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In this paper, we give a generalization of Kitaevs stabilizer code based on chain complex theory of bicommutative Hopf algebras. Due to the bicommutativity, the Kitaevs stabilizer code extends to a broader class of spaces, e.g. finite CW-complexes ; more generally short abstract complex over a commutative unital ring R which is introduced in this paper. Given a finite-dimensional bisemisimple bicommutative Hopf algebra with an R-action, we introduce some analogues of A-stabilizers, B-stabilizers and the local Hamiltonian, which we call by the (+)-stabilizers, the (-)-stabilizers and the elementary operator respectively. We prove that the eigenspaces of the elementary operator give an orthogonal decomposition and the ground-state space is isomorphic to the homology Hopf algebra. In application to topology, we propose a formulation of topological local stabilizer models in a functorial way. It is known that the ground-state spaces of Kitaevs stabilizer code extends to Turaev-Viro TQFT. We prove that the 0-eigenspaces of a topological local stabilizer model extends to a projective TQFT which is improved to a TQFT in typical examples. Furthermore, we give a generalization of the duality in the literature based on the Poincare-Lefschetz duality of R-oriented manifolds.
The codomain category of a generalized homology theory is the category of modules over a ring. For an abelian category A, an A-valued (generalized) homology theory is defined by formally replacing the category of modules with the category A. It is kn
This paper provides motivation as well as a method of construction for Hopf algebras, starting from an associative algebra. The dualization technique involved relies heavily on the use of Sweedlers dual.
The Minkowski spacetime quantum Clifford algebra structure associated with the conformal group and the Clifford-Hopf alternative k-deformed quantum Poincare algebra is investigated in the Atiyah-Bott-Shapiro mod 8 theorem context. The resulting algeb
We present a general algorithm constructing a discretization of a classical field theory from a Lagrangian. We prove a new discrete Noether theorem relating symmetries to conservation laws and an energy conservation theorem not based on any symmetry.
The purpose of this paper is to establish meromorphy properties of the partial scattering amplitude T(lambda,k) associated with physically relevant classes N_{w,alpha}^gamma of nonlocal potentials in corresponding domains D_{gamma,alpha}^delta of the