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**Dialgebras.**
*(English)*
Zbl 0999.17002

Loday, Jean-Louis (ed.) et al., Dialgebras and related operads. Berlin: Springer. Lect. Notes Math. 1763, 7-66 (2001).

The present paper, as well as the other three papers in the same volume of the Springer Lecture Notes in Mathematics (see the following three reviews Zbl 0999.17003, Zbl 0999.17004, Zbl 0999.17005), is a part of a long-standing project whose ultimate aim is to study periodicity phenomena in algebraic \(K\)-theory. The concrete purpose of the article is to introduce and study a new class of algebras which produce Leibniz algebras in the same way as one obtains Lie algebras from associative algebras. The Leibniz algebras are characterized by the property that the multiplication (called a bracket) from the right is a derivation but the bracket is no more skew-symmetric as for Lie algebras.

The idea is to start with two distinct operations for the products \(xy\) and \(yx\) and to consider a vector space (called an associative dialgebra) which has two binary multiplications \(\dashv\) and \(\vdash\) subject to some associativity conditions.

In the first two sections of the paper the author introduces the notion of dimonoid, develops the calculus in a dimonoid, describes the free dimonoid and introduces the notion of dialgebra. He also gives an explicit description of the free dialgebra.

In Section 3 the author constructs a (co)homology theory for dialgebras and proves that the (co)homology groups vanish on the free dialgebra. As a consequence, he gets also a new (co)homology theory for ordinary associative algebras. It is surprising that in the construction of the chain complex the combinatorics of planar binary trees is involved.

Section 4 deals with the relationship between dialgebras and Leibniz algebras: If \((D,\vdash,\dashv)\) is a dialgebra, then \((D,[x,y]=x\dashv y-y\vdash x)\) is a Leibniz algebra. The functor \((D,\vdash,\dashv)\to (D,[x,y])\) has a left adjoint, which is the universal enveloping dialgebra of a Leibniz algebra.

In Section 5 the author introduces another type of algebra with two operations \(\prec\) and \(\succ\) (called dendriform algebras) such that the product made of the sum \(x\prec y+y\succ x\) is associative. The main result in the section is the explicit description of the free dendriform algebra by means of binary trees. In Section 6 the author constructs (co)homology groups for dendriform algebras which, as in the case of dialgebras, vanish on the free objects. The next section relates dendriform algebras with dual-Leibniz algebras and associative algebras based on the relationship between binary trees and permutations.

The results’ intertwining associative dialgebras and dendriform algebras are expressed in the framework of algebraic operads in Section 8. The last section is devoted to strong homotopy associative dialgebras.

For the entire collection see [Zbl 0970.00010].

The idea is to start with two distinct operations for the products \(xy\) and \(yx\) and to consider a vector space (called an associative dialgebra) which has two binary multiplications \(\dashv\) and \(\vdash\) subject to some associativity conditions.

In the first two sections of the paper the author introduces the notion of dimonoid, develops the calculus in a dimonoid, describes the free dimonoid and introduces the notion of dialgebra. He also gives an explicit description of the free dialgebra.

In Section 3 the author constructs a (co)homology theory for dialgebras and proves that the (co)homology groups vanish on the free dialgebra. As a consequence, he gets also a new (co)homology theory for ordinary associative algebras. It is surprising that in the construction of the chain complex the combinatorics of planar binary trees is involved.

Section 4 deals with the relationship between dialgebras and Leibniz algebras: If \((D,\vdash,\dashv)\) is a dialgebra, then \((D,[x,y]=x\dashv y-y\vdash x)\) is a Leibniz algebra. The functor \((D,\vdash,\dashv)\to (D,[x,y])\) has a left adjoint, which is the universal enveloping dialgebra of a Leibniz algebra.

In Section 5 the author introduces another type of algebra with two operations \(\prec\) and \(\succ\) (called dendriform algebras) such that the product made of the sum \(x\prec y+y\succ x\) is associative. The main result in the section is the explicit description of the free dendriform algebra by means of binary trees. In Section 6 the author constructs (co)homology groups for dendriform algebras which, as in the case of dialgebras, vanish on the free objects. The next section relates dendriform algebras with dual-Leibniz algebras and associative algebras based on the relationship between binary trees and permutations.

The results’ intertwining associative dialgebras and dendriform algebras are expressed in the framework of algebraic operads in Section 8. The last section is devoted to strong homotopy associative dialgebras.

For the entire collection see [Zbl 0970.00010].

Reviewer: Vesselin Drensky (Sofia)