Abstract
Abstractly defined super-derivations on Fermionic systems on a lattice are studied. The existence and uniqueness of the associated standard super-potential are shown for every super-derivation with the subalgebra of all local operators as its domain. The relation between the standard super-potential of a super-derivation and the standard potential for the square of the super-potential (which is shown to be a derivation in the case of finite range super-potentials) is obtained (by use of local super-Hamiltonian for the super-derivation and local Hamiltonian for the square). As a consequence, a necessary and sufficient condition for a super-derivation to be nilpotent is obtained in terms of the corresponding standard super potential. Examples of translation invariant nilpotent super-derivations are given in the case of super-potentials of finite ranges on a one-dimensional lattice.
A merit of considering the super-potential associated with a super-derivation is that the former can be used as free parameters for the latter.
Similar content being viewed by others
References
Araki H. (2004). Conditional expectations relative to a product state and the corresponding standard potentials. Commun. Math. Phys. 246: 113–132
Araki H. (2006). Standard potentials for non-even dynamics. Commun. Math. Phys. 262: 161–176
Araki H., Moriya H. (2003). Equilibrium statistical mechanics of Fermion lattice systems. Rev. Math. Phys. 15: 93–198
Jaffe A., Lesniewski A., Osterwalder K. (1988). Quantum K-theory. Commun. Math. Phys. 118: 1–14
Jaffe A., Lesniewski A., Osterwalder K. (1989). On super-KMS functionals and entire cyclic cohomology. K-Theory, 2: 675–682
Jaffe A., Lesniewski A., Wisniowski M. (1989). Deformation of super-KMS functionals. Commun. Math. Phys. 121: 527–540
Kishimoto A., Nakamura H. (1994). Super-derivations. Commun. Math. Phys. 159: 15–27
Sakai S. (1976). On one-parameter subgrouops of *-automorphisms on operator algebras and the corresponding unbounded derivations. Amer. J. Math. 98: 427–440
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Y. Kawahigashi.
Rights and permissions
About this article
Cite this article
Araki, H. Super-Derivations and Associated Standard Super-Potentials. Commun. Math. Phys. 276, 261–285 (2007). https://doi.org/10.1007/s00220-007-0314-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-007-0314-3