Combinatorics/Algebra Seminar
Spring 2014 Archive:

April 28

Sung-Yell Song: On relation graphs of 3-class nonsymmetric association schemes.

April 14, 21

Cliff Bergman: Verbose rings.

Abstract: An ideal  I  of a ring  R  is called fully invariant if  I  is preserved by every endomorphism of  R. For a subset  S  of

 Z< x₁, x₂, . . . >,

the ideal  I  is the verbal ideal induced by  S  if  I  is generated by

{ p(a₁, . . . , a_n) : p in S,  a₁, . . . , a_n in R }.

It is easy to show that every verbal ideal is fully invariant. The converse is false in general. We shall show that if  R  satisfies  x^k = x 
for some k > 1, then every fully invariant ideal of  R  is verbal. The proof is a nice application of natural duality as developed by Davey and Werner.

April 7

William DeMeo (University of South Carolina): From subgroup lattice intervals to group properties.

Abstract: We explain how to use the shape of an interval in a subgroup lattice to deduce properties of the underlying group. We describe a few properties that are "interval enforceable" in this sense, and a few that are not, and we connect these ideas to an old open problem in universal algebra.

March 31

Jonathan Smith: Report from the AMS Spring Meeting, Knoxville, TN.

March 17, 24

Spring break (and aftermath): no seminar.

February 24

Anna Romanowska (Warsaw University of Technology): Duality for some classes of convex sets.

Abstract: There is a well known self-duality for the category of finite-dimensional (real) vector spaces which can be adjusted to the category of corresponding affine spaces. This duality can be extended to the category of all affine spaces. However, it cannot be restricted to provide a duality for the category of convex subsets of (real) affine spaces. We will discuss the problem of duality for some classes of convex sets. We consider convex subsets of real affine spaces as abstract algebras, so-called barycentric algebras. Dualities for some classes of barycentric algebras will be described. In constructing some of these dualities, we use methods of dualizing so-called Płonka sums of algebras.

February 17, March 3, 10

Jonathan Smith: Quantum quasigroups, I, II, III.

Abstract: Quantum quasigroups (and quantum loops) are nonassociative extensions of Hopf algebras, with a much simpler axiomatization. They include previous attempts at extending Hopf algebra structure to nonassociative situations, in particular the Moufang-Hopf algebras of Benkart et al., and the H-bialgebras of Pérez-Izquierdo. We show that Hopf algebras form quantum quasigroups, and provide sufficient conditions for a quantum quasigroup to augment to a Hopf algebra. A key example of a quantum quasigroup provides an algebra of rooted binary trees, serving as indices for nonassociative powers.

January 27; February 3, 10

Candidate colloquia: no seminar.

January 20

King Holiday: no seminar.

Archive of earlier seminars

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