dorsal/arxiv
View SchemaOn the q-analog of homological algebra
| Authors | M. M. Kapranov |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9611005 |
| URL | https://arxiv.org/abs/q-alg/9611005 |
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.
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"abstract": "This is an attempt to generalize some basic facts of homological algebra to\nthe case of \"complexes\" in which the differential satisfies the condition\n$d^N=0$ instead of the usual $d^2=0$. Instead of familiar sign factors, the\nconstructions related to such \"N-complexes\" involve powers of q where q is a\nprimitive Nth root of 1. We show that the homology (in a natural sense) of an\nN-complex is an $(N-1)$-complex which is $(N-1)$-exact, and the role of the\nEuler characteristic is played by the trigonometric sum $\\sum q^i \\dim(C^i)$.\nBy q-deforming the de Rham differential we develop a version of the theory of\ndifferential forms which is coordinate-dependent but covariant with respect to\na natural Hopf algebra. In particular, there is a meaningful formalism of\nconnections with the curvature being an N-form given by the N th power of the\ncovariant derivative. For $N=3$ the expression for the curvature is very\nsimilar to the Chern-Simons functional. This text was written in 1991.",
"arxiv_id": "q-alg/9611005",
"authors": [
"M. M. Kapranov"
],
"categories": [
"q-alg",
"math.QA"
],
"title": "On the q-analog of homological algebra",
"url": "https://arxiv.org/abs/q-alg/9611005"
},
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