The study of Maps (X,G), the group of polynomial maps of a complex algebraic variety X into a complex algebraic group G, and its representations is only well developed in the case that X is a complex torus C. In this case Maps (X,G) is a loop group and the corresponding Lie-algebra Maps (X, ◦ G) is the loop algebra C[t, t]⊗ ◦ G. Here the representation comes to life only after one replaces Maps...

We show that there are precisely two, up to conjugation, anti-involutions σ± of the algebra of differential operators on the circle preserving the principal gradation. We classify the irreducible quasifinite highest weight representations of the central extension D̂± of the Lie subalgebra of this algebra fixed by −σ±, and find the unitary ones. We realize them in terms of highest weight represen...

We begin to study the Lie theoretical analogs of symplectic reflection algebras for Γ a finite cyclic group, which we call “cyclic double affine Lie algebra”. We focus on type A : in the finite (resp. affine, double affine) case, we prove that these structures are finite (resp. affine, toroidal) type Lie algebras, but the gradings differ. The case which is essentially new is sln(C[u, v]⋊Γ). We ...

Non-commutative Poisson algebras are the algebras having an associative algebra structure and a Lie algebra structure together with the Leibniz law. For finite-dimensional ones we show that if they are semisimple as associative algebras then they are standard, on the other hand, if they are semisimple as Lie algebras then their associative products are trivial. We also give the descriptions of ...

Introduction 2 1. Lie algebras: recollections 3 1.1. The basics 3 1.2. Scaling the structure 3 1.3. Filtrations 4 1.4. The Chevalley complex 4 1.5. The functor of primitives 6 1.6. The enhanced adjunction 6 1.7. The symmetric Hopf algebra 8 2. Looping Lie algebras 9 2.1. Group-Lie algebras 10 2.2. Forgetting to group structure 10 2.3. Chevalley complex of group-Lie algebras 11 2.4. Chevalley co...

We begin to study the Lie theoretical analogs of symplectic reflection algebras for Γ a finite cyclic group, which we call “cyclic double affine Lie algebra”. We focus on type A : in the finite (resp. affine, double affine) case, we prove that these structures are finite (resp. affine, toroidal) type Lie algebras, but the gradings differ. The case which is essentially new is sln(C[u, v] o Γ). W...

Let An be the free associative algebra with n generators over C, consider the Lie algebra A1 of its outer derivations (the derivations modulo the inner derivations). Let A0 be its quasi-classical limit, that it the Lie algebra of outer derivations of the free Poisson algebra with n generators over C. Boris Feigin conjectured around 1998 that the Lie algebras A0 and A1 are isomorphic. It turns o...

The Lie algebra gl(λ) with λ ∈ C, introduced by B L Feigin, can be embedded into the Lie algebra of differential operators on the real line (see [7]). We give an explicit formula of the embedding of gl(λ) into the algebra D λ of differential operators on the space of tensor densities of degree λ on R. Our main tool is the notion of projectively equivariant symbol of a differential operator.

We develop a complete theory of non-formal deformation quantization on the cotangent bundle weakly exponential Lie group. An appropriate integral formula for star-product is introduced together with suitable space functions which well defined. This becomes Fréchet algebra as pre-C*-algebra. Basic properties are proved, and extension to Hilbert an distributions given. A C*-algebra observables st...

We establish that the Lie algebra of weight one states in a (strongly) rational vertex operator algebra is reductive, and that its Lie rank l is bounded above by the effective central charge c̃. We show that lattice vertex operator algebras may be characterized by the equalities c̃ = l = c, and in particular holomorphic lattice theories may be characterized among all holomorphic vertex operator a...