Let G be a simple complex factorizable Poisson algebraic group. Let U (g) be the corresponding quantum group. We study the U (g)-equivariant quantization C [G] of the affine coordinate ring C[G] along the Semenov-Tian-Shansky bracket. For a simply connected group G, we give an elementary proof for the analog of the Kostant–Richardson theorem stating that C [G] is a free module over its center.

Abstract Let ???? be a finite-dimensional Lie algebra. The symmetric algebra (????) is equipped with the standard Lie–Poisson bracket. In this paper, we elaborate on surprising observation that one naturally associates second compatible Poisson bracket to any finite order automorphism ? of ????. We study related Poisson-commutative subalgebras (????; ?) ????(????) and associated contractions To...

Let G be a connected reductive group acting on a finite dimensional vector space V . Assume that V is equipped with a G-invariant symplectic form. Then the ring O(V ) of polynomial functions becomes a Poisson algebra. The ring O(V ) of invariants is a sub-Poisson algebra. We call V multiplicity free if O(V ) is Poisson commutative, i.e., if {f, g} = 0 for all invariants f and g. Alternatively, ...

We first study some properties‎ ‎of $A$-module homomorphisms $theta:Xrightarrow Y$‎, ‎where $X$‎ ‎and $Y$ are Fréchet $A$-modules and $A$ is a unital‎ ‎Fréchet algebra‎. ‎Then we show that if there exists a‎ ‎continued bisection of the identity for $A$‎, ‎then $theta$ is‎ ‎automatically continuous under certain condition on $X$‎. ‎In‎ ‎particular‎, ‎every homomorphism from $A$ into certain‎ ‎Fr...

We study Lagrangian subalgebras of a semisimple Lie algebra with respect to the imaginary part of the Killing form. We show that the variety L of Lagrangian subalgebras carries a natural Poisson structure Π. We determine the irreducible components of L, and we show that each irreducible component is a smooth fiber bundle over a generalized flag variety, and that the fiber is the product of the ...

We consider Nambu-Poisson 3-algebras on three dimensional manifolds M3, such as the Euclidean 3-space R, the 3-sphere S as well as the 3-torus T . We demonstrate that in the Clebsch-Monge gauge, the Lie algebra of volume preserving diffeomorphisms SDiff(M3) is identical to the Nambu-Poisson algebra on M3. Moreover the fundamental identity for the Nambu 3-bracket is just the commutation relation...

Let P be a Poisson homogeneous G-space. In [Dr2], Drinfeld shows that corresponding to each p ∈ P , there is a maximal isotropic Lie subalgebra lp of the Lie algebra d, the double Lie algebra of the tangent Lie bialgebra (g, g∗) of G. Moreover, for g ∈ G, the two Lie algebras lp and lgp are related by lgp = Adg lp via the Adjoint action of G on d. In particular, they are isomorphic as Lie algeb...

Let O be a nilpotent orbit of complex semisimple Lie algebra g and let π:X→O¯ the finite covering associated with universal O. In previous article [14] we have explicitly constructed Q-factorial terminalization X˜ X when is classical. this count how many non-isomorphic terminalizations has. We construct Poisson deformation over H2(X˜,C) look at action Weyl group W(X) on H2(X˜,C). The main resul...

Let M(n) be the algebra (both Lie and associative) of n×n matrices over C. Then M(n) inherits a Poisson structure from its dual using the bilinear form (x, y) = −tr xy. The Gl(n) adjoint orbits are the symplectic leaves and the algebra, P (n), of polynomial functions on M(n) is a Poisson algebra. In particular if f ∈ P (n) then there is a corresponding vector field ξf on M(n). If m ≤ n then M(m...