization of the theorem of Chevalley-Shephard-Todd to the rings of invariants of pseudoreflection groups over Dedekind domains. In the special case of a principal ideal domain in which the group order is invertible it is proved that this ring of invariants is isomorphic to a polynomial ring. An intermediate result is that every finitely generated regular graded algebra over a Dedekind domain is isomorphic to a tensor product of blowup algebras.

Injective Tensor Products in Strict Deformation Quantization

The aim of this paper is twofold. Firstly we provide necessary and sufficient criteria for the existence of a strict deformation quantization of algebraic tensor products of Poisson algebras, and secondly we discuss the existence of products of KMS states. As an application, we discuss the correspondence between quantum and classical Hamiltonians in spin systems and we provide a relation between the resolvent of Schödinger operators for non-interacting many particle systems and quantization maps.

A diamond lemma for Hecke-type algebras

In this paper we give a version of Bergman’s diamond lemma which applies to certain monoidal categories presented by generators and relations. In particular, it applies to: the Coxeter presentation of the symmetric groups, the quiver Hecke algebras of Khovanov-Lauda-Rouquier, the Webster tensor product algebras, and various generalizations of these. We also give an extension of Manin-Schechtmann theory to non-reduced expressions.

Twisted tensor products of DG algebras

Twisted tensor products of DG algebras

Homotopy liftings and Hochschild cohomology of some twisted tensor products

The Hochschild cohomology of a tensor product of algebras is isomorphic to a graded tensor product of Hochschild cohomology algebras, as a Gerstenhaber algebra. A similar result holds when the tensor product is twisted by a bicharacter. We present new proofs of these isomorphisms, using Volkov’s homotopy liftings that were introduced for handling Gerstenhaber brackets expressed on arbitrary bimodule resolutions. Our results illustrate the utility of homotopy liftings for theoretical purposes.

On Arens regularity of projective tensor product of Schatten spaces

We discuss Arens regularity of the projective tensor product of Banach algebras consisting of Schatten class operators. More precisely, exploiting Ülger’s biregularity criterion, we prove that for any Hilbert space ℋ, the Banach algebras Sp(ℋ)⊗γ Sq (ℋ) and B(S2(ℋ))⊗γ S2(ℋ) are not Arens regular for every pair 1≤p,q≤2; whereas, when ℋ is separable and S2(ℋ) is equipped with the Schur product, we show that S2(ℋ) is completely continuous and that S2(ℋ)⊗γ S2(ℋ) is Arens regular.

Decomposition of topological Azumaya algebras

AbstractLet $\mathscr {A}$ be a topological Azumaya algebra of degree $mn$ over a CW complex X. We give conditions for the positive integers m and n, and the space X so that $\mathscr {A}$ can be decomposed as the tensor product of topological Azumaya algebras of degrees m and n. Then we prove that if $m&lt;n$ and the dimension of X is higher than $2m+1$ , $\mathscr {A}$ may not have such decomposition.

Hyperquaternion Conformal Groups

The paper gives a new representation of conformal groups in n dimensions in terms of hyperquaternions defined as tensor products of quaternion algebras (or a subalgebra thereof). Being Clifford algebras, hyperquaternions provide a good representation of pseudo-orthogonal groups such as $$O(p+1,q+1)$$ isomorphic to the nD conformal group with $$n=p+q.$$ The representation yields simple expressions of the generators, independently of matrices or operators. The canonical decomposition and the invariants are discussed. As application, the 4D relativistic conformal group is detailed together with a worked example. Finally, the formalism is compared to the operator representation. Potential uses include in particular, conformal geometry, computer graphics and conformal field theory.

κ -Poincaré comodules, braided tensor products, and noncommutative quantum field theory

We discuss the obstruction to the construction of a multiparticle field theory on a $\kappa$-Minkowski noncommutative spacetime: the existence of multilocal functions which respect the deformed symmetries of the problem. This construction is only possible for a light-like version of the commutation relations, if one requires invariance of the tensor product algebra under the coaction of the $\kappa$-Poincar\'e group. This necessitates a braided tensor product. We study the representations of this product, and prove that $\kappa$-Poincar\'e-invariant N-point functions belong to an Abelian subalgebra, and are therefore commutative. We use this construction to define the 2-point Whightman and Pauli--Jordan functions, which turn out to be identical to the undeformed ones. We finally outline how to construct a free scalar $\kappa$-Poincar\'e-invariant quantum field theory, and identify some open problems.

On the triple tensor product of nilpotent Lie algebras

In this paper, we give the explicit structure of and where H is a generalized Heisenberg Lie algebra of rank at most 2. Moreover, for a non-abelian nilpotent Lie algebra L, we obtain an upper bound for the dimension of

Derivations and automorphisms of locally matrix algebras

We describe derivations and automorphisms of infinite tensor products of matrix algebras. Using this description we show that for a countable–dimensional locally matrix algebra A over a field F the dimension of the Lie algebra of outer derivations of A and the order of the group of outer automorphisms of A are both equal to |F|ℵ0, where |F| is the cardinality of the field F.

Dual Hyperquaternion Poincaré Groups

A new representation of the Poincaré groups in n dimensions via dual hyperquaternions is developed, hyperquaternions being defined as a tensor product of quaternion algebras (or a subalgebra thereof). This formalism yields a uniquely defined product and simple expressions of the Poincaré generators, with immediate physical meaning, revealing the algebraic structure independently of matrices or operators. An extended multivector calculus is introduced (allowing a possible sign change of the metric or of the exterior product). The Poincaré groups are formulated as a dual extension of hyperquaternion pseudo-orthogonal groups. The canonical decomposition and the invariants are discussed. As concrete example, the 4D Poincaré group is examined together with a numerical application. Finally, the hyperquaternion representation is compared to the quantum mechanical one. Potential applications include in particular, moving reference frames and computer graphics.

Algebraic Tensor Products Revisited: Axiomatic Approach

This is an expository paper on tensor products where the standard approaches for constructing concrete instances of algebraic tensor products of linear spaces, via quotient spaces or via linear maps of bilinear maps, are reviewed by reducing them to different but isomorphic interpretations of an abstract notion, viz. the universal property, which is based on a pair of axioms.

Twisted tensor products of DG algebras

Twisted tensor products of DG algebras

Jordan Left {g, h}-Derivation over Some Algebras

In this article, left {g, h}-derivation and Jordan left {g, h}-derivation on algebras are introduced. It is shown that there is no Jordan left {g, h}-derivation over $${{\cal M}_n}\left(C \right)$$ and ℍℝ, for g ≠ h. Examples are given which show that every Jordan left {g, h}-derivation over $${{\cal T}_n}\left(C \right)$$ , $${{\cal M}_n}\left(C \right)$$ and ℍℝ are not left {g, h}-derivations. Also, the Jordan left {g, h}-derivations over $${{\cal T}_n}\left(C \right)$$ , $${{\cal M}_n}\left(C \right)$$ and ℍℝ are right centralizers, where C is a 2-torsionfree commutative ring. Moreover, we prove the result of Jordan left {g, h}-derivation to be a left {g, h}-derivation over tensor products of algebras as well as for algebra of polynomials.

Tensor Product of Dimension Effect Algebras

Dimension effect algebras were introduced in (A. Jencova, S. Pulmannova, Rep. Math. Phys. 62 (2008), 205-218), and it was proved that they are unit intervals in dimension groups. We prove that the effect algebra tensor product of dimension effect algebras is a dimension effect algebra, which is the unit interval in the unital abelian po-groups tensor product of the corresponding dimension groups.