10 research outputs found
Algebras and groups defined by permutation relations of alternating type
The class of finitely presented algebras over a field with a set of
generators and defined by homogeneous relations of the form
, where
runs through \Alt_{n}, the alternating group, is considered. The
associated group, defined by the same (group) presentation, is described. A
description of the radical of the algebra is found. It turns out that the
radical is a finitely generated ideal that is nilpotent and it is determined by
a congruence on the underlying monoid, defined by the same presentation
Finitely Presented Monoids and Algebras defined by Permutation Relations of Abelian Type, II
The class of finitely presented algebras A over a field K with a set of
generators x_{1},...,x_{n} and defined by homogeneous relations of the form
x_{i_1}x_{i_2}...x_{i_l}=x_{sigma(i_1)}x_{sigma(i_2)}...x_{sigma(i_l)}, where l
geq 2 is a given integer and sigma runs through a subgroup H of Sym_n, is
considered. It is shown that the underlying monoid S_{n,l}(H)=
<x_1,x_2,...,x_n|x_{i_1}x_{i_2}...x_{i_l}=x_{sigma(i_1)}x_{sigma(i_2)}...x_{\sigma
(i_l)}, sigma in H, i_1,...,i_l in {1,...,n}> is cancellative if and only if H
is semiregular and abelian. In this case S_{n,l}(H) is a submonoid of its
universal group G. If, furthermore, H is transitive then the periodic elements
T(G) of G form a finite abelian subgroup, G is periodic-by-cyclic and it is a
central localization of S_{n,l}(H), and the Jacobson radical of the algebra A
is determined by the Jacobson radical of the group algebra K[T(G)]. Finally, it
is shown that if H is an arbitrary group that is transitive then K[S_{n,l}(H)]
is a Noetherian PI-algebra of Gelfand-Kirillov dimension one; if furthermore H
is abelian then often K[G] is a principal ideal ring. In case H is not
transitive then K[S_{n,l}(H)] is of exponential growth.Comment: 8 page
Structure monoids of set-theoretic solutions of the Yang-Baxter equation
Given a set-theoretic solution of the Yang--Baxter equation, we
denote by the structure monoid and by , respectively
, the left, respectively right, derived structure monoid of
. It is shown that there exist a left action of on and a right
action of on and 1-cocycles and of with coefficients
in and in with respect to these actions respectively. We investigate
when the 1-cocycles are injective, surjective or bijective. In case is
finite, it turns out that is bijective if and only if is left
non-degenerate, and is bijective if and only if is right
non-degenerate. In case is left non-degenerate, in particular is
bijective, we define a semi-truss structure on and then we show that
this naturally induces a set-theoretic solution on the least
cancellative image of . In case is naturally
embedded in , for example when is irretractable, then
is an extension of . It also is shown that non-degenerate
irretractable solutions necessarily are bijective.Comment: 21 pages. Some minor changes have been implemented. To appear in
Publicacions Matem\`atiques Some additional minor changes were implemente
Braces and symmetric groups with special conditions
We study symmetric groups and left braces satisfying special conditions, or
identities. We are particularly interested in the impact of conditions like
and on the properties of the symmetric group and
its associated brace. We show that the symmetric group associated to
a nontrivial solution has multipermutation level if and only if
satisfies . In the special case of a two-sided brace we express
each of the conditions and as identities on the
associated radical ring . We apply these to construct examples of
two-sided braces satisfying some prescribed conditions. In particular we
construct a finite two-sided brace with condition which does
not satisfy . (It is known that condition implies
). We show that a finitely generated two-sided brace which
satisfies \textbf{lri} has a finite multipermutation level which is bounded by
the number of its generators.Comment: 20 page