Analogue Zhelobenko invariants, Bernstein–Gelfand–Gelfand operators and the Kostant Clifford algebra conjecture

Let $ \mathfrak{g} $ be a complex simple Lie algebra. The Kostant Clifford algebra conjecture can be formulated and somewhat extended as a question [7, Conj. 1.3] concerning the Harish-Chandra map for the enveloping algebra of $ \mathfrak{g} $ . In that work [7, Cor. 8.8] an analogue Kostant conjecture, obtained by replacing the Harish-Chandra map by a ??generalized Harish-Chandra?? map, was proved using a careful analysis of Zhelobenko invariants which describe the image of this map. In the present work we establish [7, Conj. 1.3] by showing that there are analogue Zhelobenko invariants which describe the image of the Harish-Chandra map. Following this a similar proof to that of [7, Cor. 8.8] goes through. In the last section a rather precise form of the Kostant Clifford algebra conjecture is established.     

Gelfand–Kirillov Dimension and Local Finiteness of Jordan Superpairs Covered by Grids and Their Associated Lie Superalgebras

Abstract

In this paper we show that a Lie superalgebra L graded by a 3-graded irreducible root system has Gelfand–Kirillov dimension equal to the Gelfand–Kirillov dimension of its coordinate superalgebra A, and that L is locally finite if and only A is so. Since these Lie superalgebras are coverings of Tits–Kantor–Koecher superalgebras of Jordan superpairs covered by a connected grid, we obtain our theorem by combining two other results. Firstly, we study the transfer of the Gelfand–Kirillov dimension and of local finiteness between these Lie superalgebras and their associated Jordan superpairs, and secondly, we prove the analogous result for Jordan superpairs: the Gelfand–Kirillov dimension of a Jordan superpair V covered by a connected grid coincides with the Gelfand– Kirillov dimension of its coordinate superalgebra A, and V is locally finite if and only if A is so.     

On the Auslander–Reiten conjecture for Cohen–Macaulay rings and path algebras

In this note, it is shown that the validity of the Auslander–Reiten conjecture for a given d-dimensional Cohen–Macaulay local ring R depends on its validity for all direct summands of d-th syzygy of R-modules of finite length, provided R is an isolated singularity. Based on this result, it is shown that under a mild assumption on the base ring R, satisfying the Auslander–Reiten conjecture behaves well under completion and reduction modulo regular elements. In addition, it will turn out that, if R is a commutative Noetherian ring and 𝒬 a finite acyclic quiver, then the Auslander–Reiten conjecture holds true for the path algebra R𝒬, whenever so does R. Using this result, examples of algebras satisfying the Auslander–Reiten conjecture are presented.     

Free Lie Rota–Baxter algebras

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

On Fomin–Kirillov algebras for complex reflection groups

In this note, we apply classification results for finite-dimensional Nichols algebras to generalizations of Fomin–Kirillov algebras to complex reflection groups. First, we focus on the case of cyclic groups where the corresponding Nichols algebras are only finite-dimensional up to order four, and we include results about the existence of Weyl groupoids and finite-dimensional Nichols subalgebras for this class. Second, recent results by Heckenberger–Vendramin [ArXiv e-prints, 1412.0857 (December 2014)] on the classification of Nichols algebras of semisimple group type can be used to find that these algebras are infinite-dimensional for many non-exceptional complex reflection groups in the Shephard–Todd classification.     

Gelfand–Kirillov dimension of generalized Weyl algebras (In memory of Guenter Rudolf Krause (1941–2015))

Given a generalized Weyl algebra A of degree 1 with the base algebra D, we prove that the difference of the Gelfand–Kirillov dimension of A and that of D could be any positive integer or infinity. Under mild conditions, this difference is exactly 1. As applications, we calculate the Gelfand–Kirillov dimensions of various algebras of interest, including the (quantized) Weyl algebras, ambiskew polynomial rings, noetherian (generalized) down-up algebras, iterated Ore extensions, quantum Heisenberg algebras, universal enveloping algebras of Lie algebras, quantum GWAs, etc.     

Lie–Poisson structures over differential algebras

Based on key elements of Olver’s approach to partial differential equations for Hamiltonian evolution, we propose an algebraic construction appropriate for Hamiltonian evolutionary systems with constraints.     

Gelfand–Kirillov Dimension of Algebras with a Locally Finite and Multiparameter Indexed Filtration

Let A have a locally finite and multiparameter indexed filtration ?, and let B be a homomorphic image of A. Thus B has the locally finite and multiparameter indexed filtration induced from ?. Here we study a relation between the associated graded algebra of A and that of B and use this result to calculate the Gelfand–Kirillov dimension of several algebras related to quantized algebras and Poisson enveloping algebras.     

Gelfand–Kirillov Dimension for Automorphism Groups of Relatively Free Algebras

Using ideas of our recent work on automorphisms of residually nilpotent relatively free groups, we introduce a new growth function for subgroups of the automorphism groups of relatively free algebras Fⁿ(V) over a field of characteristic zero and the related notion of Gelfand-Kirillov dimension, and study their behavior. We prove that, under some natural restrictions, the Gelfand-Kirillov dimension of the group of tame automorphisms of Fⁿ(V) is equal to the Gelfand-Kirillov dimension of the algebra Fⁿ(V). We show that, in some cases, the Gelfand-Kirillov dimension of the group of tame automorphisms of Fⁿ(V) is smaller than the Gelfand-Kirillov dimension of the whole automorphism group, and calculate the Gelfand-Kirillov dimension of the automorphism group of Fⁿ(V) for some important varieties V.Partially supported by Grant MM605/96 of the Bulgarian Foundation for Scientific Research.2000 Mathematics Subject Classification: primary 16R10, 16P90; secondary 16W20, 17B01, 17B30, 17B40     

Modular framed vertex operator algebras and ■-forms

We investigate modular framed vertex operator algebras over an algebraically closed field F whose characteristic is different from 2 and 7.In particular,the rationality of modular framed vertex operator algebras is established.For a modular code vertex operator algebra,the irreducible modules are constructed and classified.Moreover,a ■-form for any framed vertex operator algebra over C is constructed.As a result,one can obtain a modular framed vertex operator algebra from any framed vertex operator algebra over C.