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## Vertex algebras and coordinate rings of semi-infinite flags

The direct sum of irreducible level one integrable representations of affine Kac-Moody Lie algebra of (affine) type *ADE* carries a structure of *P*/*Q*-graded vertex operator algebra. There exists a filtration on this direct sum studied by Kato and Loktev such that the corresponding graded vector space is a direct sum of global Weyl modules. The associated graded space with respect to the dual filtration is isomorphic to the homogenous coordinate ring of semi-infinite flag variety. We describe the ring structure in terms of vertex operators and endow the homogenous coordinate ring with a structure of *P*/*Q*-graded vertex operator algebra. We use the vertex algebra approach to derive semi-infinite Plücker-type relations in the homogeneous coordinate ring.