Defining relations for tame automorphism groups of polynomial rings and wild automorphisms of free associative algebras

2006 ◽  
Vol 73 (2) ◽  
pp. 229-233
Author(s):  
U. U. Umirbaev
Keyword(s):  
Automorphism Groups ◽  
Polynomial Rings ◽  
Associative Algebras ◽  
Defining Relations ◽  
Tame Automorphism ◽  
Wild Automorphisms

scholarly journals Distributive laws between the operads Lie and Com

2020 ◽  
Vol 30 (08) ◽  
pp. 1565-1576
Author(s):  
Murray Bremner ◽  
Vladimir Dotsenko
Keyword(s):  
Lie Algebras ◽  
Computer Algebra ◽  
Gröbner Bases ◽  
Classification Problem ◽  
Grobner Bases ◽  
Polynomial Rings ◽  
Associative Algebras ◽  
Distributive Law ◽  
Free Modules

Using methods of computer algebra, especially, Gröbner bases for submodules of free modules over polynomial rings, we solve a classification problem in theory of algebraic operads: we show that the only nontrivial (possibly inhomogeneous) distributive law between the operad of Lie algebras and the operad of commutative associative algebras is given by the Livernet–Loday formula deforming the Poisson operad into the associative operad.

WEYL TYPE ALGEBRAS FROM QUANTUM TORI

2006 ◽  
Vol 08 (02) ◽  
pp. 135-165 ◽  
Author(s):  
KAIMING ZHAO
Keyword(s):  
Differential Operator ◽  
Lie Algebra ◽  
Associative Algebra ◽  
Operator Algebra ◽  
Laurent Polynomials ◽  
Associative Algebras ◽  
Defining Relations ◽  
Quantum Tori ◽  
Quantum Version ◽  
Algebra Over A Field

We introduce and study the quantum version of the differential operator algebra on Laurent polynomials and its associated Lie algebra over a field F of characteristic 0. The q-quantum torus Fq is the unital associative algebra over F generated by [Formula: see text] subject to the defining relations titj = qi,jtjti, where qi,i = 1, [Formula: see text]. Let D be a subspace of [Formula: see text] where ∂i is the derivation on Fq sending [Formula: see text] to [Formula: see text]. Then, the quantum differential operator algebra is the associative algebra Fq[D]. Assume that Fq[D] is simple as an associative algebra. We compute explicitly all 2-cocycles of Fq[D], viewed as a Lie algebra. More precisely, we show that the second cohomology group of Fq[D] has dimension n if D = 0, dimension 1 if dim D = 1, and dimension 0 if dim D > 1. We also determine all isomorphisms and anti-isomorphisms Fq[D] → Fq′[D′] of simple associative algebras, and all isomorphisms Fq[D]/F → Fq′[D′]/F of simple Lie algebras.

On the augmentation topology of automorphism groups of affine spaces and algebras

2018 ◽  
Vol 28 (08) ◽  
pp. 1449-1485 ◽  
Author(s):  
Alexei Kanel-Belov ◽  
Jie-Tai Yu ◽  
Andrey Elishev
Keyword(s):  
Automorphism Group ◽  
Associative Algebra ◽  
Automorphism Groups ◽  
Jacobian Conjecture ◽  
Free Associative Algebra ◽  
Topological Properties ◽  
Negative Solution ◽  
Base Field ◽  
Associative Algebras ◽  
Affine Spaces

We study topological properties of Ind-groups [Formula: see text] and [Formula: see text] of automorphisms of polynomial and free associative algebras via Ind-schemes, toric varieties, approximations, and singularities. We obtain a number of properties of [Formula: see text], where [Formula: see text] is the polynomial or free associative algebra over the base field [Formula: see text]. We prove that all Ind-scheme automorphisms of [Formula: see text] are inner for [Formula: see text], and all Ind-scheme automorphisms of [Formula: see text] are semi-inner. As an application, we prove that [Formula: see text] cannot be embedded into [Formula: see text] by the natural abelianization. In other words, the Automorphism Group Lifting Problem has a negative solution. We explore close connection between the above results and the Jacobian conjecture, as well as the Kanel-Belov–Kontsevich conjecture, and formulate the Jacobian conjecture for fields of any characteristic. We make use of results of Bodnarchuk and Rips, and we also consider automorphisms of tame groups preserving the origin and obtain a modification of said results in the tame setting.

scholarly journals KAZHDAN CONSTANTS OF GROUP EXTENSIONS

2010 ◽  
Vol 20 (05) ◽  
pp. 671-688
Author(s):  
UZY HADAD
Keyword(s):  
Free Group ◽  
Finite Groups ◽  
Automorphism Groups ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Group Extensions ◽  
Abelian Extensions ◽  
Tame Automorphism ◽  
Relatively Free Group

We give bounds on Kazhdan constants of abelian extensions of (finite) groups. As a corollary, we improved known results of Kazhdan constants for some meta-abelian groups and for the relatively free group in the variety of p-groups of lower p-series of class 2. Furthermore, we calculate Kazhdan constants of the tame automorphism groups of the free nilpotent groups.

Synergy in the Theories of Gröbner Bases and Path Algebras

1993 ◽  
Vol 45 (4) ◽  
pp. 727-739 ◽  
Author(s):  
Daniel R. Farkas ◽  
C. D. Feustel ◽  
Edward L. Green
Keyword(s):  
General Theory ◽  
Gröbner Basis ◽  
Gröbner Bases ◽  
Grobner Bases ◽  
Polynomial Rings ◽  
Grobner Basis ◽  
Associative Algebras ◽  
Path Algebras

AbstractA general theory for Grôbner basis in path algebras is introduced which extends the known theory for commutative polynomial rings and free associative algebras.

A wild automorphism of Usl(2)

1976 ◽  
Vol 80 (1) ◽  
pp. 61-65 ◽  
Author(s):  
A. Joseph
Keyword(s):  
Simple Form ◽  
Automorphism Groups ◽  
Associative Algebras ◽  
Enveloping Algebra ◽  
Locally Nilpotent ◽  
Wild Automorphism ◽  
Torsion Free

AbstractIt is known that the automorphism groups of various torsion-free associative algebras over two generators take a particularly simple form. It has been suggested (10), though never proved, that this fails over three or more generators. Here it is shown that this is indeed the case for the enveloping algebra of sl(2), a result which answers a question implicitly raised in (4). A weaker hypothesis is proposed for such automorphism groups and this is related to the structure of locally nilpotent derivations.

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