The gl(1|1) Lie superbialgebras

Abstract

By direct calculations of matrix form of super Jacobi and mixed super Jacobi identities which are obtained from adjoint representation, and using the automorphism supergroup of the gl(1|1) Lie superalgebra, we determine and classify all gl(1|1) Lie superbialgebras. Then, by calculating their classical r-matrices, the gl(1j1) coboundary Lie superbialgebras and their types (triangular, quasi-triangular or factorizable) are determined, furthermore in this way super Poisson structures on the GL(1|1) Lie supergroup are obtained. Also, we classify Drinfeld superdoubles based on the gl(1|1) as a theorem. Afterwards, as a physical application of the coboundary Lie superbialgebras, we construct a new integrable system on the homogeneous superspace OSp(1|2)/U(1). Finally, we make use of the Lyakhovsky and Mudrov formalism in order to build up the deformed gl(1|1) Lie superalgebra related to all gl(1|1) coboundary Lie superbialgebras. For one case, the quantization at the supergroup level is also provided, including its quantum R-matrix.

Section 8 and 2 references have added

References in corpus (3)

• The Classical r-matrix of AdS/CFT and its Lie Bialgebra Structure
• Poisson-Lie T-dual sigma models on supermanifolds
• String cosmology from Poisson-Lie T-dual sigma models on supermanifolds

Cited by in corpus (11)

• Super Poisson-Lie symmetry of the GL(1|1) WZNW model and worldsheet boundary conditions
• WZW models as mutual super Poisson-Lie T-dual sigma models
• Invariant Poisson-Nijenhuis structures on Lie groups and classification
• Yang-Baxter deformations of WZW model on the Heisenberg Lie group
• Yang-Baxter deformation of WZW model based on Lie supergroups: The cases of $GL(1|1)$ and $(C^3+A)$
• Classical r-matrices of real low-dimensional Jacobi-Lie bialgebras and their Jacobi-Lie groups
• Lie superbialgebra structures on the Lie superalgebra $({\mathcal{C}}^3+\mathcal{A})$ and deformation of related integrable Hamiltonian systems
• A hierarchy of WZW models related to super Poisson-Lie T-duality
• Cosmological string backgrounds from super Poisson-Lie T-plurality
• T-duality/plurality of BTZ black hole metric coupled to two fermionic fields
• More on (gauged) WZW models over low-dimensional Lie supergroups and their integrable deformations