Abstract:
We study self-distributive algebraic structures: algebras,
bialgebras, additional structures on them, relations of these
structures with Hopf algebras, Lie algebras, Leibnitz algebras,
etc. The basic example of such structures is given by rack and
quandle bialgebras. But we go further to the general coassociative
comultiplication. The principal motivation for this work is the development of linear algebra related to the notion of a quandle in
analogy with the ubiquitous role of group algebras in the category
of groups with possible applications to the theory of knot
invariants. We describe self-distributive algebras and show that
some quandle algebras and some Novikov algebras are
self-distributive. We also give a full classification of counital
self-distributive bialgebras in dimension $2$ over $\mathbb{C}$.