Induced modules of split Iwahori–Hecke algebras are reducible

We show that the modules of an Iwahori–Hecke algebra obtained via induction from a proper parabolic subalgebra are always reducible.     

The first cohomology of N = 1 super Virasoro algebras

Abstract

In this paper, we determine the first cohomology groups of centerless N?=?1 super Virasoro algebras with coefficients in their tensor density modules. As applications, we classify all central extensions and derivations of the complete spectrum-generating superalgebra.     

Endomorphism algebras and Igusa–Todorov algebras

Let A be an Artin algebra. If $V\in \operatorname{mod} A$ such that the global dimension of  $\operatorname{End}_{A}V$ is at most 3, then for any ${M\in \operatorname{add}_{A}V}$ , both B and B ^op are 2-Igusa–Todorov algebras, where ${B=\operatorname{End}_{A}M}$ . Let ${P\in \operatorname{mod} A}$ be projective and ${B=\operatorname{End}_{A}P}$ such that the projective dimension of P as a right B-module is at most n(<∞). If A is an m-syzygy-finite algebra (resp. an m-Igusa–Todorov algebra), then B is an (m+n)-syzygy-finite algebra (resp. an (m+n)-Igusa–Todorov algebra); in particular, the finitistic dimension of B is finite in both cases. Some applications of these results are given.     

Cohen–Macaulay Auslander algebras of skewed-gentle algebras

For any skewed-gentle algebra, we characterize its indecomposable Gorenstein projective modules explicitly and describe its Cohen–Macaulay Auslander algebra. We prove that skewed-gentle algebras are always Gorenstein, which is independent of the characteristic of the ground field, and the Cohen–Macaulay Auslander algebras of skewed-gentle algebras are also skewed-gentle algebras.     

s-Representations for involutions¶on affine Kac–Moody algebras are polar

Let denote the eigenspace decomposition of a twisted affine Kac–Moody algebra with respect to an involution , where is a twisted loop algebra, is the center and d is the scaling element of . We endow with the standard bilinear symmetrical form.Then with and carries a Lorentzian signature. Let denote the group that corresponds to , then the adjoint representation of on can be restricted to and this submanifold is isometrical to the Hilbert space E _ε, where is the decomposition of the twisted loop algebra with respect to the induced involutionρ₀.We thus obtain an affine representation on E _ε and we show that this representation is polar, i. e., there exists a submanifold that intersects all orbits, and intersects them orthogonally. Received: 16 February 2000 RID=" ID="Supported by a DFG grant.     

Nonlinear self-adjointness of the Krichever–Novikov equation

It is known that the classification of third-order evolutionary equations with the constant separant possessing a nontrivial Lie–Bäcklund algebra (in other words, integrable equations) results in the linear equation, the KdV equation and the Krichever–Novikov equation. The first two of these equations are nonlinearly self-adjoint. This property allows to associate conservation laws of the equations in question with their symmetries. The problem on nonlinear self-adjointness of the Krichever–Novikov equation has not been solved yet. In the present paper we solve this problem and find the explicit form of the differential substitution providing the nonlinear self-adjointness.     

Almost Koszul algebras and stably Calabi–Yau algebras

We discuss different properties of Frobenius almost Koszul algebras of periodic type. We describe their bimodule projective resolutions and their relations with twisted superpotentials. We give a sufficient condition for a Frobenius almost Koszul algebra of periodic type to be stably Calabi–Yau. We also discuss the stably Calabi–Yau property of skew group algebras.     

Leavitt path algebras are Bézout

We study mapping class group orbits of homotopy and isotopy classes of curves with self-intersections. We exhibit the asymptotics of the number of such orbits of curves with a bounded number of self-intersections, as the complexity of the surface tends to infinity.We also consider the minimal genus of a subsurface that contains the curve. We determine the asymptotic number of orbits of curves with a fixed minimal genus and a bounded self-intersection number, as the complexity of the surface tends to infinity.As a corollary of our methods, we obtain that most curves that are homotopic are also isotopic. Furthermore, using a theorem by Basmajian, we get a bound on the number of mapping class group orbits on a given hyperbolic surface that can contain short curves. For a fixed length, this bound is polynomial in the signature of the surface.The arguments we use are based on counting embeddings of ribbon graphs.     

How accessible are categories of algebras?

For locally finitely presentable categories it is well known that categories of F-algebras, where F is a finitary endofunctor, are also locally finitely presentable. We prove that this generalizes to locally finitely multipresentable categories. But it fails, in general, for finitely accessible categories: we even present an example of a strongly finitary functor F (one that preserves finitely presentable objects) whose category of F-algebras is not finitely accessible. On the other hand, categories of F-algebras are proved to be ω₁-accessible for all strongly finitary functors—and it is an open problem whether this holds for all finitary functors.     

Commuting Krichever–Novikov differential operators with polynomial coefficients

Under study are some commuting rank 2 differential operators with polynomial coefficients. We prove that, for every spectral curve of the form w² = z³+c₂z2+c₁z+c₀ with arbitrary coefficients c_i, there exist commuting nonselfadjoint operators of orders 4 and 6 with polynomial coefficients of arbitrary degree.     

Quantizations of Kac–Moody algebras

We analyze the extent to which a quantum universal enveloping algebra of a Kac–Moody algebra g$\mathfrak{g}$ is defined by multidegrees of its defining relations. To this end, we consider a class of character Hopf algebras defined by the same number of defining relations of the same degrees as the Kac–Moody algebra g$\mathfrak{g}$. We demonstrate that if the generalized Cartan matrix A$A$ of g$\mathfrak{g}$ is connected then the algebraic structure, up to a finite number of exceptional cases, is defined by just one “continuous” parameter q$q$ related to a symmetrization of A$A$, and one “discrete” parameter m$\mathfrak{m}$ related to the modular symmetrizations of A$A$. The Hopf algebra structure is defined by n(n−1)/2$n(n-1)/2$ additional “continuous” parameters. We also consider the exceptional cases for Cartan matrices of finite or affine types in more detail, establishing the number of exceptional parameter values in terms of the Fibonacci sequence.     

Localization of Rota–Baxter algebras

A commutative Rota–Baxter algebra can be regarded as a commutative algebra that carries an abstraction of the integral operator. With the motivation of generalizing the study of algebraic geometry to Rota–Baxter algebras, we extend the central concept of localization for commutative algebras to commutative Rota–Baxter algebras. The existence of such a localization is proved and, under mild conditions, its explicit construction is obtained. The existence of tensor products of commutative Rota–Baxter algebras is also proved and the compatibility of localization and the tensor product of Rota–Baxter algebras is established. We further study Rota–Baxter coverings and show that they form a Grothendieck topology.     

Free Lie Rota–Baxter algebras

We construct a free Lie algebra with a Rota–Baxter operator.