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1. CJM 2017 (vol 69 pp. 992)

 Classification of Regular Parametrized One-relation Operads Jean-Louis Loday introduced a class of symmetric operads generated by one bilinear operation subject to one relation making each left-normed product of three elements equal to a linear combination of right-normed products: $(a_1a_2)a_3=\sum_{\sigma\in S_3}x_\sigma\, a_{\sigma(1)}(a_{\sigma(2)}a_{\sigma(3)})\ ;$ such an operad is called a parametrized one-relation operad. For a particular choice of parameters $\{x_\sigma\}$, this operad is said to be regular if each of its components is the regular representation of the symmetric group; equivalently, the corresponding free algebra on a vector space $V$ is, as a graded vector space, isomorphic to the tensor algebra of $V$. We classify, over an algebraically closed field of characteristic zero, all regular parametrized one-relation operads. In fact, we prove that each such operad is isomorphic to one of the following five operads: the left-nilpotent operad defined by the relation $((a_1a_2)a_3)=0$, the associative operad, the Leibniz operad, the dual Leibniz (Zinbiel) operad, and the Poisson operad. Our computational methods combine linear algebra over polynomial rings, representation theory of the symmetric group, and GrÃ¶bner bases for determinantal ideals and their radicals. Keywords:parametrized one-relation algebra, algebraic operad, Koszul duality, representation theory of the symmetric group, determinantal ideal, GrÃ¶bner basisCategories:18D50, 13B25, 13P10, 13P15, 15A54, 17-04, , , , , 17A30, 17A50, 20C30, 68W30
 Twisted Vertex Operators and Unitary Lie Algebras A representation of the central extension of the unitary Lie algebra coordinated with a skew Laurent polynomial ring is constructed using vertex operators over an integral $\mathbb Z_2$-lattice. The irreducible decomposition of the representation is explicitly computed and described. As a by-product, some fundamental representations of affine Kac-Moody Lie algebra of type $A_n^{(2)}$ are recovered by the new method. Keywords:Lie algebra, vertex operator, representation theoryCategories:17B60, 17B69