AGeNT Seminar: Classification of Nonsymmetric Operads using Computational Linear and Commutative Algebra (Dr. Murray Bremner)

Monday, March 20, at 3pm in Mclean 242.1

Speaker: Murray Bremner

Title: Classification of Nonsymmetric Operads using Computational Linear and Commutative Algebra

Abstract: The talk will begin with a review of basic definitions and notations for algebraic operads(that is, operads in the symmetric monoidal category of vector spaces over a field of characteristic 0).  I will discuss both symmetric and nonsymmetric operads, and their bases in terms of both classical monomials and labelled trees.  I will then focus on free operads and operad ideals generated by quadratic relations, and explain how many computations in this setting can be reduced to elementary linear algebra.  Attempting to classify all nonsymmetric operads with two binary operations and quadratic relations leads to the problem of understanding how the rank of a matrix with entries in a polynomial ring depends on the values of the indeterminates.  Determinantal ideals are not helpful, since even in the simplest cases they are far too complex (for example, we obtain an ideal in 13 variables generated by 10^25 determinants of size 22).  Instead, we use what we call the "partial Smith form" algorithm which attempts to compute the Smith form of the matrix but terminates when there are no more nonzero scalars.  (The Smith form is only guaranteed to exist over Euclidean domains, so as soon as we have polynomials in more than one indeterminate, the theory becomes much more complex, requiring for example Gröbner bases for submodules of free modules over polynomial rings.)  Using these methods we have been able to discover new one- and two-parameter deformations of the diassociative operad introduced by Loday in the mid-1990's; this operad governs diassociative algebras (associative dialgebras) and is noteworthy for having dimension n in arity n.  We conclude with some comments on the symmetry group of the diassociative relations.