Gelfand-Dorfman algebras, derived identities, and the Manin product of operads. (English) Zbl 1448.17023
The paper is about relations between Novikov algebras, Poisson algebras, Novikov-Poisson algebras, Gelfand-Dorfman algebras, dual Gelfand-Dorfman algebras and some other structures.
A Gelfand-Dorfman algebra (GD-algebra) is a system \((A,\circ,[\,])\) with two bilinear operations such that \((A, \circ)\) is a Novikov algebra, \((A, [\,])\) is a Lie algebra, and the following additional identity holds: \[ [a, b \circ c] - [c, b \circ a]+[b, a] \circ c - [b, c] \circ a -b \circ [a, c]=0.\] Every Poisson algebra \((\mathcal{P}, \cdot, [\, ])\) with a derivation \(d\) gives a GD-algebra relative to the operations \([\, ]\) and \(\circ\) \(\left(\ x \circ y = x \cdot d(y) \ \right).\) Let us say that a GD-algebra \(A\) is special if it can be embedded into a differential Poisson algebra.
The first result of the paper is regarding the calculation of defined identities of dual Gelfand-Dorfman algebras and a description of a linear basis of the free dual Gelfand-Dorfman algebras (GD\(^!\)-algebras), which give a proper sub-variety of the variety of Novikov-Poisson algebras. After that, the authors defined the variety BiCom of algebras with two associative and commutative operations \(*\) and \(\odot,\) such that \((x \odot y) * z = x \odot (y * z).\) Every BiCom algebra with a derivation \(d\) gives a GD\(^!\)-algebras relative to the operations \(*\) and a new multiplication \( \star\) \(\left( \ x \star y = d(x) \odot y \ \right).\) Let us say that a GD\(^!\)-algebra \(B\) is special if it can be embedded into an appropriate differential BiCom-algebra.
The main part of the rest of the paper is devoted to a study of special GD-algebras. Namely, the authors constructed the linear basis of the special GD-algebra [Theorem 10], and proved that there are no special identities of degree 3. In the last section, they have found two independent special identities of degree 4 [Propositions 2 and 5]. In the end of the paper, it was conjectured that all GD-algebras \((A,\circ, [\,]),\) where \((A, \circ)\) is Novikov and \([x , y ]= x \circ y - y \circ x,\) are special.
A Gelfand-Dorfman algebra (GD-algebra) is a system \((A,\circ,[\,])\) with two bilinear operations such that \((A, \circ)\) is a Novikov algebra, \((A, [\,])\) is a Lie algebra, and the following additional identity holds: \[ [a, b \circ c] - [c, b \circ a]+[b, a] \circ c - [b, c] \circ a -b \circ [a, c]=0.\] Every Poisson algebra \((\mathcal{P}, \cdot, [\, ])\) with a derivation \(d\) gives a GD-algebra relative to the operations \([\, ]\) and \(\circ\) \(\left(\ x \circ y = x \cdot d(y) \ \right).\) Let us say that a GD-algebra \(A\) is special if it can be embedded into a differential Poisson algebra.
The first result of the paper is regarding the calculation of defined identities of dual Gelfand-Dorfman algebras and a description of a linear basis of the free dual Gelfand-Dorfman algebras (GD\(^!\)-algebras), which give a proper sub-variety of the variety of Novikov-Poisson algebras. After that, the authors defined the variety BiCom of algebras with two associative and commutative operations \(*\) and \(\odot,\) such that \((x \odot y) * z = x \odot (y * z).\) Every BiCom algebra with a derivation \(d\) gives a GD\(^!\)-algebras relative to the operations \(*\) and a new multiplication \( \star\) \(\left( \ x \star y = d(x) \odot y \ \right).\) Let us say that a GD\(^!\)-algebra \(B\) is special if it can be embedded into an appropriate differential BiCom-algebra.
The main part of the rest of the paper is devoted to a study of special GD-algebras. Namely, the authors constructed the linear basis of the special GD-algebra [Theorem 10], and proved that there are no special identities of degree 3. In the last section, they have found two independent special identities of degree 4 [Propositions 2 and 5]. In the end of the paper, it was conjectured that all GD-algebras \((A,\circ, [\,]),\) where \((A, \circ)\) is Novikov and \([x , y ]= x \circ y - y \circ x,\) are special.
Reviewer: Ivan Kaygorodov (Novosibirsk)
MSC:
| 17B63 | Poisson algebras |
| 37K30 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures |
| 08B20 | Free algebras |
Software:
SINGULARReferences:
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