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# Homology, Homotopy and Applications

## Volume 4 (2002)

### Number 2

### The Roos Festschrift volume

### Trees, free right-symmetric algebras, free Novikov algebras and identities

Pages: 165 – 190

DOI: http://dx.doi.org/10.4310/HHA.2002.v4.n2.a8

#### Authors

#### Abstract

An algebra with the identity $x\circ (y\circ z-z\circ y)= (x\circ y)\circ z-(x\circ z)\circ y$ is called right-symmetric. A right-symmetric algebra with the identity $x\circ(y\circ z)= y\circ(x\circ z)$ is called Novikov. We describe bases of free right-symmetric algebras and free Novikov algebras and give realizations of them in terms of trees. The free Lie algebra is obtained as a Lie subalgebra of the free right-symmetric algebra. We use our methods to study identities of Witt algebras.

#### Keywords

rooted trees, Lie-admissable algebras, right-symmetric algebras, Novikov algebras, vector fields algebras, identities, free basis

#### 2010 Mathematics Subject Classification

17B01, 17B66, 17D25