This is a report on an ongoing project with B. Leclerc and J. Schroeer. Ou r aim is to extend Lusztig'\;s constructionof a semicanonical basis for the enveloping algebra of the positive part of a symmetric Kac-Moody Lie algebra\,which is in terms of the preprojective algebra of the correspondi ng quiver over the complex numbers\, to the morenatural case of symmetriza ble Kac-Moody Lie algebras. To this end we study certain quivers\, which u sually containloops\, together with a potential and some nilpotency condit ions. Most of the basic constructions carry over to thisnew setup with som e modifications. In particular\, the components of maximal dimension of ou r generalized nilpotentvarieties have the structure of a B(\\infty)-crysta l of the corresponding type\, and we can construct semicanonical functions associated to those components. To conclude\, we would have to show that the constructible functionswhich have support with positive codimension\, form an ideal.In the second part we can give some more details about the p roofs and discuss the case B_2\, which supports ourconjecture.

LOCATION:Big Seminar room Ground floor / Office Bldg West (I21.EG.101)\, IS T Austria ORGANIZER:jdeanton@ist.ac.at SUMMARY:Quivers with relations for symmetrizable Cartan matrices and semica nonical functions URL:https://talks-calendar.app.ist.ac.at/events/1207 END:VEVENT END:VCALENDAR