i[member of]V]), and the corresponding closed subscheme is the projective space P "without the edge [mu]"; the coordinate ring is [F.

In this presentation, an edge corresponds to a relation, and we construct a coordinate ring for [THETA]([GAMMA]) = S([GAMMA]) by deleting all relations of the ambient space P([GAMMA]) which are defined by edges in the complement of [GAMMA].

r], be the multi-graded coordinate ring of G/B with respect to [L.

The results of this paper says that the multi-graded coordinate ring of a Schubert variety in G/B with respect to r ample line bundles [L.

Then R is called multi-homogeneous coordinate ring of X with respect to [L.

ii) The

coordinate ring A of X with respect to ([L.

Previous approaches to these problems for constructing (quantum) cluster algebra structures on (quantum)

coordinate rings arising in Lie theory were done on a case-by-case basis, relying on the combinatorics of each concrete family, they say, and these findings will make that unnecessary.

1078 174184 Free resolutions of

coordinate rings of projective varieties and related topics (Kyoto 1998).

In recent works, the applicant developed new techniques from combinatorics/graph theory in order to study the representation theory of quantised

coordinate rings, whereas the host has been developing algorithmic methods in order to study the representation theory of these noncommutative algebras.

Among cluster algebras are

coordinate rings of many algebraic varieties that play a prominent rule in representation theory, invariant theory, the study of total positivity, and other areas, but the discussion here is limited to the relations of cluster algebra theory to Poisson geometry and the theory of integrable systems.

The subject of Quantum Groups is a rapidly diversifying field of mathematics and mathematical physics, originally launched by developments in theoretical physics and statistical mechanics involving quantum analogues of Lie algebras and coordinate rings of algebraic groups.

The aim of the course is to provide young researchers with the necessary tools to tackle open problems in the subject area, giving them the opportunity to learn the most recent results on the structure and representation theory of quantized coordinate rings and quantized enveloping algebras.