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Register a new userRepresentation theory of type B and C standard Levi W-algebras
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We classify the finite dimensional irreducible representations with integral central character of finite $W$-algebras $U(mathfrak g,e)$ associated to standard Levi nilpotent orbits in classical Lie algebras of types B and C. This classification is given explicitly in terms of the highest weight theory for finite $W$-algebras.
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For each natural number n greater than 1, we define an algebra satisfying many properties that one might expect to hold for a Brauer algebra of type Cn. The monomials of this algebra correspond to scalar multiples of symmetric Brauer diagrams on 2n strands. The algebra is shown to be free of rank the number of such diagrams and cellular, in the sense of Graham and Lehrer.
For each n>0, we define an algebra having many properties that one might expect to hold for a Brauer algebra of type Bn. It is defined by means of a presentation by generators and relations. We show that this algebra is a subalgebra of the Brauer algebra of type Dn+1 and point out a cellular structure in it. This work is a natural sequel to the introduction of Brauer algebras of type Cn, which are subalgebras of classical Brauer algebras of type A2n-1 and differ from the current ones for n>2.
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We introduce a new family of affine $W$-algebras associated with the centralizers of arbitrary nilpotent elements in $mathfrak{gl}_N$. We define them by using a version of the BRST complex of the quantum Drinfeld--Sokolov reduction. A family of free generators of the new algebras is produced in an explicit form. We also give an analogue of the Fateev--Lukyanov realization for these algebras by applying a Miura-type map.
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