dorsal/arxiv
View SchemaOn Some Generalizations of Batalin-Vilkovisky Algebras
| Authors | Füsun Akman |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9506027 |
| URL | https://arxiv.org/abs/q-alg/9506027 |
Abstract
We define the concept of higher order differential operators on a general noncommutative, nonassociative superalgebra A, and show that a vertex operator superalgebra has plenty of them, namely modes of vertex operators. A linear operator \Delta is a differential operator of order at most r if an inductively defined (r+1)-form \Phi_{\Delta}^{r+1} is identically zero. When A is supercommutative and associative and \Delta is an odd, square zero, second order differential operator on A, \Phi_{\Delta}^2 defines a "Batalin-Vilkovisky algebra" structure on A. We generalize this notion to any superalgebra with an odd, square zero, second order differential operator, and show that all properties of the classical BV bracket but one continue to hold "on the nose". We also point out connections to Leibniz algebras and the noncommutative homology theory of Loday. Taking the generalization process one step further, we remove all conditions on \Delta to examine the changes in the basic properties of the bracket. We see that a topological chiral algebra with a mild restriction yields a classical BV algebra in the cohomology. Finally, we investigate the quantum BV master equation and relate it to deformations of differential graded algebras.
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"abstract": "We define the concept of higher order differential operators on a general\nnoncommutative, nonassociative superalgebra A, and show that a vertex operator\nsuperalgebra has plenty of them, namely modes of vertex operators. A linear\noperator \\Delta is a differential operator of order at most r if an inductively\ndefined (r+1)-form \\Phi_{\\Delta}^{r+1} is identically zero. When A is\nsupercommutative and associative and \\Delta is an odd, square zero, second\norder differential operator on A, \\Phi_{\\Delta}^2 defines a \"Batalin-Vilkovisky\nalgebra\" structure on A. We generalize this notion to any superalgebra with an\nodd, square zero, second order differential operator, and show that all\nproperties of the classical BV bracket but one continue to hold \"on the nose\".\nWe also point out connections to Leibniz algebras and the noncommutative\nhomology theory of Loday. Taking the generalization process one step further,\nwe remove all conditions on \\Delta to examine the changes in the basic\nproperties of the bracket. We see that a topological chiral algebra with a mild\nrestriction yields a classical BV algebra in the cohomology. Finally, we\ninvestigate the quantum BV master equation and relate it to deformations of\ndifferential graded algebras.",
"arxiv_id": "q-alg/9506027",
"authors": [
"F\u00fcsun Akman"
],
"categories": [
"q-alg",
"hep-th",
"math.QA"
],
"title": "On Some Generalizations of Batalin-Vilkovisky Algebras",
"url": "https://arxiv.org/abs/q-alg/9506027"
},
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