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Register a new userDouble cross biproduct and bi-cycle bicrossproduct Lie bialgebras
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Authors
Tao Zhang
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We construct double cross biproduct and bi-cycle bicrossproduct Lie bialgebras from braided Lie bialgebras. The main result generalizes Majids matched pairs of Lie algebras, Drinfelds quantum double, and Masuokas cross product Lie bialgebras.
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We study (quasi-)twilled pre-Lie algebras and the associated $L_infty$-algebras and differential graded Lie algebras. Then we show that certain twisting transformations on (quasi-)twilled pre-Lie algbras can be characterized by the solutions of Maurer-Cartan equations of the associated differential graded Lie algebras ($L_infty$-algebras). Furthermore, we show that $mathcal{O}$-operators and twisted $mathcal{O}$-operators are solutions of the Maurer-Cartan equations. As applications, we study (quasi-)pre-Lie bialgebras using the associated differential graded Lie algebras ($L_infty$-algebras) and the twisting theory of (quasi-)twilled pre-Lie algebras. In particular, we give a construction of quasi-pre-Lie bialgebras using symplectic Lie algebras, which is parallel to that a Cartan $3$-form on a semi-simple Lie algebra gives a quasi-Lie bialgebra.
For any integer $d$ we introduce a prop $RHra_d$ of oriented ribbon hypergraphs (in which edges can connect more than two vertices) and prove that it admits a canonical morphism of props, $$ Holieb_d^diamond longrightarrow RHra_d, $$ $Holieb_d^diamond$ being the (degree shifted) minimal resolution of prop of involutive Lie bialgebras, which is non-trivial on every generator of $Holieb_d^diamond$. We obtain two applications of this general construction. As a first application we show that for any graded vector space $W$ equipped with a family of cyclically (skew)symmetric higher products the associated vector space of cyclic words in elements of $W$ has a combinatorial $Holieb_d^diamond$-structure. As an illustration we construct for each natural number $Ngeq 1$ an explicit combinatorial strongly homotopy involutive Lie bialgebra structure on the vector space of cyclic words in $N$ graded letters which extends the well-known Schedlers necklace Lie bialgebra structure from the formality theory of the Goldman-Turaev Lie bialgebra in genus zero. Second, we introduced new (in general, non-trivial) operations in string topology. Given any closed connected and simply connected manifold $M$ of dimension $geq 4$. We show that the reduced equivariant homology $bar{H}_bullet^{S^1}(LM)$ of the space $LM$ of free loops in $M$ carries a canonical representation of the dg prop $Holieb_{2-n}^diamond$ on $bar{H}_bullet^{S^1}(LM)$ controlled by four ribbon hypergraphs explicitly shown in this paper.
We describe $L_infty$-algebras governing homotopy relative Rota-Baxter Lie algebras and triangular $L_infty$-bialgebras, and establish a map between them. Our formulas are based on a functorial approach to Voronovs higher derived brackets construction which is of independent interest.
The $n$-Lie bialgebras are studied. In Section 2, the $n$-Lie coalgebra with rank $r$ is defined, and the structure of it is discussed. In Section 3, the $n$-Lie bialgebra is introduced. A triple $(L, mu, Delta)$ is an $n$-Lie bialgebra if and only if $Delta$ is a conformal $1$-cocycle on the $n$-Lie algebra $L$ associated to $L$-modules $(L^{otimes n}, rho_s^{mu})$, $1leq sleq n$, and the structure of $n$-Lie bialgebras is investigated by the structural constants. In Section 4, two-dimensional extension of finite dimensional $n$-Lie bialgebras are studied. For an $m$ dimensional $n$-Lie bialgebra $(L, mu, Delta)$, and an $ad_{mu}$-invariant symmetric bilinear form on $L$, the $m+2$ dimensional $(n+1)$-Lie bialgebra is constructed. In the last section, the bialgebra structure on the finite dimensional simple $n$-Lie algebra $A_n$ is discussed. It is proved that only bialgebra structures on the simple $n$-Lie algebra $A_n$ are rank zero, and rank two.
We introduce fusion bialgebras and their duals and systematically study their Fourier analysis. As an application, we discover new efficient analytic obstructions on the unitary categorification of fusion rings. We prove the Hausdorff-Young inequality, uncertainty principles for fusion bialgebras and their duals. We show that the Schur product property, Youngs inequality and the sum-set estimate hold for fusion bialgebras, but not always on their duals. If the fusion ring is the Grothendieck ring of a unitary fusion category, then these inequalities hold on the duals. Therefore, these inequalities are analytic obstructions of categorification. We classify simple integral fusion rings of Frobenius type up to rank 8 and of Frobenius-Perron dimension less than 4080. We find 34 ones, 4 of which are group-like and 28 of which can be eliminated by applying the Schur product property on the dual. In general, these inequalities are obstructions to subfactorize fusion bialgebras.
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