Twisted Tensor Research Articles

A cubical model for a fibration

In the paper the notion of truncating twisting function from a simplicial set to a cubical set and the corresponding notion of twisted Cartesian product of these sets are introduced. The latter becomes a cubical set. Using this construction together with the theory of twisted tensor products for homotopy G-algebras a strictly associative multiplicative model for a fibration is obtained.

Defects in gradient micropolar elasticity—II: edge dislocation and wedge disclination

A theory of gradient micropolar elasticity based on first gradients of distortion and bend–twist tensors for an isotropic micropolar medium has been proposed in Part I of this paper. Gradient micropolar elasticity is an extension of micropolar elasticity such that in addition to double stresses double couple stresses also appear. The strain energy depends on the micropolar distortion and bend–twist terms as well as on distortion and bend–twist gradients. We use a version of this gradient theory which can be connected to Eringen's nonlocal micropolar elasticity. The theory is used to study a straight-edge dislocation and a straight-wedge disclination. As one important result, we obtained nonsingular expressions for the force and couple stresses. For the edge dislocation the components of the force stress have extremum values near the dislocation line and those of the couple stress have extremum values at the dislocation line and for the wedge disclination the components of the force stress have extremum values at the disclination line and those of the couple stress have extremum values near the disclination line.

On the finitistic dimension conjecture I: related to representation-finite algebras

We use the class of representation-finite algebras to investigate the finitistic dimension conjecture. In this way we obtain a large class of algebras for which the finitistic dimension conjecture holds. The main results in this paper are: (1) Let A be an artin algebra and let I j ,1⩽ j⩽ n be a family of ideals in A with I 1 I 2⋯ I n =0, such that proj.dim( A I j )<∞ and proj.dim( I j ) A =0 for all j⩾3. If A/ I 1 and A/ I 2 are representation-finite and if A/ I j has finite finitistic dimension for j⩾3, then the finitistic dimension of A is finite. In particular, the finitistic dimension conjecture is true for algebras obtained from representation-finite algebras by forming dual extensions, trivially twisted extensions, Hochschild extensions, matrix algebras and tensor products with algebras of radical-square-zero. (2) Let A, B and C be three artin algebras with the same identity such that (i) C⊆ B⊆ A, and (ii) the Jacobson radical of C is a left ideal of B and the Jacobson radical of B is a left ideal of A. If A is representation-finite, then C has finite finitistic dimension. We also provide a way to construct algebras satisfying all conditions in (2), and this leads to a new reformulation of the finitistic dimension conjecture.

Distinguished representations and poles of twisted tensor 𝐿-functions

Let E / F E/F be a quadratic extension of p p -adic fields. If π \pi is an admissible representation of G L n ( E ) GL_n(E) that is parabolically induced from discrete series representations, then we prove that the space of P n ( F ) P_n(F) -invariant linear functionals on π \pi has dimension one, where P n ( F ) P_n(F) is the mirabolic subgroup. As a corollary, it is deduced that if π \pi is distinguished by G L n ( F ) GL_n(F) , then the twisted tensor L L -function associated to π \pi has a pole at s = 0 s=0 . It follows that if π \pi is a discrete series representation, then at most one of the representations π \pi and π ⊗ χ \pi \otimes \chi is distinguished, where χ \chi is an extension of the local class field theory character associated to E / F E/F . This is in agreement with a conjecture of Flicker and Rallis that relates the set of distinguished representations with the image of base change from a suitable unitary group.

Twisted tensor product group embeddings and complete partial ovoids on quadrics in PG(2 t−1, q)

We study the geometry of the groups SL(2, q t ) and Sp(2 m, q t ), m, t⩾2, q even, in their twisted tensor product representation. We construct families of complete partial ovoids of hyperbolic quadrics in PG(2 t −1, q), all attaining the Blokhuis–Moorhouse bound. We identify the stabilizers of the complete partial ovoids.

Twisted tensor product of multiplier Hopf (∗-)algebras

We combine two (not necessarily commutative or co-commutative) multiplier Hopf (∗-)algebras to a new multiplier Hopf (∗-)algebra. We use the construction of a twisted tensor product for algebras, as introduced by A. Van Daele. We then proceed to find sufficient conditions on the twist map for this twisted tensor product to be a multiplier Hopf (∗-)algebra with the natural comultiplication. For usual Hopf algebras, we find that Majid's double crossed product by a matched pair of Hopf algebras is exactly the twisted tensor product Hopf algebra according to an appropriate twist map. Starting from two dually paired multiplier Hopf (∗-)algebras we construct the Drinfel'd double multiplier Hopf algebra in the framework of twisted tensor products.

A twisted tensor product on symbolic dynamical systems and the Ashley's problem

We define the notion of fiber bundle via a twisted tensor product on the transition matrices. We define the notion of topological conjugacy and shift equivalence in this bundle context and show that topological conjugacy implies shift equivalence. We show that the $\Sigma_A$ fits into our fiber bundle context. We introduce another system $\Sigma_W$, topologically conjugate to the full $2-$shift, which has the same base space and fiber as the Ashley system, but is constructed with a different twisting. We show that $\Sigma_A$ and $\Sigma_W$ are shift equivalent but not bundle isomorphic.

On a universal solution to the reflection equation

For a given quasi-triangular Hopf algebra \(\mathcal{H}\), we study relations between the braided group \(\tilde {\mathcal{H}}^* \) and Drinfeld's twist. We show that the braided bialgebra structure of \(\tilde {\mathcal{H}}^* \) is naturally described by means of twisted tensor powers of \(\mathcal{H}\) and their module algebras. We introduce a universal solution to the reflection equation (RE) and deduce a fusion prescription for RE-matrices.

Extensions of modules for SL(2,K)

In this paper, we consider the induced modules ∇ and the Weyl modules Δ for the algebraic group G=SL(2,K) where K is an algebraically closed field of characteristic p>0. We determine the G-modules Hi(G1,∇(s)⊗∇(t)) for all i⩾0, where G1 is the first Frobenius kernel of G. We then use it to find the Ext1-spaces between twisted tensor products of Weyl modules and induced modules for G. Moreover, we describe explicitly the non-split extensions corresponding to ∇'s.

FREE FIELD AND PARAFERMIONIC REALIZATIONS OF TWISTED $su(3)^{(2)}_k$ CURRENT ALGEBRA

Free field and twisted parafermionic representations of twisted [Formula: see text] current algebra are obtained. The corresponding twisted Sugawara energy-momentum tensor is given in terms of three (β,γ) pairs and two scalar fields and also in terms of twisted parafermionic currents and one scalar field. Two screening currents of the first kind are presented in terms of the free fields.

Reflection equation- and FRT-type algebras

We introduce two types of algebras which include respectively the well known reflection equation (RE) and Faddeev-Reshetikhin-Takhtayan algebras associated with a quasitriangular Hopf algebraH. We show that these two types of algebras are twist-equivalent. It follows that a RE algebra is a module algebra over a twisted tensor square ofH. We present some applications to the equivariant quantization.

Twisted sl(3,C)(2)k current algebra: free field representation and screening currents

Motivated by application of twisted current algebra in description of the entropy of Ads 3 black hole, we investigate the simplest twisted current algebra sl (3, C ) (2) k . Free field representation of the twisted algebra, and the corresponding twisted Sugawara energy-momentum tensor are obtained by using three ( β , γ ) pairs and two scalar fields. Primary fields and two screening currents of the first kind are presented.

Invariant ideals of abelian group algebras and representations of groups of Lie type

This paper contributes to the general study of ideal lattices in group algebras of infinite groups. In recent years, the second author has extensively studied this problem for G G an infinite locally finite simple group. It now appears that the next stage in the general problem is the case of abelian-by-simple groups. Some basic results reduce this problem to that of characterizing the ideals of abelian group algebras stable under certain (simple) automorphism groups. Here we begin the analysis in the case where the abelian group A A is the additive group of a finite-dimensional vector space V V over a locally finite field F F of prime characteristic p p , and the automorphism group G G is a simple infinite absolutely irreducible subgroup of G L ( V ) GL(V) . Thus G G is isomorphic to an infinite simple periodic group of Lie type, and G G is realized in G L ( V ) GL(V) via a twisted tensor product ϕ \phi of infinitesimally irreducible representations. If S S is a Sylow p p -subgroup of G G and if ⟨ v ⟩ \langle v\rangle is the unique line in V V stabilized by S S , then the approach here requires a precise understanding of the linear character associated with the action of a maximal torus T G T_G on ⟨ v ⟩ \langle v\rangle . At present, we are able to handle the case where ϕ \phi is a rational representation with character field equal to F F .

Hochschild Homology of Twisted Tensor Products

Hochschild Homology of Twisted Tensor Products

On Nahm's transformation with twisted boundary conditions

Following two different tracks, we arrive at a definition of Nahm's transformation valid for self-dual fields on the four-dimensional torus T 4 with non-zero twist tensor. The transform is again a self-dual gauge field defined on a new torus T ^ 4 and with non-zero twist tensor. It preserves the property of being an involution.

Twisted tensor product subgroups of finite classical groups

If Cln(qr) denotes a classical group with natural module W of dimensioa n over Fqr , then the twisted tensor product module is realised over Fq, and yields an embedding . These embeddings play a significant role in the subgroup structure of classical groups; for example, Seitz [18] shows that any maximal absolutely irreducible subgroup defined over a proper extension field of Fq is of this form. In this paper we study the precise nature of these embeddings, and go on to investigate their maximality or otherwise. We show that the normaliser of Cln(qr) is usually maximal, with an explicit list of just 4 families of exceptions.

Alge`bres de cochaiˆnes quasi-commutatives et fibrations alge´briques

Let k be a field of any characteristic p and k-dga the category of connected cochain algebras. In [3] Dupont and Hess defined notions of twisted tensor product, twisted extension, algebraic fibration and acyclic closure. In this paper we define the notion of quasi-commutative cochain algebras and their morphisms and we prove that every morphism of quasi-commutative cochain algebras is an algebraic fibration. Consequently, every such a cochain algebra admits an acyclic closure.

An Algebraic Model of Fibration with the Fiber K(π, n)-Space

Abstract For a fibration with the fiber K(π, n)-space, the algebraic model as a twisted tensor product of chains of the base with standard chains of K(π, n)-complex is given which preserves multiplicative structure as well. In terms of this model the action of the n-cohomology of the base with coefficients in π on the homology of fibration is described.

An algebraic model of fibration with the fiberK(π,n)-space

For a fibration with the fiberK(π,n)-space, the algebraic model as a twisted tensor product of chains of the base with standard chains ofK(π,n)-complex is given which preserves multiplicative structure as well. In terms of this model the action of then-cohomology of the base with coefficients in π on the homology of fibration is described.

On the Analysis of Twisted Director Configurations

Abstract A twist tensor (T-tensor) is introduced, which is defined for differentiable vector and director fields. Its eigensystem describes the local helical structure of the underlying field. It can have up to two nonzero eigenvalues, which indicate whether the local structure is untwisted, helical or double-twisted. The eigenvalues q i, if real valued, are the helical wave numbers, and the corresponding eigenvectors represent the local twist axes. The T-tensor can serve as a tool to analyze director configurations in chiral nematic liquid crystals, and applications in computational fluid dynamics seem feasible.