Noncommutative Leibniz-Poisson algebras. (English) Zbl 1136.17003

The paper under review proposes a notion of a noncommutative Poisson algebra, where the bracket is that of a Leibniz algebra and the derivation property for the product of two elements occurs in the first (left) entry of the bracket. More specifically, let \(k\) be a commutative ring with unit. Then a Leibniz algebra [J.-L. Loday and T. Pirashvili, Math. Ann. 296, 139–158 (1993; Zbl 0821.17022)] over \(k\) is a \(k\)-module \(A\) together with a \(k\)-linear map \([\;, \;]: A \otimes A \to A\) satisfying the Leibniz identity (which the reviewer writes as)
\[ [[a, b], c] = [[a,c], b] + [a, [b, c]]. \]
A noncommutative Leibniz-Poisson algebra (NLP) is an associative \(k\)-algebra \(P\) together with a bracket \([\;, \;]: P \otimes P \to P\) so that (1) \((P,\;[\;, \;])\) is a Leibniz algebra and
\[ [a \cdot b,c] = a\cdot [b,c] + [a,c]\cdot b . \tag{2} \]
The paper then continues with a description of free NLP-algebras, and definitions of NLP-representations, actions, crossed modules, extensions, operad-dual NLP-algebras, and NLP-algebra cohomology. A representation over an NLP-algebra \(P\) is a \(P\)-\(P\)-bimodule \(M\) together with two \(k\)-linear maps
\[ [\;, \;]: P \otimes M \to M, \qquad [\;, \;]: M \otimes P \to P \]
such that
\[ \begin{aligned} [[p_1,p_2],m] &= [[p_1,m],p_2]+ [p_1,[p_2,m]],\\ [[p_1,m],p_2] &= [[p_1,p_2],m]+ [p_1,[m,p_2]],\\ [[m,p_1],p_2] &= [[m,p_2],p_1]+ [m,[p_1,p_2]],\\ [p_1\cdot m, p_2] &= p_1\cdot [m,p_2]+ [p_1,p_2]\cdot m \\ [m\cdot p_1,p_2] &= m\cdot [p_1,p_2]+ [m,p_2]\cdot p_1,\\ [p_1\cdot p_2,m] &= p_1\cdot [p_2,m]+ [p_1,m]\cdot p_2. \end{aligned} \]
Let \(k\) be a field, \((C^*_H(P,M),\partial_H)\) the chain complex for Hochschild cohomology, and \((C^*_L(P, M), \partial_L)\) the chain complex for Leibniz cohomology, where
\[ C^n_H (P,M)= C^n_L (P,M)= \operatorname{Hom}(P^{\otimes n},M). \]
Let \(M^e = \operatorname{Hom}(P, M)\). The authors define cochain maps \(\beta^*: C^n_L(P,M) \to C^n_H(P,M^e)\), \(n\geq 1\), and \(\alpha^*: C^n_H (P,M) \to C^n_H (P,M^e )\), \( n \geq 1\), where the latter is given in [J. M. Casas and T. Pirashvili, Manuscr. Math. 119, No. 1, 1–15 (2006; Zbl 1100.18005)]. A new cochain complex \(C^*(P,M)\) is defined using the mapping cones on \(\alpha^*\) and \(-\beta^*\). In particular,
\[ C^*(P,M)= \text{cone}(\alpha^*) \coprod_{(i_1,i_2)} \text{cone}(-\beta^*), \]
where \(i_1\) and \(i_2\) are inclusions
\[ \text{cone}(\alpha^*) @>{i_1}>> \overline{C}{}^{*-1}_H (P,M^e) @>{i_2}>> \text{cone}(-\beta^*). \]
By definition the NLP-cohomology of \(P\) with coefficients in \(M\), \(H^*_{\text{NLP}}(P,M)\), is the homology of the cochain complex \(C^*_{\text{NLP}}(P, \, M)\), where
\[ \begin{aligned} C^0_{\text{NLP}}(P,M) &= 0, \\ C^1_{\text{NLP}}(P,M) &= \operatorname{Hom}(P, M), \\ C^n_{\text{NLP}}(P,M) &= C^n(P,M), \qquad n \geq 2 . \end{aligned} \]
It is proved is that if \(P\) is a free NLP-algebra, then \(H^n_{\text{NLP}} (P,{}\_ ) = 0 \) for \(n > 2\).


17A32 Leibniz algebras
17B63 Poisson algebras
18D50 Operads (MSC2010)
Full Text: DOI


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