Let G = SpecA be an affine functor of monoids. We prove that A∗ is the enveloping functor of algebras of G and that the category of Gmodules is equivalent to the category of A∗-modules. Moreover, we prove that the category of affine functors of monoids is anti-equivalent to the category of functors of affine bialgebras. Applications of these results include Cartier duality, neutral Tannakian du...

We consider the centers of affine vertex algebras at critical level associated with simple Lie algebras. derive new formulas for generators in classical types. also give a formula Capelli-type determinant symplectic and calculate Harish-Chandra images Casimir elements arising from characteristic polynomial matrix each algebra.

Although the importance of developing a mathematics which transcends practical use was already understood by the Greeks over 2000 years ago, it is heartening even today when mathematical ideas created for their abstract interest are found to be useful in formulating descriptions of nature. Historically, the idea of symmetry has its scientific origin in the Greeks' discovery of the five regular ...

We give a combinatorial construction, not involving a presentation, of almost all untwisted affine Kac–Moody algebras modulo their onedimensional centres in terms of signed raising and lowering operators on a certain distributive lattice B. The lattice B is constructed combinatorially as a set of ideals of a “full heap” over the Dynkin diagram, which leads to a kind of categorification of the p...

We explain the construction of minimal tilting complexes for objects highest weight categories and we study in detail standard simple objects. For certain representations complex Lie algebras, affine Kac-Moody algebras quantum groups at roots unity, relate multiplicities indecomposable appearing terms these to coefficients Kazhdan-Lusztig polynomials.

We find necessary and sufficient conditions of irreducibility of vacuum modules over affine Lie algebras and superalgebras. From this we derive conditions of simplicity of minimal W -algebras. Moreover, in the case of the Virasoro and Neveu–Schwarz algebras we obtain explicit formulas for the vacuum determinants. © 2006 Elsevier Inc. All rights reserved.