부산대학교 도서관

A Leibniz $n$-algebra is a vector space equipped with an $n$-linear operation $[-,\dots ,-]$ satisfying the $n$-Leibniz identity $$\multline [[x\sb 1,\dots,x\sb n],y\sb 1,\dots ,y\sb {n-1}]=\\ \sum \sp n\sb {i=1} [x\sb 1,\dots ,x\sb {i-1},[x\sb i,y\sb 1,\dots ,y\sb {n-1}],x\sb {i+1},\dots ,x\sb n];\endmultline$$ this identity means that any ${\rm ad}(y\sb 1,\dots ,y\sb {n-1})\coloneq [-,y\sb 1,\dots ,y\sb {n-1}]$ is a derivation with respect to $[-,\dots,-]$. The case $n=2$ is that of Leibniz algebras, J.-L.\ Loday's ``noncommutative Lie algebras'' [in {\it Cyclic cohomology and noncommutative geometry (Waterloo, ON, 1995)}, 91--102, Amer. Math. Soc., Providence, RI, 1997; MR1478704 (98j:17001)]. The case $n=3$ includes Lie triple systems [W. G. Lister, Trans. Amer. Math. Soc. {\bf 72} (1952), 217--242; MR0045702 (13,619d)]. For $n\geq 3$, skew-symmetric Leibniz $n$-algebras appear in Nambu mechanics [Y.\ L. Daletskiĭ\ and L. A. Takhtajan, Lett. Math. Phys. {\bf 39} (1997), no.~2, 127--141; MR1437747 (98e:17003)]. \par The paper under review defines Leibniz $n$-algebras and gives several examples. The free Leibniz $n$-algebra on a vector space is described. A functor is constructed, from Leibniz $(n+1)$-algebras to Leibniz algebras, which sends an object $L$ to the vector space $L\sp {\otimes n}$ equipped with the Leibniz bracket $$\multline [x\sb 1\otimes \dots \otimes x\sb n,y\sb 1\otimes \dots \otimes y\sb n]\coloneq\\ \sum \sb {i=1}\sp n x\sb 1\otimes \dots \otimes [x\sb i,y\sb 1,\dots ,y\sb n]\otimes \dots \otimes x\sb n.\endmultline$$ The main result of the paper states that this functor sends free objects to free objects. \par The authors also define representations of an $n$-Leibniz algebra $L$, and ``Quillen homology'' of $L$ with coefficients in a representation $M$: $H\sp *\sb {\rm Quillen}(L,M) \coloneq H\sp * ({\rm Der}(P\sb *,M))$, where Der is the module of derivations and $P\sb * \mapsto L$ is a weak equivalence, with $P\sb *$ a simplicial Leibniz $n$-algebra such that any $P\sb k$ is a free Leibniz $n$-algebra. The main theorem above allows them to define a small complex that computes $H\sp *\sb {\rm Quillen}(L,M)$.