Preprojective representations of valued quivers and reduced words in the Weyl group of a Kac-Moody algebra

This paper studies connections between the preprojective representations of a valued quiver, the (+)-admissible sequences of vertices, and the Weyl group by associating to each preprojective representation a canonical (+)-admissible sequence. A (+)-admissible sequence is the canonical sequence of some preprojective representation if and only if the product of simple reflections associated to the vertices of the sequence is a reduced word in the Weyl group. As a consequence, for any Coxeter element of the Weyl group associated to an indecomposable symmetrizable generalized Cartan matrix, the group is infinite if and only if the powers of the element are reduced words. The latter strengthens known results of Howlett, Fomin-Zelevinsky, and the authors

Finite-dimensional algebras with smallest resolutions of simple modules

Let $X$ be a finitely generated left module over a left artinian ring $R$, and let $p(X)=\{l_i\}$ be the infinite sequence of nonnegative integers where $l_i$ is the length of the $i$-th term of the minimal projective resolution of $X$. We introduce a preorder relation $\le$ on the set $\{p(X)\}$ and characterize the elementary finite-dimensional algebras $\Lambda$ with the following property. Let $S$ be a simple $\Lambda$-module, and let $T$ be a finitely generated module over an arbitrary left artinian ring $R$. If the projective dimension of $S$ does not exceed the projective dimension of $T$, then $p(S)\le p(T)$. We characterize the indicated algebras by quivers with relations.Comment: Minor revisions, to appear in Journal of Algebr

Almost Split Morphisms, Preprojective Algebras and Multiplication Maps of Maximal Rank

With a grading previously introduced by the second-named author, the multiplication maps in the preprojective algebra satisfy a maximal rank property that is similar to the maximal rank property proven by Hochster and Laksov for the multiplication maps in the commutative polynomial ring. The result follows from a more general theorem about the maximal rank property of a minimal almost split morphism, which also yields a quadratic inequality for the dimensions of indecomposable modules involved

Sequences of reflection functors and the preprojective component of a valued quiver

This paper concerns preprojective representations of a finite connected valued quiver without oriented cycles. For each such representation, an explicit formula in terms of the geometry of the quiver gives a unique, up to a certain equivalence, shortest (+)-admissible sequence such that the corresponding composition of reflection functors annihilates the representation. The set of equivalence classes of the above sequences is a partially ordered set that contains a great deal of information about the preprojective component of the Auslander-Reiten quiver. The results apply to the study of reduced words in the Weyl group associated to an indecomposable symmetrizable generalized Cartan matrix

Adjoint Functors, Projectivization, and Differentiation Algorithms for Representations of Partially Ordered Sets

Adjoint functors and projectivization in representation theory of partially ordered sets are used to generalize the algorithms of differentiation by a maximal and by a minimal point. Conceptual explanations are given for the combinatorial construction of the derived set and for the differentiation functor

Mapping a Continental Shelf and Slope in the 1990s: A Tale of Three Multibeams

Increasing societal pressures on the U.S. continental shelves adjacent to dense population centers have brought to light the lack of accurate base maps in these areas. Existing bathymetric maps and random sidescan sonar surveys are either not accurate enough or do not provide the coverage necessary to make policy decisions. Until the mid 1990s, it was not financially prudent nor technically efficient to map the shallow shelves. However, the availability of high-resolution multibeam mapping systems now allow efficient and accurate mapping of the continental margins. In 1996 the U.S. Geological Survey began a large-scale seafloor mapping campaign on the continental shelf and slope adjacent to Los Angeles, CA. The first survey used a Kongsberg Simrad EM1000 (95 kHz). The survey continued in 1998 by mapping the slope and proximal basins from Newport to Long Beach, CA, using a Kongsberg Simrad EM300 (30 kHz). The area was completed in May 1999 by mapping the entire shelf adjacent to Long Beach, CA using an EM3000D (a dual-headed 300-kHz system). The mapping used both INS from the vehicle motion sensor and DGPS to provide position accuracies of ~1 m. All the data were processed in the field in near realtime using software developed at the Univ. of New Brunswick. Because of the different systems used and the range of water depths, the spatial resolution of the processed data varies from \u3c0.5 m on the inner shelf to 8 m on the basin floors. Perspective overviews of backscatter draped over bathymetry reveals a host of geological features unknown to exist in this area. These features include shallow, linear gullys, barchan dunes, small-scale bedforms in shallow troughs, major canyon system complexes, large- and smallscale mass movements, faults, and large areas of outcrop. The effects on sediment transport of man-made features, such as sewer outfall pipes and dredge-disposal fields, are clearly delineated on the new maps. The maps provide the fundamental base maps for studies as varied as those involving benthic habitats, marine disposal sites, sediment transport, and tectonic ma

Local Theory of Almost Split Sequences for Comodules

We show that almost split sequences in the category of comodules over a coalgebra with finite-dimensional right-hand term are direct limits of almost split sequences over finite dimensional subcoalgebras. In previous work we showed that such almost split sequences exist if the right hand term has a quasifinitely copresented linear dual. Conversely, taking limits of almost split sequences over finte-dimensional comodule categories, we then show that, for countable-dimensional coalgebras, certain exact sequences exist which satisfy a condition weaker than being almost split, which we call ``finitely almost split\u27\u27. Under additional assumptions, these sequences are shown to be almost split in the appropriate category

Edward Frenkel’s “Love & Math: the Heart of Hidden Reality” - a review

The autobiography in order to convey to the general audience both the human and professional aspects of mathematics. He explores the human aspect of mathematics by describing his work on several projects. He shows that, in addition to perseverance and hard work, solving a difficult mathematical problem requires imagination and original ideas, and that beauty and elegance usually characterize important mathematical results. The creative process of a mathematician in many ways resembles that of an artist or musician and generates the whole gamut of emotions, the most important of which is love