Selecta Mathematica-new Series 

About: Selecta Mathematica-new Series is an academic journal published by Springer Science+Business Media. The journal publishes majorly in the area(s): Lie algebra & Cohomology. It has an ISSN identifier of 1022-1824. Over the lifetime, 958 publications have been published receiving 25197 citations. The journal is also known as: SM.

Quivers with potentials and their representations I: Mutations

Abstract: We study quivers with relations given by noncommutative analogs of Jacobian ideals in the complete path algebra. This framework allows us to give a representation-theoretic interpretation of quiver mutations at arbitrary vertices. This gives a far-reaching generalization of Bernstein–Gelfand–Ponomarev reflection functors. The motivations for this work come from several sources: superpotentials in physics, Calabi–Yau algebras, cluster algebras.

Wonderful models of subspace arrangements

Abstract: The motivation stems from our attempt to understand Drinfeld's construction (el. [Dr2]) of special solutions of the Khniznik-Zamolodchikov equation (of. [K-Z]) with some prescribed asymptotic behavior and its consequences for some universal constructions associated to braiding: universal unipotent monodromy representations of braid groups, the construction of a universal Vassiliev invariant for knots, braided categories etc. The K-Z connection is a special flat meromorphic connection on C ~ with simple poles on a family of hyperplanes. It turns out that the prescription of the asymptotic behavior for such connections is controlled by the geometry of a suitable modification of C ~ in which the union of the polar hyperplanes is replaced by a divisor with normal crossings. In the process of developing this geometry we realized that our constructions could be developed more generally for subspace arrangements and became aware of the paper of Fulton-MaePherson [F-M] in which a Hironaka model is described for the complement of the big diagonal in the power of a smooth variety X. It became clear to us that our techniques were quite similar to theirs and so we adopted their notation of nested set in the appropriate general form. Although we work in a linear subspaces setting it is clear that the methods are essentially local and one can recover their results from our analysis applied to certain special configurations of subspaces. In fact the theory can be applied whenever we have a subvariety of a smooth variety which locally (in the gtale topology) appears as a union of subspaces.

Stable bundles, representation theory and Hermitian operators

Abstract: We give an interpretation and a solution of the classical problem of the spectrum of the sum of Hermitian matrices in terms of stable bundles on the projective plane.

Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory

Abstract: In this paper, we construct a naturalC*-dynamical system whose partition function is the Riemann ζ function. Our construction is general and associates to an inclusion of rings (under a suitable finiteness assumption) an inclusion of discrete groups (the associated ax+b groups) and the corresponding Hecke algebras of bi-invariant functions. The latter algebra is endowed with a canonical one parameter group of automorphisms measuring the lack of normality of the subgroup. The inclusion of rings ℤ⊂ℚ provides the desiredC*-dynamical system, which admits the ζ function as partition function and the Galois group Gal(ℚcycl/ℚ) of the cyclotomic extension ℚcycl of ℚ as symmetry group. Moreover, it exhibits a phase transition with spontaneous symmetry breaking at inverse temperature β=1 (cf. [Bos-C]). The original motivation for these results comes from the work of B. Julia [J] (cf. also [Spe]).