scholarly journals Combinatorics and structure of Hecke–Kiselman algebras

2020 ◽  
Vol 22 (07) ◽  
pp. 2050022
Author(s):  
Jan Okniński ◽  
Magdalena Wiertel
Keyword(s):  
Crucial Role ◽  
Matrix Type ◽  
Oriented Graphs ◽  
Dimension One ◽  
Kirillov Dimension ◽  
Noetherian Algebra

Hecke–Kiselman monoids [Formula: see text] and their algebras [Formula: see text], over a field [Formula: see text], associated to finite oriented graphs [Formula: see text] are studied. In the case [Formula: see text] is a cycle of length [Formula: see text], a hierarchy of certain unexpected structures of matrix type is discovered within the monoid [Formula: see text] and this hierarchy is used to describe the structure and the properties of the algebra [Formula: see text]. In particular, it is shown that [Formula: see text] is a right and left Noetherian algebra, while it has been known that it is a PI-algebra of Gelfand–Kirillov dimension one. This is used to characterize all Noetherian algebras [Formula: see text] in terms of the graphs [Formula: see text]. The strategy of our approach is based on the crucial role played by submonoids of the form [Formula: see text] in combinatorics and structure of arbitrary Hecke–Kiselman monoids [Formula: see text].

scholarly journals Leavitt path algebras satisfying a polynomial identity

2016 ◽  
Vol 15 (05) ◽  
pp. 1650084 ◽  
Author(s):  
Jason P. Bell ◽  
T. H. Lenagan ◽  
Kulumani M. Rangaswamy
Keyword(s):  
Polynomial Identity ◽  
Finite Graph ◽  
Infinite Graphs ◽  
Leavitt Path Algebra ◽  
Arbitrary Graph ◽  
Graph Theoretic ◽  
Path Algebras ◽  
Leavitt Path Algebras ◽  
Dimension One ◽  
Kirillov Dimension

Leavitt path algebras [Formula: see text] of an arbitrary graph [Formula: see text] over a field [Formula: see text] satisfying a polynomial identity are completely characterized both in graph-theoretic and algebraic terms. When [Formula: see text] is a finite graph, [Formula: see text] satisfying a polynomial identity is shown to be equivalent to the Gelfand–Kirillov dimension of [Formula: see text] being at most one, though this is no longer true for infinite graphs. It is shown that, for an arbitrary graph [Formula: see text], the Leavitt path algebra [Formula: see text] has Gelfand–Kirillov dimension zero if and only if [Formula: see text] has no cycles. Likewise, [Formula: see text] has Gelfand–Kirillov dimension one if and only if [Formula: see text] contains at least one cycle, but no cycle in [Formula: see text] has an exit.

On Algebras With Gelfand-Kirillov Dimension One

1993 ◽  
Vol 119 (4) ◽  
pp. 1095
Author(s):  
Shigeru Kobayashi ◽  
Yuji Kobayashi
Keyword(s):  
Dimension One ◽  
Kirillov Dimension

Affine algebras of Gelfand-Kirillov dimension one are PI

1985 ◽  
Vol 97 (3) ◽  
pp. 407-414 ◽  
Author(s):  
L. W. Small ◽  
J. T. Stafford ◽  
R. B. Warfield
Keyword(s):  
Polynomial Identity ◽  
Prime Radical ◽  
Finitely Generated ◽  
Finite Module ◽  
Affine Algebras ◽  
Dimension One ◽  
Kirillov Dimension ◽  
Algebra Over A Field

The aim of this paper is to prove:Theorem.Let R be an affine (finitely generated) algebra over a field k and of Gelfand-Kirillov dimension one. Then R satisfies a polynomial identity. Consequently, if N is the prime radical of R, then N is nilpotent and R/N is a finite module over its Noetherian centre.

scholarly journals On algebras with Gel\cprime fand-Kirillov dimension one

1993 ◽  
Vol 119 (4) ◽  
pp. 1095-1095
Author(s):  
Shigeru Kobayashi ◽  
Yuji Kobayashi
Keyword(s):  
Dimension One ◽  
Kirillov Dimension

SEMIPRIME QUADRATIC ALGEBRAS OF GELFAND–KIRILLOV DIMENSION ONE

2004 ◽  
Vol 03 (03) ◽  
pp. 283-300 ◽  
Author(s):  
FERRAN CEDÓ ◽  
ERIC JESPERS ◽  
JAN OKNIŃSKI
Keyword(s):  
Positive Integer ◽  
Characteristic Zero ◽  
Prime Ideals ◽  
Quadratic Algebras ◽  
Degeneracy Condition ◽  
Minimal Prime ◽  
Dimension One ◽  
Kirillov Dimension

We consider algebras over a field K with a presentation K<x1,…,xn:R>, where R consists of [Formula: see text] square-free relations of the form xixj=xkxl with every monomial xixj, i≠j, appearing in one of the relations. The description of all four generated algebras of this type that satisfy a certain non-degeneracy condition is given. The structure of one of these algebras is described in detail. In particular, we prove that the Gelfand–Kirillov dimension is one while the algebra is noetherian PI and semiprime in case when the field K has characteristic zero. All minimal prime ideals of the algebra are described. It is also shown that the underlying monoid is a semilattice of cancellative semigroups and its structure is described. For any positive integer m, we construct non-degenerate algebras of the considered type on 4m generators that have Gelfand–Kirillov dimension one and are semiprime noetherian PI algebras.

scholarly journals Gelfand-Kirillov dimension of multi-filtered algebras

1999 ◽  
Vol 42 (1) ◽  
pp. 155-168 ◽  
Author(s):  
José Gómez Torrecillas
Keyword(s):  
Graded Algebra ◽  
Associative Algebras ◽  
Enveloping Algebra ◽  
Quantum Matrices ◽  
Weyl Algebras ◽  
Finite Dimensional ◽  
Ore Extensions ◽  
Filtered Algebras ◽  
Kirillov Dimension ◽  
Noetherian Algebra

We consider associative algebras filtered by the additive monoid ℕp. We prove that, under quite general conditions, the study of Gelfand-Kirillov dimension of modules over a multi-filtered algebra R can be reduced to the associated ℕp-graded algebra G(R). As a consequence, we show the exactness of the Gelfand-Kirillov dimension when the multi-filtration is finite-dimensional and G(R) is a finitely generated noetherian algebra. Our methods apply to examples like iterated Ore extensions with arbitrary derivations and “homothetic” automorphisms (e.g. quantum matrices, quantum Weyl algebras) and the quantum enveloping algebra of sl(v + 1)

scholarly journals Jordan Algebras of Gelfand–Kirillov Dimension One

1996 ◽  
Vol 180 (1) ◽  
pp. 211-238 ◽  
Author(s):  
C. Martinez ◽  
E. Zelmanov
Keyword(s):  
Jordan Algebras ◽  
Dimension One ◽  
Kirillov Dimension

scholarly journals Prime affine algebras of Gelfand—Kirillov dimension one

1984 ◽  
Vol 91 (2) ◽  
pp. 386-389 ◽  
Author(s):  
L.W Small ◽  
R.B Warfield
Keyword(s):  
Affine Algebras ◽  
Dimension One ◽  
Kirillov Dimension

The observation of solutions in Ni3Nb

1988 ◽  
Vol 46 ◽  
pp. 540-541
Author(s):  
Z.M. Wang ◽  
J.P. Zhang
Keyword(s):  
Electron Microscopy ◽  
High Resolution ◽  
Phase Modulation ◽  
High Resolution Electron Microscopy ◽  
Antiphase Domain ◽  
Boundary Area ◽  
Matrix Type ◽  
Resolution Electron ◽  
Domain Boundaries ◽  
Kontorova Model

High resolution electron microscopy reveals that antiphase domain boundaries in β-Ni3Nb have a hexagonal unit cell with lattice parameters ah=aβ and ch=bβ, where aβ and bβ are of the orthogonal β matrix. (See Figure 1.) Some of these boundaries can creep “upstairs” leaving an incoherent area, as shown in region P. When the stepped boundaries meet each other, they do not lose their own character. Our consideration in this work is to estimate the influnce of the natural misfit δ{(ab-aβ)/aβ≠0}. Defining the displacement field at the boundary as a phase modulation Φ(x), following the Frenkel-Kontorova model [2], we consider the boundary area to be made up of a two unit chain, the upper portion of which can move and the lower portion of the β matrix type, assumed to be fixed. (See the schematic pattern in Figure 2(a)).

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