## 9 Citations

### Classical $W$-algebras and Frobenius manifolds related to Liouville completely integrable systems

- Mathematics
- 2019

We proved that the local bihamiltonian structure obtained by generalized Drinfeld-Sokolov reduction associated to a nilpotent element of semisimple type is reduced by Dirac reduction to the loop…

### Algebraic classical W-algebras and Frobenius manifolds

- MathematicsLetters in Mathematical Physics
- 2021

We consider Drinfeld-Sokolov bihamiltonian structure associated to a distinguished nilpotent elements of semisimple type and the space of common equilibrium points defined by its leading term. On…

### Dirac reductions and Classical W-algebras

- Mathematics
- 2022

. In the ﬁrst part of this paper, we generalize Dirac reduction to the extent of non-local Poisson vertex superalgebra and non-local SUSY Poisson vertex algebra cases. Next, we modify this reduction…

### Boussinesq hierarchy and bi-Hamiltonian geometry

- Mathematics, PhysicsJournal of Mathematical Physics
- 2021

We study the Boussinesq hierarchy in the geometric context of the theory of bi-Hamiltonian manifolds. First, we recall how its bi-Hamiltonian structure can be obtained by means of a process called…

### Poisson pencils: Reduction, exactness, and invariants

- MathematicsJournal of Geometry and Physics
- 2019

### Affine Kac-Moody Algebras and Tau-Functions for the Drinfeld-Sokolov Hierarchies: the Matrix-Resolvent Method

- MathematicsSymmetry, Integrability and Geometry: Methods and Applications
- 2022

. For each affine Kac–Moody algebra X ( r ) n of rank ℓ , r = 1 , 2, or 3, and for every choice of a vertex c m , m = 0 , . . . , ℓ , of the corresponding Dynkin diagram, by using the…

### 7 F eb 2 01 9 Poisson pencils : reduction , exactness , and invariants

- Mathematics
- 2019

We study the invariants (in particular, the central invariants) of suitable Poisson pencils from the point of view of the theory of bi-Hamiltonian reduction, paying a particular attention to the case…

## References

SHOWING 1-10 OF 32 REFERENCES

### Generalized Drinfeld-Sokolov Hierarchies II: The Hamiltonian Structures

- Mathematics
- 1991

In this paper we examine the bi-Hamiltonian structure of the generalized KdV-hierarchies. We verify that both Hamiltonian structures take the form of Kirillov brackets on the Kac-Moody algebra, and…

### TRANSVERSE POISSON STRUCTURES TO ADJOINT ORBITS IN SEMISIMPLE LIE ALGEBRAS

- Mathematics
- 2006

We study the transverse Poisson structure to adjoint orbits in a complex semisimple Lie algebra. The problem is first reduced to the case of nilpotent orbits. We prove then that in suitably chosen…

### Frobenius Manifolds and Central Invariants for the Drinfeld - Sokolov Bihamiltonian Structures

- Mathematics
- 2007

### Generalized Drinfel'd-Sokolov hierarchies

- Mathematics
- 1993

In this paper we examine the bi-Hamiltonian structure of the generalized KdV-hierarchies. We verify that both Hamiltonian structures take the form of Kirillov brackets on the Kac-Moody algebra, and…

### Drinfeld-Sokolov reduction on a simple lie algebra from the bihamiltonian point of view

- Mathematics
- 1992

We show that the Drinfeld-Sokolov reduction can be framed in the general theory of bihamiltonian manifolds, with the help of a specialized version of a reduction theorem for Poisson manifolds by…

### Equivalence of the Drinfeld-Sokolov reduction to a bi-Hamiltonian reduction

- Mathematics
- 1995

We show that the Drinfeld-Sokolov reduction is equivalent to a bi-Hamiltonian reduction, in the sense that these two reductions, although different, lead to the same reduced Poisson (more correctly,…

### On the completeness of the set of classical W-algebras obtained from DS reductions

- Mathematics
- 1993

We clarify the notion of the DS -- generalized Drinfeld-Sokolov -- reduction approach to classical ${\cal W}$-algebras. We first strengthen an earlier theorem which showed that an $sl(2)$ embedding…

### Generalized Drinfel'd-Sokolov hierarchies

- Mathematics
- 1992

A general approach is adopted to the construction of integrable hierarchies of partial differential equations. A series of hierarchies associated to untwisted Kac-Moody algebras, and conjugacy…

### Nilpotent orbits and finite W-algebras

- Mathematics
- 2009

In recent years, the finite W-algebras associated to a semisimple Lie algebra and its nilpotent element have been studied intensively from different viewpoints. In this lecture series, we shall…