On dynamical r-matrices obtained from Dirac reduction and their generalizations to affine Lie algebras

According to Etingof and Varchenko, the classical dynamical Yang-Baxter equation is a guarantee for the consistency of the Poisson bracket on certain Poisson-Lie groupoids. Here it is noticed that Dirac reductions of these Poisson manifolds give rise to a mapping from dynamical r-matrices on a pair \L\subset \A to those on another pair \K\subset \A, where \K\subset \L\subset \A is a chain of Lie algebras for which \L admits a reductive decomposition as \L=\K+\M. Several known dynamical r-matrices appear naturally in this setting, and its application provides new r-matrices, too. In particular, we exhibit a family of r-matrices for which the dynamical variable lies in the grade zero subalgebra of an extended affine Lie algebra obtained from a twisted loop algebra based on an arbitrary finite dimensional self-dual Lie algebra.Comment: 19 pages, LaTeX, added a reference and a footnote and removed some typo

Translated tori in the characteristic varieties of complex hyperplane arrangements

We give examples of complex hyperplane arrangements for which the top characteristic variety contains positive-dimensional irreducible components that do not pass through the origin of the character torus. These examples answer several questions of Libgober and Yuzvinsky. As an application, we exhibit a pair of arrangements for which the resonance varieties of the Orlik-Solomon algebra are (abstractly) isomorphic, yet whose characteristic varieties are not isomorphic. The difference comes from translated components, which are not detected by the tangent cone at the origin.Comment: Revised and expanded; 16 pages, 10 figures; to appear in Topology and its Application

Reconstructing multisets over commutative groupoids and affine functions over nonassociative semirings

A reconstruction problem is formulated for multisets over commutative groupoids. The cards of a multiset are obtained by replacing a pair of its elements by their sum. Necessary and sufficient conditions for the reconstructibility of multisets are determined. These results find an application in a different kind of reconstruction problem for functions of several arguments and identification minors: classes of linear or affine functions over nonassociative semirings are shown to be weakly reconstructible. Moreover, affine functions of sufficiently large arity over finite fields are reconstructible.Comment: 18 pages. Int. J. Algebra Comput. (2014

More nonexistence results for symmetric pair coverings

A $(v,k,\lambda)$-covering is a pair $(V, \mathcal{B})$, where $V$ is a $v$-set of points and $\mathcal{B}$ is a collection of $k$-subsets of $V$ (called blocks), such that every unordered pair of points in $V$ is contained in at least $\lambda$ blocks in $\mathcal{B}$. The excess of such a covering is the multigraph on vertex set $V$ in which the edge between vertices $x$ and $y$ has multiplicity $r_{xy}-\lambda$, where $r_{xy}$ is the number of blocks which contain the pair $\{x,y\}$. A covering is symmetric if it has the same number of blocks as points. Bryant et al.(2011) adapted the determinant related arguments used in the proof of the Bruck-Ryser-Chowla theorem to establish the nonexistence of certain symmetric coverings with $2$-regular excesses. Here, we adapt the arguments related to rational congruence of matrices and show that they imply the nonexistence of some cyclic symmetric coverings and of various symmetric coverings with specified excesses.Comment: Submitted on May 22, 2015 to the Journal of Linear Algebra and its Application