Cohomology and deformations of oriented dialgebras
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Cohomology and deformations of oriented dialgebras Ali N. A. Koam1 · Ripan Saha2 Received: 25 March 2020 / Accepted: 2 September 2020 © Tbilisi Centre for Mathematical Sciences 2020
Abstract We introduce a notion of oriented dialgebra and develop a cohomology theory for oriented dialgebras by mixing the standard chain complexes computing group cohomology and associative dialgebra cohomology. We also introduce a formal deformation theory for oriented dialgebras and show that cohomology of oriented dialgebras controls such deformations. Keywords Oriented dialgebras · Oriented dialgebra cohomology · One-parameter formal deformations · Singular extensions · Involutions Mathematics Subject Classification 16E30 · 16E40 · 18G55
1 Introduction The notion of dialgebras was introduced by Loday [13] while studying the periodicity phenomena in algebraic K -theory. The associative dialgebras or simply dialgebras are generalizations of associative algebras, whose structure is determined by two associative operations intertwined by some relations. A (co)homology theory of dialgebras using planar binary trees has been introduced by Loday, and dialgebra (co)homology with coefficients was introduced by Frabetti [5]. The notion of an oriented algebra was introduced by Koam and Pirashvili [11] with the aim of developing the equivariant version of Hochschild cohomology. Oriented algebras generalize both G-algebras and involutive associative algebras. The authors
Communicated by Jim Stasheff.
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Ripan Saha [email protected] Ali N. A. Koam [email protected]
1
Department of Mathematics, Faculty of Sciences, Jazan University, Jizan, Saudi Arabia
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Department of Mathematics, Raiganj University, Raiganj, West Bengal 733134, India
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A. N. A. Koam, R. Saha
developed a cohomology theory of oriented algebras and showed how extensions and deformations are related to the cohomology. Algebras with involution play an important role in the study of algebras arising in certain geometric contexts [1–3,12]. For example, the de Rham cohomology of a manifold with an involution carries an involutive A∞ -algebra structure [12]. Costello [3] showed that cyclic A∞ -algebras are equivalent to open topological conformal field theories. Following Costello, Braun [1,2] showed that cyclic involutive A∞ -algebras are equivalent to open Klein topological conformal field theories, and studied Hochschild cohomology of involutive associative algebras (more generally of involutive A∞ algebras). In the present paper, we introduce a notion of oriented dialgebra, which is related to that of an oriented algebra as a dialgebra is to an associative algebra. Oriented dialgebras are more general than G-algebras and involutive dialgebras. In this paper, we aim to develop an equivariant version of dialgebra cohomology. Let ODias be the category oriented dialgebras. Similarly to the non-oriented case, we establish a commutative diagram of functors, which relates ODias to other categories of Loday-type algebras in the oriented setting. We mix the standard chain
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