185 research outputs found
The partially alternating ternary sum in an associative dialgebra
The alternating ternary sum in an associative algebra, , gives rise to the partially alternating ternary sum in an
associative dialgebra with products and by making the
argument the center of each term: . We use computer algebra to determine the polynomial identities in
degree satisfied by this new trilinear operation. In degrees 3 and 5 we
obtain and ; these identities define a new variety of partially alternating ternary
algebras. We show that there is a 49-dimensional space of multilinear
identities in degree 7, and we find equivalent nonlinear identities. We use the
representation theory of the symmetric group to show that there are no new
identities in degree 9.Comment: 14 page
Computations involving differential operators and their actions on functions
The algorithms derived by Grossmann and Larson (1989) are further developed for rewriting expressions involving differential operators. The differential operators involved arise in the local analysis of nonlinear dynamical systems. These algorithms are extended in two different directions: the algorithms are generalized so that they apply to differential operators on groups and the data structures and algorithms are developed to compute symbolically the action of differential operators on functions. Both of these generalizations are needed for applications
Finite Type Invariants of w-Knotted Objects II: Tangles, Foams and the Kashiwara-Vergne Problem
This is the second in a series of papers dedicated to studying w-knots, and
more generally, w-knotted objects (w-braids, w-tangles, etc.). These are
classes of knotted objects that are wider but weaker than their "usual"
counterparts. To get (say) w-knots from usual knots (or u-knots), one has to
allow non-planar "virtual" knot diagrams, hence enlarging the the base set of
knots. But then one imposes a new relation beyond the ordinary collection of
Reidemeister moves, called the "overcrossings commute" relation, making
w-knotted objects a bit weaker once again. Satoh studied several classes of
w-knotted objects (under the name "weakly-virtual") and has shown them to be
closely related to certain classes of knotted surfaces in R4. In this article
we study finite type invariants of w-tangles and w-trivalent graphs (also
referred to as w-tangled foams). Much as the spaces A of chord diagrams for
ordinary knotted objects are related to metrized Lie algebras, the spaces Aw of
"arrow diagrams" for w-knotted objects are related to not-necessarily-metrized
Lie algebras. Many questions concerning w-knotted objects turn out to be
equivalent to questions about Lie algebras. Most notably we find that a
homomorphic universal finite type invariant of w-foams is essentially the same
as a solution of the Kashiwara-Vergne conjecture and much of the
Alekseev-Torossian work on Drinfel'd associators and Kashiwara-Vergne can be
re-interpreted as a study of w-foams.Comment: 57 pages. Improvements to the exposition following a referee repor
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