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.

scholarly journals Some properties of the Levitzki radical in alternative rings

1982 ◽  
Vol 32 (1) ◽  
pp. 52-60 ◽  
Author(s):  
Michael Rich
Keyword(s):  
Nilpotent Ideal ◽  
Finitely Generated ◽  
Alternative Ring ◽  
Locally Nilpotent ◽  
Torsion Free

AbstractTwo local nilpotent properties of an associative or alternative ringAcontaining an idempotent are shown. First, ifA=A11+A10+A01+A00is the Peirce decomposition ofArelative toethen ifais associative or semiprime alternative and 3-torsion free then any locally nilpotent idealBofAiigenerates a locally nilpotent ideal 〈B〉 ofA. As a consequenceL(Aii) =Aii∩L(A)for the Levitzki radicalL. Also bounds are given for the index of nilpotency of any finitely generated subring of 〈B〉. Second, ifA(x)denotes a homotope ofAthenL(A)⊆L(A(x))and, in particular, ifA(x)is an isotope ofAthenL(A)=L(A(x)).

scholarly journals Locally nilpotent derivations and automorphism groups of certain Danielewski surfaces

2017 ◽  
Vol 469 ◽  
pp. 96-108 ◽  
Author(s):  
Angelo Calil Bianchi ◽  
Marcelo Oliveira Veloso
Keyword(s):  
Automorphism Groups ◽  
Locally Nilpotent

Note on locally nilpotent torsion-free groups

1971 ◽  
Vol 11 (6) ◽  
pp. 1033-1034
Author(s):  
E. M. Levich ◽  
A. I. Tokarenko
Keyword(s):  
Free Groups ◽  
Locally Nilpotent ◽  
Torsion Free

Automorphism groups of models of Peano arithmetic

2002 ◽  
Vol 67 (4) ◽  
pp. 1249-1264 ◽  
Author(s):  
James H. Schmerl
Keyword(s):  
Topological Group ◽  
Automorphism Group ◽  
Closed Subgroup ◽  
Automorphism Groups ◽  
Ordered Set ◽  
Peano Arithmetic ◽  
Ordered Sets ◽  
Torsion Free ◽  
Linearly Ordered Set

Which groups are isomorphic to automorphism groups of models of Peano Arithmetic? It will be shown here that any group that has half a chance of being isomorphic to the automorphism group of some model of Peano Arithmetic actually is.For any structure , let Aut() be its automorphism group. There are groups which are not isomorphic to any model = (N, +, ·, 0, 1, ≤) of PA. For example, it is clear that Aut(N), being a subgroup of Aut((, <)), must be torsion-free. However, as will be proved in this paper, if (A, <) is a linearly ordered set and G is a subgroup of Aut((A, <)), then there are models of PA such that Aut() ≅ G.If is a structure, then its automorphism group can be considered as a topological group by letting the stabilizers of finite subsets of A be the basic open subgroups. If ′ is an expansion of , then Aut(′) is a closed subgroup of Aut(). Conversely, for any closed subgroup G ≤ Aut() there is an expansion ′ of such that Aut(′) = G. Thus, if is a model of PA, then Aut() is not only a subgroup of Aut((N, <)), but it is even a closed subgroup of Aut((N, ′)).There is a characterization, due to Cohn [2] and to Conrad [3], of those groups G which are isomorphic to closed subgroups of automorphism groups of linearly ordered sets.

Classical Radicals of Invariants of Algebraic Skew Derivations

2022 ◽  
Vol 29 (01) ◽  
pp. 53-66
Author(s):  
Jeffrey Bergen ◽  
Piotr Grzeszczuk
Keyword(s):  
Hopf Algebra ◽  
Lie Algebra ◽  
Jacobson Radical ◽  
Prime Radical ◽  
Nilpotent Radical ◽  
Enveloping Algebra ◽  
Locally Nilpotent ◽  
Nil Radical

Let [Formula: see text] be an automorphism and[Formula: see text] be a [Formula: see text]-skew [Formula: see text]-derivation of an [Formula: see text]-algebra [Formula: see text]. We prove that if [Formula: see text] is semiprimitive and [Formula: see text] is algebraic, then the subalgebra [Formula: see text] has nilpotent Jacobson radical. Using this result, we obtain similar relations for the Baer prime radical, the Levitzki locally nilpotent radical, and the Köthe nil radical when the field [Formula: see text] is uncountable. Then we apply it to actions of the [Formula: see text]-dimensional Taft Hopf algebra [Formula: see text] and the [Formula: see text]-analogue [Formula: see text] of the enveloping algebra of the Lie algebra [Formula: see text].

scholarly journals Automorphisms of nilpotent groups of class two with small rank

1985 ◽  
Vol 39 (1) ◽  
pp. 121-131 ◽  
Author(s):  
Thomas A. Fournelle
Keyword(s):  
Automorphism Groups ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Factor Group ◽  
Direct Sums ◽  
Matrix Computations ◽  
Central Factor ◽  
Rank One ◽  
Free Abelian Groups ◽  
Torsion Free

AbstractRational abelian groups, that is, torsion-free abelian groups of rank one, are characterized by their types. This paper characterizes rational nilpotent groups of class two, that is, nilpotent groups of class two in which the center and central factor group are direct sums of rational abelian groups. This characterization is done according to the types of the summands of the center and the central factor group. Using these types and some cohomological techniques it is possible to determine the automorphism group of the nilpotent group in question by performing essentially matrix computations.In particular, the automorphism groups of rational nilpotent groups of class two and rank three are completely described. Specific examples are given of semicomplete and pseudocomplete nilpotent groups.

An independence theorem for automorphisms of torsion-free groups

1981 ◽  
Vol 90 (3) ◽  
pp. 403-409
Author(s):  
U. H. M. Webb
Keyword(s):  
Automorphism Group ◽  
Nilpotent Group ◽  
Free Group ◽  
Automorphism Groups ◽  
Torsion Free Group ◽  
Free Nilpotent Group ◽  
Free Abelian Groups ◽  
Independence Theorem ◽  
The Relationship ◽  
Torsion Free

This paper considers the relationship between the automorphism group of a torsion-free nilpotent group and the automorphism groups of its subgroups and factor groups. If G2 is the derived group of the group G let Aut (G, G2) be the group of automorphisms of G which induce the identity on G/G2, and if B is a subgroup of Aut G let B¯ be the image of B in Aut G/Aut (G, G2). A p–group or torsion-free group G is said to be special if G2 coincides with Z(G), the centre of G, and G/G2 and G2 are both elementary abelian p–groups or free abelian groups.

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