On pseudo-bialgebras

We study pseudoalgebras from the point of view of pseudo-dual of classical Lie coalgebra structures. We define the notions of Lie H-coalgebra and Lie pseudo-bialgebra. We obtain the analog of the CYBE, the Manin triples and Drinfeld's double for Lie pseudo-bialgebras. We also get a natural description of the annihilation algebra associated to a pseudoalgebra as a convolution algebra, clarifying this constructions in the theory of pseudoalgebras.Comment: arXiv admin note: substantial text overlap with arXiv:math/000712

Quasifinite Representations of Classical Lie subalgebras of W∞,pW_{\infty,p}

We show that there are exactly two anti-involution $\sigma_{\pm}$ of the algebra of differential operators on the circle that are a multiple of $p(t\partial_t)$ preserving the principal gradation (p\in\CC[x] non-constant). We classify the irreducible quasifinite highest weight representations of the central extension \hat{\D}_p^{\pm} of the Lie subalgebra fixed by $-\sigma_{\pm}$. The most important cases are the subalgebras \hat{\D}_x^{\pm} of $W_{\infty}$, that are obtained when $p(x)=x$. In these cases we realize the irreducible quasifinite highest weight modules in terms of highest weight representation of the central extension of the Lie algebra of infinite matrices with finitely many non-zero diagonals over the algebra \CC[u]/(u^{m+1}) and its classical Lie subalgebras of $C$ and $D$ types.Comment: arXiv admin note: text overlap with arXiv:math/9801136 by other author

On irreducibIe infinite conformaI algebras

On irreducibIe infinite conformaI algebra

Irreducible modules over finite simple Lie conformal superalgebras of type K

We construct all finite irreducible modules over Lie conformal superalgebras of type KComment: Accepted for publication in J. Math. Phys

Classification of finite irreducible modules over the Lie conformal superalgebra CK6

We classify all continuous degenerate irreducible modules over the exceptional linearly compact Lie superalgebra E(1, 6), and all finite degenerate irreducible modules over the exceptional Lie conformal superalgebra CK6, for which E(1, 6) is the annihilation algebra

Vertex Coalgebras, Coassociator, and Cocommutator Formulas

Based on the definition of vertex coalgebra introduced by Hubbard, 2009, we prove that this notion can be reformulated using coskew symmetry, coassociator and cocommutator formulas without restrictions on the grading. We also prove that a vertex coalgebra can be defined in terms of dual versions of the axioms of Lie conformal algebra and differential algebra