# Local matrix generalizations of $W$-algebras

@article{Zuo2014LocalMG, title={Local matrix generalizations of \$W\$-algebras}, author={Dafeng Zuo}, journal={arXiv: Mathematical Physics}, year={2014} }

In this paper, we propose local matrix generalizations of the classical $W$-algebras based on the second Hamiltonian structure of the $\mathcal{Z}_m$-valued KP hierarchy, where $\mathcal{Z}_m$ is a maximal commutative subalgebra of $gl(m,\mathbb{C})$.

## 12 Citations

### Hamiltonian Structures and Integrability of Frobenius Algebra-Valued (n, m)th KdV Hierarchy

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We introduce Frobenius algebra ℱ-valued (n, m)th KdV hierarchy and construct its bi-Hamiltonian structures by employing ℱ-valued pseudo-differential operators. As an illustrative example, the (1,…

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We construct the extended flow equations of a new ZN-Toda hierarchy taking values in a commutative subalgebra ZN of gl(N,C). We give the Hirota bilinear equations and tau function of this new…

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### Gauge transformation and symmetries of the commutative multicomponent BKP hierarchy

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In this paper, we defined a new multi-component B type Kadomtsev-Petviashvili (BKP) hierarchy that takes values in a commutative subalgebra of gl ( N , C ) . ?> After this, we give the gauge…

### Integrability of the Frobenius algebra-valued Kadomtsev-Petviashvili hierarchy

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- 2015

We introduce a Frobenius algebra-valued Kadomtsev-Petviashvili (KP) hierarchy and show the existence of Frobenius algebra-valued τ-function for this hierarchy. In addition, we construct its…

### Three Types Generalized Zn-Heisenberg Ferromagnet Models

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By taking values in a commutative subalgebra , we construct a new generalized - Heisenberg ferromagnet model in (1+1)-dimensions. The corresponding geometrical equivalence between the generalized -…

### Conical defects in higher spin theories

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A bstractWe study conical defect geometries in the SL(N,$ \mathbb{R} $) × SL(N,$ \mathbb{R} $) (and SL(N,$ \mathbb{C} $)) Chern-Simons formulation of higher spin gauge theories in AdS3. We argue that…

### The extended ZN-Toda hierarchy

- MathematicsTheoretical and Mathematical Physics
- 2015

We construct the extended flow equations of a new ZN-Toda hierarchy taking values in a commutative subalgebra ZN of gl(N,C). We give the Hirota bilinear equations and tau function of this new…

## References

SHOWING 1-10 OF 48 REFERENCES

### New integrable hierarchies from vertex operator representations of polynomial Lie algebras

- Mathematics
- 2004

### Poisson Structures for Dispersionless Integrable Systems and Associated W-Algebras

- Mathematics
- 1996

AbstractIn analogy to the KP theory, the second Poisson structure for the dispersionless KP hierarchy can be defined on the space of commutative pseudodifferential operators
$$L = p^n +…

### $\mathcal {W}$-constraints for the total descendant potential of a simple singularity

- MathematicsCompositio Mathematica
- 2013

Abstract Simple, or Kleinian, singularities are classified by Dynkin diagrams of type $ADE$. Let $\mathfrak {g}$ be the corresponding finite-dimensional Lie algebra, and $W$ its Weyl group. The set…

### Higher spin fields and the Gelfand-Dickey algebra

- Mathematics
- 1989

We show that in 2-dimensional field theory, higher spin algebras are contained in the algebra of formal pseudodifferential operators introduced by Gelfand and Dickey to describe integrable nonlinear…

### Lectures on Classical W-Algebras

- Mathematics
- 1997

These are lecture notes of lectures given in 1993 in Cortona, Italy. W-algebras appeared in the conformal field theory as extensions of the Virasoro algebra. They are closely connected with…

### Extended Classical Conformal Algebras and the Second Hamiltonian Structure of Lax Equations

- Mathematics, Physics
- 1988

### Non-local matrix generalizations ofW-algebras

- Mathematics
- 1994

There is a standard way to define two symplectic (hamiltonian) structures, the first and second Gelfand-Dikii brackets, on the space of ordinarymth-order linear differential…

### Classical r-matrices and compatible Poisson brackets for coupled KdV systems

- Mathematics
- 1989

AbstractThe formalism of classical r-matrices is used to construct families of compatible Poisson brackets for some nonlinear integrable systems connected with Virasoro algebras. We recover the…