746 research outputs found
Commutators and Squares in Free Nilpotent Groups
In a free group no nontrivial commutator is a square. And in the
free group F2=F(x1,x2) freely generated by x1,x2 the commutator [x1,x2] is never the product of two squares in F2, although it is always the product of three squares. Let F2,3=〈x1,x2〉 be a free nilpotent group of rank 2 and class
3 freely generated by x1,x2. We prove that in F2,3=〈x1,x2〉, it is possible
to write certain commutators as a square. We denote by Sq(γ) the minimal
number of squares which is required to write γ as a product of squares in group G. And we define Sq(G)=sup{Sq(γ);γ∈G′}.
We discuss the question of when the square length of a given commutator of
F2,3 is equal to 1 or 2 or 3. The precise formulas for expressing any commutator of F2,3 as the minimal number of squares are given. Finally as an application of these results we prove that Sq(F′2,3)=3
On the cohomology of Galois groups determined by Witt rings
Let F denote a field of characteristic different from two. In this paper we
describe the mod 2 cohomology of a Galois group which is determined by the Witt
ring WF
Nilpotent deformations of N=2 superspace
We investigate deformations of four-dimensional N=(1,1) euclidean superspace
induced by nonanticommuting fermionic coordinates. We essentially use the
harmonic superspace approach and consider nilpotent bi-differential Poisson
operators only. One variant of such deformations (termed chiral nilpotent)
directly generalizes the recently studied chiral deformation of N=(1/2,1/2)
superspace. It preserves chirality and harmonic analyticity but generically
breaks N=(1,1) to N=(1,0) supersymmetry. Yet, for degenerate choices of the
constant deformation matrix N=(1,1/2) supersymmetry can be retained, i.e. a
fraction of 3/4. An alternative version (termed analytic nilpotent) imposes
minimal nonanticommutativity on the analytic coordinates of harmonic
superspace. It does not affect the analytic subspace and respects all
supersymmetries, at the expense of chirality however. For a chiral nilpotent
deformation, we present non(anti)commutative euclidean analogs of N=2 Maxwell
and hypermultiplet off-shell actions.Comment: 1+16 pages; v2: discussion of (pseudo)conjugations extended, version
to appear in JHE
Irrationality of generic quotient varieties via Bogomolov multipliers
The Bogomolov multiplier of a group is the unramified Brauer group associated
to the quotient variety of a faithful representation of the group. This object
is an obstruction for the quotient variety to be stably rational. The purpose
of this paper is to study these multipliers associated to nilpotent pro-
groups by transporting them to their associated Lie algebras. Special focus is
set on the case of -adic Lie groups of nilpotency class , where we
analyse the moduli space. This is then applied to give information on
asymptotic behaviour of multipliers of finite images of such groups of exponent
. We show that with fixed and increasing , a positive proportion of
these groups of order have trivial multipliers. On the other hand, we
show that by fixing and increasing , log-generic groups of order
have non-trivial multipliers. Whence quotient varieties of faithful
representations of log-generic -groups are not stably rational. Applications
in non-commutative Iwasawa theory are developed.Comment: 34 pages; improved expositio
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