Novikov commutator algebras are special
A Novikov algebra \((V,\circ)\) is a nonassociative algebra satisfying the identities \((x\circ y)\circ z = (x\circ z)\circ y\) and \((x, y, z) = (y, x, z)\), where \((x, y, z) = (x\circ y) \circ z- x \circ (y\circ z)\) is the associator. A Gelfand-Dorfman algebra (or a GD-algebra) is a Lie algebra \((V,[\cdot,\cdot])\) with an additional multiplication \(\circ\) such that \((V,\circ)\) is a Novikov algebra and \([x, y\circ z] - [x, y] \circ z = [z, y\circ x] - [z, y] \circ x + y \circ [x, z]\), \(x,y,z\in V\). If \(P\) is a differential Poisson algebra, i.e. a commutative associative algebra with Poisson bracket \(\{\cdot,\cdot\}\) and a derivation \(d\), then it is a GD-algebra with operations \([\cdot,\cdot]=\{\cdot,\cdot\}\) and \(x\circ y=xd(y)\). The GD-algebra \(V\) is special if it can be embedded into an appropriate differential Poisson algebra \(P\). Using implicit arguments from algebraic geometry, it was established in [\textit{P. S. Kolesnikov} et al., J. Algebra 539, 260--284 (2019; Zbl 1448.17023)] that not all GD-algebras are special. In the paper under review the authors give an elegant explicit example of a nonspecial GD-algebra. This is the algebra \(V = sl_2\) with the standard linear basis \(e, f, h\) and the nonzero \(\circ\)-products of the basis elements \(e\circ h = 4f\), \(h\circ e = -2f\) and \(e\circ e = -h\). \par It is known that every Novikov algebra \((V,\circ)\) becomes a GD-algebra \(V^{(-)}\) with the Lie bracket \([x, y] = x\circ y- y\circ x\), \(x, y \in V\). The authors use results from [\textit{L. A. Bokut} et al., J. Algebra Appl. 16, No. 1, Article ID 1750001, 22 p. (2017; Zbl 1405.17060)] and give a short proof that the GD-algebra \(V^{(-)}\) is special.
scientific article; zbMATH DE number 7226859
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