scholarly journals Actions of the additive group Ga on certain noncommutative deformations of the plane

2021 ◽  
Vol 29 (2) ◽  
pp. 269-279
Author(s):  
Ivan Kaygorodov ◽  
Samuel A. Lopes ◽  
Farukh Mashurov
Keyword(s):  
Automorphism Group ◽  
Polynomial Ring ◽  
Characteristic Zero ◽  
Weyl Algebra ◽  
Additive Group ◽  
Prime Characteristic ◽  
Arbitrary Characteristic ◽  
Arbitrary Polynomial ◽  
Comodule Algebra ◽  
The Family

Abstract We connect the theorems of Rentschler [18] and Dixmier [10] on locally nilpotent derivations and automorphisms of the polynomial ring A 0 and of the Weyl algebra A 1, both over a field of characteristic zero, by establishing the same type of results for the family of algebras A h = 〈 x , y | y x − x y = h ( x ) 〉 , {A_h} = \left\langle {x,y|yx - xy = h\left( x \right)} \right\rangle , , where h is an arbitrary polynomial in x. In the second part of the paper we consider a field 𝔽 of prime characteristic and study 𝔽[t]-comodule algebra structures on Ah . We also compute the Makar-Limanov invariant of absolute constants of Ah over a field of arbitrary characteristic and show how this subalgebra determines the automorphism group of Ah .

scholarly journals ALGEBRAIC CURVES, RIEMANN SURFACES AND KLEIN SURFACES WITH NO NON-TRIVIAL AUTOMORPHISMS OR SYMMETRIES

2002 ◽  
Vol 45 (1) ◽  
pp. 141-148 ◽  
Author(s):  
Peter Turbek
Keyword(s):  
Automorphism Group ◽  
Finite Number ◽  
Characteristic Zero ◽  
Riemann Surfaces ◽  
Algebraic Curves ◽  
Klein Surfaces ◽  
New Family ◽  
Family Of Curves ◽  
Primary 14H37 ◽  
The Family

AbstractThe explicit defining equations of a new family of curves whose members have a trivial automorphism group are given. Each member is defined for characteristic zero and all but a finite number of characteristics greater than zero. This family, in conjunction with a previously appearing family of the author’s, provides explicit examples of algebraic curves which possess only the trivial automorphism for each genus greater than three. The family is then used to construct Riemann surfaces without anticonformal automorphisms and Klein surfaces with no non-trivial automorphisms.AMS 2000 Mathematics subject classification: Primary 14H37; 30F50; 30F99

scholarly journals On the Automorphism Group of the First Weyl Algebra

2013 ◽  
Vol 42 (1) ◽  
pp. 81-95 ◽  
Author(s):  
M. K. Kouakou ◽  
A. Tchoudjem
Keyword(s):  
Automorphism Group ◽  
Weyl Algebra

scholarly journals Algebraic Surfaces of General Type with Small c21 in Positive Characteristic

2008 ◽  
Vol 191 ◽  
pp. 111-134 ◽  
Author(s):  
Christian Liedtke
Keyword(s):  
Characteristic Zero ◽  
General Type ◽  
Positive Characteristic ◽  
Algebraic Surfaces ◽  
Surfaces Of General Type ◽  
Arbitrary Characteristic ◽  
Numerical Invariants ◽  
Characteristic 2

AbstractWe establish Noether’s inequality for surfaces of general type in positive characteristic. Then we extend Enriques’ and Horikawa’s classification of surfaces on the Noether line, the so-called Horikawa surfaces. We construct examples for all possible numerical invariants and in arbitrary characteristic, where we need foliations and deformation techniques to handle characteristic 2. Finally, we show that Horikawa surfaces lift to characteristic zero.

Interpolation in the Automorphism Group of a Polynomial Ring

2020 ◽  
Vol 27 (03) ◽  
pp. 587-598
Author(s):  
M’hammed El Kahoui ◽  
Najoua Essamaoui ◽  
Mustapha Ouali
Keyword(s):  
Automorphism Group ◽  
Commutative Ring ◽  
Polynomial Ring ◽  
Proper Ideal ◽  
Polynomial Rings ◽  
Group Homomorphism ◽  
Volume Preserving

Let R be a commutative ring with unity and SAn(R) be the group of volume-preserving automorphisms of the polynomial R-algebra R[n]. Given a proper ideal 𝔞 of R, we address in this paper the question of whether the canonical group homomorphism SAn(R) → SAn(R/𝔞) is surjective. As an application, we retrieve and generalize, in a unified way, several known results on residual coordinates in polynomial rings.

scholarly journals A Remark on the Dixmier Conjecture

2019 ◽  
Vol 63 (1) ◽  
pp. 6-12
Author(s):  
V. V. Bavula ◽  
V. Levandovskyy
Keyword(s):  
Characteristic Zero ◽  
Weyl Algebra ◽  
Graded Algebra

AbstractThe Dixmier Conjecture says that every endomorphism of the (first) Weyl algebra $A_{1}$ (over a field of characteristic zero) is an automorphism, i.e., if $PQ-QP=1$ for some $P,Q\in A_{1}$, then $A_{1}=K\langle P,Q\rangle$. The Weyl algebra $A_{1}$ is a $\mathbb{Z}$-graded algebra. We prove that the Dixmier Conjecture holds if the elements $P$ and $Q$ are sums of no more than two homogeneous elements of $A_{1}$ (there is no restriction on the total degrees of $P$ and $Q$).

On the automorphism group of the integral group ring of the infinite dihedral group

1988 ◽  
Vol 108 (1-2) ◽  
pp. 1-10
Author(s):  
D. A. R. Wallace
Keyword(s):  
Automorphism Group ◽  
Normal Subgroup ◽  
Group Ring ◽  
Polynomial Ring ◽  
Dihedral Group ◽  
Integral Group Ring ◽  
Integral Group ◽  
Index 2 ◽  
Skolem Theorem ◽  
Inner Automorphisms

SynopsisLet ℤ(G) be the integral group ring of the infinite dihedral groupG; the aim is to calculate Autℤ(G), the group of Z-linear automorphisms of ℤ(G). It is shown that Aut ℤ(G), the subgroup of Aut *ℤ(G) consisting of those automorphisms that preserve elementwise the centre of ℤ(G), is a normal subgroup of Aut *ℤ(G) of index 2 and that Aut*ℤ(G) may be embedded monomorphically intoM2(ℤ[t]), the ring of 2 × 2 matrices over a polynomial ring ℤ[t]From this embedding and by the Noether—Skolem Theorem it is shown that Inn (G), the group of inner automorphisms of ℤ(G) induced by the units of ℤ(G), is a normal subgroup of Aut*ℤ(G)/ such that Aut*ℤ(G)/Inn (G) is isomorphic to the Klein four-group.

scholarly journals Metric spaces related to Abelian groups

2021 ◽  
Vol 22 (1) ◽  
pp. 169
Author(s):  
Amir Veisi ◽  
Ali Delbaznasab
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Additive Group ◽  
Ordered Group ◽  
Infinite Cyclic Group ◽  
Ordered Groups ◽  
The Family ◽  
Nonempty Compact ◽  
Metric Topology ◽  
Induced Topology

<p>When working with a metric space, we are dealing with the additive group (R, +). Replacing (R, +) with an Abelian group (G, ∗), offers a new structure of a metric space. We call it a G-metric space and the induced topology is called the G-metric topology. In this paper, we are studying G-metric spaces based on L-groups (i.e., partially ordered groups which are lattices). Some results in G-metric spaces are obtained. The G-metric topology is defined which is further studied for its topological properties. We prove that if G is a densely ordered group or an infinite cyclic group, then every G-metric space is Hausdorff. It is shown that if G is a Dedekind-complete densely ordered group, (X, d) a G-metric space, A ⊆ X and d is bounded, then f : X → G with f(x) = d(x, A) := inf{d(x, a) : a ∈ A} is continuous and further x ∈ cl<sub>X</sub>A if and only if f(x) = e (the identity element in G). Moreover, we show that if G is a densely ordered group and further a closed subset of R, K(X) is the family of nonempty compact subsets of X, e &lt; g ∈ G and d is bounded, then d′ (A, B) &lt; g if and only if A ⊆ N<sub>d</sub>(B, g) and B ⊆ N<sub>d</sub>(A, g), where N<sub>d</sub>(A, g) = {x ∈ X : d(x, A) &lt; g}, d<sub>B</sub>(A) = sup{d(a, B) : a ∈ A} and d′ (A, B) = sup{d<sub>A</sub>(B), d<sub>B</sub>(A)}.</p>

scholarly journals Algebraic approach to Rump’s results on relations between braces and pre-lie algebras

2020 ◽  
pp. 2250054
Author(s):  
Agata Smoktunowicz
Keyword(s):  
Lie Algebras ◽  
Characteristic Zero ◽  
Algebraic Approach ◽  
Geometric Approach ◽  
Prime Characteristic ◽  
Affine Manifolds ◽  
Algebraic Interpretation

In 2014, Wolfgang Rump showed that there exists a correspondence between left nilpotent right [Formula: see text]-braces and pre-Lie algebras. This correspondence, established using a geometric approach related to flat affine manifolds and affine torsors, works locally. In this paper, we explain Rump’s correspondence using only algebraic formulae. An algebraic interpretation of the correspondence works for fields of sufficiently large prime characteristic as well as for fields of characteristic zero.

SLICED Ga ACTIONS

2001 ◽  
Vol 11 (01) ◽  
pp. 19-24
Author(s):  
JAMES K. DEVENEY ◽  
DAVID R. FINSTON
Keyword(s):  
Fixed Point ◽  
Characteristic Zero ◽  
Affine Space ◽  
Additive Group ◽  
Invariant Ring ◽  
Three Space ◽  
Free Actions

Let k be a field of characteristic zero. Certain classes of fixed point free actions of the additive group of k on affine n-space over k are known to be conjugate to global translations (i.e. to admit equivariant slices). These classes include actions on complex three space for which the invariant ring contains a variable, and certain generalizations of such actions to affine space of any dimension. Methods to construct an equivariant slice for these classes are presented.

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