Automorphisms Of Algebras Research Articles

Cohomology of tails, Tate–Vogel cohomology, and noncommutative Serre duality over Koszul quiver algebras

The main result of the paper shows that, under Koszul duality between quiver algebras, cohomology of tails is identified with graded Vogel cohomology. As an application, a new proof of the noncommutative Serre duality over generalized Artin–Schelter regular Koszul quiver algebras is given. It is deduced from a similar formula over an arbitrary (i.e., not necessarily Koszul) Frobenius algebra, which turns out to be equivalent to the Auslander–Reiten formula. As another application, it is shown that, over a generalized Artin–Schelter regular Koszul quiver algebra, any algebra automorphism appearing in the noncommutative Serre duality formula is closely related, under Koszul duality, to the Nakayama automorphism of the Koszul-dual algebra.

Entropy and induced dynamics on state spaces

We consider the topological entropy of state space and quasistate space homeomorphisms induced from C*-algebra automorphisms. Our main result asserts that, for automorphisms of separable exact C*-algebras, zero Voiculescu-Brown entropy implies zero topological entropy on the quasi-state space (and also more generally on the entire unit ball of the dual). As an application we obtain a simple description of the topological Pinsker algebra in terms of local Voiculescu-Brown entropy.

Automorphisms of quantum and classical Poisson algebras

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button. We prove Pursell–Shanks type results for the Lie algebra $\mathcal{D}(M)$ of all linear differential operators of a smooth manifold M, for its Lie subalgebra $\mathcal{D}^1(M)$ of all linear first-order differential operators of M and for the Poisson algebra S(M) = Pol(T*M) of all polynomial functions on T*M, the symbols of the operators in $\mathcal{D}(M)$. Chiefly, however, we provide explicit formulas completely describing the automorphisms of the Lie algebras $\mathcal{D}^1(M)$, S(M) and $\mathcal{D}(M)$.

Newton polygons and families of polynomials

We consider a continuous family (f s ), s∈[0,1] of complex polynomials in two variables with isolated singularities, that are Newton non-degenerate. We suppose that the Euler characteristic of a generic fiber is constant. We firstly prove that the set of critical values at infinity depends continuously on s, and secondly that the degree of the f s is constant (up to an algebraic automorphism of ℂ2).

Automorphisms of associative algebras and noncommutative geometry

A class of differential calculi is explored which is determined by a set of automorphisms of the underlying associative algebra. Several examples are presented. In particular, differential calculi on the quantum plane, the $h$-deformed plane and the quantum group GLpq(2) are recovered in this way. Geometric structures like metrics and compatible linear connections are introduced.

A note on 2-local automorphisms of digraph algebras

In this note we show that every 2-local automorphism of a digraph algebra A is an automorphism if in addition A is symmetric. Moreover, we produce an example to clarify that the conclusion may not be true if A has not the extra condition of symmetry. By this means, we give a negative answer to the question about whether every 2-local automorphism of a digraph algebra is automatically an automorphism.

ON CO-HOPFIAN NILPOTENT GROUPS

Co-Hopfian finitely generated torsion-free nilpotent groups are characterized in this paper in terms of their Lie algebra automorphisms. Many examples of such groups are also constructed. 2000 Mathematics Subject Classification 17B30, 20F18 (primary).

Extensions of “thickened” Verma modules of the Virasoro algebra

Let g denote the Virasoro Lie algebra, h its Cartan subalgebra, and S( h) the symmetric algebra on h . In this paper we consider “thickened” Verma modules M(λ) which are (U( g),S( h)) -bimodules satisfying M(λ)⊗ S( h) C≅M(λ) where M( λ) is the usual Verma module with highest weight λ∈ h ∗ . We determine Ext 1( M(μ), M(λ)) to be S( h)/φ μ,λS( h) where φ μ, λ is, up to a C -algebra automorphism of S( h) , a product of irreducible factors of the determinant of the Shapovalov matrix. This result provides a conceptual explanation of the factorization of the Shapovalov determinant and implies that the inverse of the Shapovalov matrix has only simple poles.

An extension of a characterization of the automorphisms of Hilbert space effect algebras

The aim of this paper is to show that if an order preserving bijective transformation of the Hilbert space effect algebra also preserves the probability with respect to a fixed pair of mixed states, then it is an ortho-order automorphism. A similar result for the orthomodular lattice of all sharp effects (i.e., projections) is also presented.

Higher polyhedral K-groups

We define higher polyhedral K-groups for commutative rings, starting from the stable groups of elementary automorphisms of polyhedral algebras. Both Volodin's theory and Quillen's + construction are developed. In the special case of algebras associated with unit simplices one recovers the usual algebraic K-groups, while the general case of lattice polytopes reveals many new aspects, governed by polyhedral geometry. This paper is a continuation of Bruns and Gubeladze (Polyhedral K 2, Manuscr. Math.) which is devoted to the study of polyhedral aspects of the classical Steinberg relations. The present work explores the polyhedral geometry behind Suslin's well known proof of the coincidence of the classical Volodin's and Quillen's theories. We also determine all K-groups coming from two-dimensional polytopes.

Superconformal boundary conditions for the WZW model

We review the most general, local, superconformal boundary conditions for the two-dimensional N=1 and N=2 non-linear sigma models, and analyse them for the N=1 and N=2 supersymmetric WZW models. We find that the gluing map between the left and right affine currents is generalised in a very specific way as compared to the constant Lie algebra automorphisms that are known.

Spin geometry on quantum groups via covariant differential calculi

Let A be a cosemisimple Hopf ∗-algebra with antipode S and let Γ be a left-covariant first-order differential ∗-calculus over A such that Γ is self-dual (see Section 2) and invariant under the Hopf algebra automorphism S 2. A quantum Clifford algebra Cl( Γ, σ, g) is introduced which acts on Woronowicz’ external algebra Γ ∧. A minimal left ideal of Cl( Γ, σ, g) which is an A -bimodule is called a spinor module. Metrics on spinor modules are investigated. The usual notion of a linear left connection on Γ is extended to quantum Clifford algebras and also to spinor modules. The corresponding Dirac operator and connection Laplacian are defined. For the quantum group SL q (2) and its bicovariant 4 D ±-calculi these concepts are studied in detail. A generalization of Bochner's theorem is given. All invariant differential operators over a given spinor module are determined. The eigenvalues of the Dirac operator are computed.

Canonical extension of actions of locally compact quantum groups

In this paper, we generalize the notion of the canonical extension of automorphisms of von Neumann algebras to the case of actions of locally compact quantum groups (in the sense of Kustermans and Vaes). Various expected properties will be shown to hold for this new canonical extension. As an application, we describe the flow of weights of the crossed product of a type III factor by some special action of a discrete Kac algebra.

Automorphisms of real four-dimensional Lie algebras and the invariant characterization of homogeneous 4-spaces

Rows 14 and 15 of table 1, and rows 13 and 14 of table 2, in the above article have been corrected. Full details are provided in the pdf.

The Coradical Filtration for Drinfeld Double Ringel-Hall Algebras

Let 𝒟(Λ) be the Drinfeld double Ringel-Hall algebra with Λ being any finite dimensional hereditary algebra over a finite field k. We determine the coradical filtration for 𝒟(Λ). As an application, we describe the group of Hopf algebra automorphisms of the Drinfeld double Ringel composition algebra of Λ.

On Minimal Permutation Representations of Classical Simple Groups

We describe minimal permutation representations, i.e., faithful permutation representations of least degree, of classical finite simple groups as groups of automorphisms of simple Lie algebras.

Automorphisms of real four-dimensional Lie algebras and the invariant characterization of homogeneous 4-spaces

The automorphisms of all four-dimensional, real Lie algebras are presented in a comprehensive way. Their action on the space of 4 × 4, real, symmetric and positive definite matrices defines equivalence classes which are used for the invariant characterization of the four-dimensional homogeneous spaces which possess an invariant basis.

Local automorphisms of operator algebras on Banach spaces

In this paper we extend a result of Šemrl stating that every 2-local automorphism of the full operator algebra on a separable infinite dimensional Hilbert space is an automorphism. In fact, besides separable Hilbert spaces, we obtain the same conclusion for the much larger class of Banach spaces with Schauder bases. The proof rests on an analogous statement concerning the 2-local automorphisms of matrix algebras for which we present a short proof. The need to get such a proof was formulated in Šemrl’s paper.

Polyhedral K 2

Using elementary graded automorphisms of polytopal algebras (essentially the coordinate rings of projective toric varieties) polyhedral versions of the group of elementary matrices and the Steinberg and Milnor groups are defined. They coincide with the usual K-theoretic groups in the special case when the polytope is a unit simplex and can be thought of as compact/polytopal substitutes for the tame automorphism groups of polynomial algebras. Relative to the classical case, many new aspects have to be taken into account. We describe these groups explicitly when the underlying polytope is 2-dimensional. Already this low-dimensional case provides interesting classes of groups.

Finite dimensional comodules over the Hopf algebra of rooted trees

The aim of this paper is an algebraic study of the Hopf algebra H_R of rooted trees, which was introduced in \cite{Kreimer1,Connes,Broadhurst,Kreimer2}. We first construct comodules over H_R from finite families of primitive elements. Furthermore, we classify these comodules by restricting the possible families of primitive elements, and taking the quotient by the action of certain groups. In the next section, we give a formula about primitive elements of the subalgebra of ladders, and construct a projection operator on the space of primitive elements. It allows us to obtain a basis of the primitive elements by an inductive process, which answers one of the questions of \cite{Kreimer}. In the last sections, we classify the Hopf algebra endomorphisms and the coalgebras endomorphisms, using the graded Hopf algebra gr(H_R) associated to the filtration by deg_p of \cite{Kreimer}. We then prove that H_R is isomorphic to gr(H_R), and deduce a decomposition of the group of the Hopf algebra automorphisms of H_R as a semi-direct product.