110 research outputs found
Left-symmetric Algebras From Linear Functions
In this paper, some left-symmetric algebras are constructed from linear
functions. They include a kind of simple left-symmetric algebras and some
examples appearing in mathematical physics. Their complete classification is
also given, which shows that they can be regarded as generalization of certain
2-dimensional left-symmetric algebras.Comment: 16 pages, accepted for publication in Journal of Algebr
Pre-alternative algebras and pre-alternative bialgebras
We introduce a notion of pre-alternative algebra which may be seen as an
alternative algebra whose product can be decomposed into two pieces which are
compatible in a certain way. It is also the "alternative" analogue of a
dendriform dialgebra or a pre-Lie algebra. The left and right multiplication
operators of a pre-alternative algebra give a bimodule structure of the
associated alternative algebra. There exists a (coboundary) bialgebra theory
for pre-alternative algebras, namely, pre-alternative bialgebras, which
exhibits all the familiar properties of the famous Lie bialgebra theory. In
particular, a pre-alternative bialgebra is equivalent to a phase space of an
alternative algebra and our study leads to what we called -equations in a
pre-alternative algebra, which are analogues of the classical Yang-Baxter
equation.Comment: 34 page
Double constructions of Frobenius algebras, Connes cocycles and their duality
We construct an associative algebra with a decomposition into the direct sum
of the underlying vector spaces of another associative algebra and its dual
space such that both of them are subalgebras and the natural symmetric bilinear
form is invariant or the natural antisymmetric bilinear form is a Connes
cocycle. The former is called a double construction of Frobenius algebra and
the latter is called a double construction of Connes cocycle which is
interpreted in terms of dendriform algebras. Both of them are equivalent to a
kind of bialgebras, namely, antisymmetric infinitesimal bialgebras and
dendriform D-bialgebras respectively. In the coboundary cases, our study leads
to what we call associative Yang-Baxter equation in an associative algebra and
-equation in a dendriform algebra respectively, which are analogues of the
classical Yang-Baxter equation in a Lie algebra. We show that an antisymmetric
solution of associative Yang-Baxter equation corresponds to the antisymmetric
part of a certain operator called -operator which gives a double
construction of Frobenius algebra, whereas a symmetric solution of -equation
corresponds to the symmetric part of an -operator which gives a
double construction of Connes cocycle. By comparing antisymmetric infinitesimal
bialgebras and dendriform D-bialgebras, we observe that there is a clear
analogy between them. Due to the correspondences between certain symmetries and
antisymmetries appearing in the analogy, we regard it as a kind of duality.Comment: 50 pages, 2 tables; Some straightforward proofs are omitted, some
terminologies and references are change
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