L-quadri-algebras. (Chinese. English summary) Zbl 1499.17003
Summary: Quadri-algebras are a class of remarkable Loday algebras. In this paper, we introduce a notion of L-quadri-algebra with 4 operations satisfying certain generalized left-symmetry, as a Lie algebraic analogue of quadri-algebra such that the commutator of the sum of the 4 operations is a Lie algebra. Any quadri-algebra is an L-quadri-algebra. Moreover, L-quadri-algebras fit into the framework of the relationships between Loday algebras and their Lie algebraic analogues, extending the well known fact that the commutator of an associative algebra is a Lie algebra. We also give the close relationships between L-quadri-algebras and some interesting structures like Rota-Baxter operators, classical Yang-Baxter equation and some bilinear forms satisfying certain conditions.
MSC:
17A30 | Nonassociative algebras satisfying other identities |
17B60 | Lie (super)algebras associated with other structures (associative, Jordan, etc.) |
17B38 | Yang-Baxter equations and Rota-Baxter operators |
16T25 | Yang-Baxter equations |