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In this paper, we first construct the controlling algebras of embedding tensors and Lie-Leibniz triples, which turn out to be a graded Lie algebra and an $L_infty$-algebra respectively. Then we introduce representations and cohomologies of embedding tensors and Lie-Leibniz triples, and show that there is a long exact sequence connecting various cohomologies. As applications, we classify infinitesimal deformations and central extensions using the second cohomology groups. Finally, we introduce the notion of a homotopy embedding tensor which will induce a Leibniz$_infty$-algebra. We realize Kotov and Strobls construction of an $L_infty$-algebra from an embedding tensor, to a functor from the category of homotopy embedding tensors to that of Leibniz$_infty$-algebras, and a functor further to that of $L_infty$-algebras.
We find a formula to compute the number of the generators, which generate the $n$-filtered space of Hopf algebra of rooted trees, i.e. the number of equivalent classes of rooted trees with weight $n$. Applying Hopf algebra of rooted trees, we show th
The main purpose of this work is to develop the basic notions of the Lie theory for commutative algebras. We introduce a class of $mathbbZ_2$-graded commutative but not associative algebras that we call ``Lie antialgebras. These algebras are closely
We study solutions of the Bethe ansatz equations associated to the orthosymplectic Lie superalgebras $mathfrak{osp}_{2m+1|2n}$ and $mathfrak{osp}_{2m|2n}$. Given a solution, we define a reproduction procedure and use it to construct a family of new s
Rota-Baxter algebras were introduced to solve some analytic and combinatorial problems and have appeared in many fields in mathematics and mathematical physics. Rota-Baxter algebras provide a construction of pre-Lie algebras from associative algebras
The orbits of Weyl groups W(A(n)) of simple A(n) type Lie algebras are reduced to the union of orbits of the Weyl groups of maximal reductive subalgebras of A(n). Matrices transforming points of the orbits of W(An) into points of subalgebra orbits ar