包含Poisson的单运算代数及其它李容许代数
Algebras with one operation including Poisson and other Lie-admissible algebras
摘要：
我们研究一个运算的代数。我们研究了这些代数何时构成一个幺半范畴，并分析了相应运算的Koszul性和循环性。我们还介绍了一种新的对称运算，二面角，负责二面角上同调的存在。主要的技巧，我们称之为极化，将表示一个代数与一个操作没有任何特定的对称性作为代数与一个交换和一个反交换操作。我们将试图说服读者，这种视角的改变有时可能会带来新的见解和结果。这一观点被Livernet和Loday用来引入一个单参数的运算族，其在0处的特化是泊松代数的运算，而在泛点处它等于结合代数的运算。我们研究这个家庭，并解释它如何可以用来解释在一个整洁和优雅的方式变形量子化。
英文摘要：
We investigate algebras with one operation. We study when these algebras form a monoidal category and analyze Koszulness and cyclicity of the corresponding operads. We also introduce a new kind of symmetry for operads, the dihedrality, responsible for the existence of the dihedral cohomology. The main trick, which we call the polarization, will be to represent an algebra with one operation without any specific symmetry as an algebra with one commutative and one anticommutative operation. We will try to convince the reader that this change of perspective might sometimes lead to new insights and results. This point of view was used by Livernet and Loday to introduce a one-parametric family of operads whose specialization at 0 is the operad for Poisson algebras, while at a generic point it equals the operad for associative algebras. We study this family and explain how it can be used to interpret the deformation quantization in a neat and elegant way.
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