A generalized super AKNS hierarchy associated with orthosymplectic Lie superalgebra OSP(2,2) and its super bi-Hamiltonian structures

For the orthosymplectic Lie superalgebra $\text{◂⋅▸}OSP(2,2)$, we choose a set of basis matrices. A linear combination of those basis matrices presents a spatial spectral matrix. The compatible condition of the spatial part and the corresponding temporal parts of the spectral problem leads to a generalized super AKNS (GSAKNS) hierarchy. By making use of the supertrace identity, the obtained GSAKNS hierarchy can be written as the super bi-Hamiltonian structures.     

A new eight-dimensional Lie superalgebra and two corresponding hierarchies of evolution equations

A new eight-dimensional Lie superalgebra is constructed and two isospectral problems with six potentials are designed. Corresponding hierarchies of nonlinear evolution equations, as well as super-AKNS and super-Levi, are derived. Their super-Hamiltonian structures are established by making use of the supertrace identity, and they are integrable in the sense of Liouville.     

NEW SUPER-HAMILTONIAN STRUCTURES OF SUPER-DIRAC HIERARCHY

In this paper, we introduce the supertrace identity and its applications. A new eight-dimensional Lie superalgebra is constructed and the super-Dirac hierarchy is derived. By the supertrace identity, we obtain the super-bi-Hamiltonian structure of the super-Dirac hierarchy.     

Self-consistent Sources and Conservation Laws for Sup er-Geng Equation Hierarchy

Based on the matrix Lie super algebra and supertrace identity, the integrable super-Geng hierarchy with self-consistent is established. Furthermore, we establish the in-finitely many conservation laws for the integrable super-Geng hierarchy. The methods de-rived by us can be generalized to other nonlinear equation hierarchies.     

Supertrace and Superquadratic Lie Structure on the Weyl Algebra,and Applications to Formal Inverse Weyl Transform

Using the Moyal *-product and orthosymplectic supersymmetry, we construct a natural nontrivial supertrace and an associated nondegenerate invariant supersymmetric bilinear form for the Lie superalgebra structure of the Weyl algebra W. We decompose adjoint and twisted adjoint actions. We define a renormalized supertrace and a formal inverse Weyl transform in a deformation quantization framework and develop some examples Mathematics Subject Classification: 53D55, 17B05, 17B10, 17B20, 17B60, 17B65     

The number of independent traces and supertraces on symplectic reflection algebras

It is shown that A:= H₁, _η (G), the sympectic reflection algebra over ?, has T_G independent traces, where T_G is the number of conjugacy classes of elements without eigenvalue 1 belonging to the finite group G ? Sp(2N) ? End(?^2N) generated by the system of symplectic reflections.

Simultaneously, we show that the algebra A, considered as a superalgebra with a natural parity, has S_G independent supertraces, where S_G is the number of conjugacy classes of elements without eigenvalue -1 belonging to G.

We consider also A as a Lie algebra A^L and as a Lie superalgebra A^S.

It is shown that if A is a simple associative algebra, then the supercommutant [A^S, A^S] is a simple Lie superalgebra having at least S_G independent supersymmetric invariant non-degenerate bilinear forms, and the quotient [A^L, A^L]/([A^L, A^L] ∩ ?) is a simple Lie algebra having at least T_G independent symmetric invariant non-degenerate bilinear forms.