Leibniz \(n\)-algebras. (Q2769475)

The paper provides foundational material for the construction of free Leibniz \(n\)-algebras and an interpretation of Leibniz \(n\)-algebra cohomology in terms of Quillen cohomology. Motivated by generalizations of Lie algebra structures to settings with \(n\)-ary operations, the authors define a Leibniz \(n\)-algebra to be a vector space \(\mathcal{L}\) equipped with an \(n\)-linear operation NEWLINE\[NEWLINE [ \, - \, , \, - \, , \, \ldots \, , \, - \, ] : {\mathcal{L}}^{\otimes n} \to \mathcal{L} NEWLINE\]NEWLINE such that NEWLINE\[NEWLINE[[x_1, x_2,\ldots, x_n], y_1, y_2, \ldots, y_{n-1}]= \sum_{i=1}^n [x_1,\ldots, x_{i-1}, [x_i, y_1, y_2, \ldots , y_{n-1}], x_{i+1}, \ldots, x_n].NEWLINE\]NEWLINE For \(n=2\), this reduces to J.-L. Loday's definition of a Leibniz algebra [\textit{J.-L. Loday} and \textit{T. Pirashvili}, Math. Ann. 296, 139--158 (1993; Zbl 0821.17022)]. A nice construction of the paper is the realization of a free Leibniz 3-algebra in terms of the magma with one generator \(e\). Let \(Y_1 = \{ e \}\), NEWLINE\[NEWLINE Y_m = \coprod_{p+q = m} Y_p \times Y_q, \qquad m \geq 2, \quad p \geq 1, \quad q \geq 1, NEWLINE\]NEWLINE and \(Y = \coprod_{m \geq 1} Y_m\). Then \(Y\) can be identified with the set of planar binary rooted trees, and \(k[Y]\), for a field \(k\), carries the structure of a free Leibniz 3-algebra in terms of the grafting operation of trees. A similar construction is offered for higher Leibniz \(n\)-algebras. Recall that if \(\mathcal{L}\) is a Leibniz \((n+1)\)-algebra, then \(D_n ({\mathcal{L}}) = {\mathcal{L}}^{\otimes n}\) is a Leibniz algebra with bracket NEWLINE\[NEWLINE [ - \, , \, -] : D_n ({\mathcal{L}}) \otimes D_n ({\mathcal{L}}) \to D_n ({\mathcal{L}}) NEWLINE\]NEWLINE given by NEWLINE\[NEWLINE[a_1 \otimes \ldots \otimes a_n , b_1 \otimes \ldots \otimes b_n]= \sum_{i=1}^n a_1 \otimes\ldots\otimes [a_i, b_1, \ldots, b_n] \otimes \ldots \otimes a_n,NEWLINE\]NEWLINE a construction similar to one in [\textit{P. Gautheron}, Lett. Math. Phys. 37, 103--116 (1996; Zbl 0849.70014)]. Proven is that if \(\mathcal{L}\) is a free Leibniz \(n\)-algebra, then \(D_{n-1}({\mathcal{L}})\) is a free Leibniz algebra. This result is then used to prove that the Quillen cohomology, when suitably applied to \(\mathcal{L}\), is isomorphic to the Leibniz \(n\)-algebra cohomology of \(\mathcal{L}\), where both cohomology groups are taken with coefficients in a representation of \(\mathcal{L}\).