The celebrated ``tenfold way'' provides a scheme for categorizing general topological states of matter, but it does not take into account the crystal symmetries that always exist in real materials. A new method extends this organization to allow the categorization of all topologically distinct electronic band structures in materials with only crystal symmetries for any number of physically relevant dimensions.

https://link.aps.org/pdf/10.1103/PhysRevX.7.041069

Topological Classification of Crystalline Insulators through Band Structure Combinatorics

Bulk and Boundary Invariants for Complex Topological Insulators

Let n ∈ℕ and let A and B be rings. An additive map h : A → B is called an n -Jordan homomorphism if h ( a n )=( h ( a )) n for all a ∈ A . Every Jordan homomorphism is an n -Jordan homomorphism, for all n ≥2, but the converse is false in general. In this paper we investigate the n -Jordan homomorphisms on Banach algebras. Some results related to continuity are given as well.

/pdf/n-jordan-homomorphisms-53bnx1obn9.pdf

n -JORDAN HOMOMORPHISMS

Non-commutative odd Chern numbers and topological phases of disordered chiral systems

Let $n\in \Bbb N,$ and let $A,B$ be two rings. An additive map $h: A\to B$ is called n-Jordan homomorphism if $h(a^n)=(h(a))^n$ for all $a \in {A}$. Every Jordan homomorphism is an n-Jordan homomorphism, for all $n\geq 2,$ but the converse is false, in general. In this paper we investigate the n-Jordan homomorphisms on Banach algebras. Indeed some results related to continuity are given as well.

n-Jordan homomorphisms

We study the basic Laplacian on Riemannian foliations by writing the basic Laplacian in terms of the orthogonal projection from square-integrable forms to basic square-integrable forms. Using a geometric interpretation of this projection, we relate the ordinary Laplacian to the basic Laplacian. Among other results, we show the existence of the basic heat kernel and establish estimates for the eigenvalues of the basic Laplacian. Introduction. Let M be a compact oriented manifold and let be a transver- sally oriented foliation on M. A foliation is a Riemannian foliation if there exists a Riemannian metric on M with the property that the distance from one leaf of to another is locally constant; such a metric is called a bundle-like metric for . Associated to are the space of basic forms: Ω B(M )= Ω B ( M , )= Ω ( M ): i(X) =0 , i ( X ) d = 0 for all X Γ(T ) , where i(X) is the interior product with the vector field X and Γ(T ) denotes the sections of the distribution T associated to . The exterior derivative d maps basic forms to basic forms; let dB denote d restricted to Ω B (M). The basic Laplacian is the operator ΔB = dB B + BdB on basic forms, where B is the adjoint of dB on Ω B (M). The analytic and geometric properties of this operator have been studied by several researchers. In (5), the basic Laplacian was studied as an operator on basic functions (i.e., functions that are constant on leaves of ), and the author proved the existence of the heat kernel in this case. In (13), the existence of the heat kernel on basic forms was proved for the case where the mean curvature form of the foliation is basic. There are also "basic" Hodge theorems, for example (6) and (10). However, the proof of the Hodge theorem in (6) does not yield various estimates that are important in applications, while the theorem proved in (10) has the same restriction as the results in (13), namely that the authors require the mean curvature form to be basic. In this paper, we study the basic Laplacian on forms, without any restriction on the mean curvature. We prove the existence and uniqueness of the heat kernel for ΔB on forms for any Riemannian foliation, and we write down an explicit formula for the heat kernel. We also present a proof of the Hodge theorem for

/pdf/the-basic-laplacian-of-a-riemannian-foliation-6kqedbd69b.pdf

The basic Laplacian of a Riemannian foliation

A subbundle of variable dimension inside the tangent bundle of a smooth manifold is called a smooth distribution if it is the pointwise span of a family of smooth vector fields. We prove that all such distributions are finitely generated, meaning that the family may be taken to be a finite collection. Further, we show that the space of smooth sections of such distributions need not be finitely generated as a module over the smooth functions. Our results are valid in greater generality, where the tangent bundle may be replaced by an arbitrary vector bundle.

/pdf/smooth-distributions-are-finitely-generated-3n0ia70bg0.pdf

Smooth distributions are finitely generated

We formulate and prove an analog of the Hopf Index Theorem for Riemannian foliations. We compute the basic Euler characteristic of a closed Riemannian manifold as a sum of indices of a non-degenerate basic vector field at critical leaf closures. The primary tool used to establish this result is an adaptation to foliations of the Witten deformation method.

http://faculty.tcu.edu/epark/papers/Hopf_DGA.pdf

A Hopf index theorem for foliations

A Hopf Index Theorem for foliations

An n-homomorphism between algebras is a linear map Φ: A → B such that Φ(a 1 ... a n ) = Φ(a 1 ) ··· Φ(a n ) for all elements a 1 ,..., a n ∈ A. Every homomorphism is an n-homomorphism for all n > 2, but the converse is false, in general. Hejazian et al. (2005) ask: Is every *-preserving n-homomorphism between C*-algebras continuous? We answer their question in the affirmative, but the even and odd n arguments are surprisingly disjoint. We then use these results to prove stronger ones: If n > 2 is even, then Φ is just an ordinary *-homomorphism. If n > 3 is odd, then Φ is a difference of two orthogonal *-homomorphisms. Thus, there are no nontrivial *-linear n-homomorphisms between C*-algebras.

/pdf/on-the-nonexistence-of-nontrivial-involutive-n-homomorphisms-4tnr17azba.pdf

On the nonexistence of nontrivial involutive "n"-homomorphisms of "C*"-algebras

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