On the q-analog of homological algebra

Abstract

This is an attempt to generalize some basic facts of homological algebra to the case of "complexes" in which the differential satisfies the condition $d^N=0$ instead of the usual $d^2=0$. Instead of familiar sign factors, the constructions related to such "N-complexes" involve powers of q where q is a primitive Nth root of 1. We show that the homology (in a natural sense) of an N-complex is an $(N-1)$-complex which is $(N-1)$-exact, and the role of the Euler characteristic is played by the trigonometric sum $\sum q^i\dim (C^i)$. By q-deforming the de Rham differential we develop a version of the theory of differential forms which is coordinate-dependent but covariant with respect to a natural Hopf algebra. In particular, there is a meaningful formalism of connections with the curvature being an N-form given by the N th power of the covariant derivative. For $N=3$ the expression for the curvature is very similar to the Chern-Simons functional. This text was written in 1991.

15 pages, plain TEX

Cited by in corpus (27)

• Tensor fields of mixed Young symmetry type and N-complexes
• Universal differential calculus on ternary algebras
• Universal $q$-differential calculus and $q$-analog of homological algebra
• Generalized differential spaces with $d^N=0$ and the $q$-differential calculus
• Hopfological Algebra
• Generalized cohomology for irreducible tensor fields of mixed Young symmetry type
• Categorification at prime roots of unity and hopfological finiteness
• The homotopy category of N-complexes is a homotopy category
• On Parity Complexes and Non-abelian Cohomology
• On the q-meromorphic Weyl algebra
• On $N$-differential graded algebras
• On the tensor product construction for q-differential algebras
• Categorification of the ring of cyclotomic integers for products of two primes
• Hochschild DGLAs and torsion algebras
• Sheaves of G-structures and generic G-models
• Simplicial Structure on Complexes
• Some topics on the $Z_3$-graded exterior algebras
• Incidence Homology of Finite Projective Spaces
• The $q$-analog of higher order Hochschild homology and the Lie derivative
• $A_{\infty }^N$-algebras
• Irreducible representations of the symmetric groups from slash homologies of p-complexes
• Generalized homologies for the zero modes of the SU(2) WZNW model
• Cohomology of torsion and completion of N-complexes
• Cell 2-Representations and Categorification at Prime Roots of Unity
• Generalized cohomologies and the physical subspace of the $SU(2)$ WZNW model
• Non-standard cohomology for equivariant sheaves: The role of generic models
• $q$-Analog Singular Homology of Convex Spaces