Related papers: A gentle introduction to Drinfel'd associators
We investigate the Drinfel'd doubles $D(\Lambda_{n,d})$ of a certain family of Hopf algebras. We determine their simple modules and their indecomposable projective modules, and we obtain a presentation by quiver and relations of these…
In our paper [Markl, Shnider: Drinfel'd Algebra Deformations and the Associahedra, IMRN 1994, no. 4, 169-176] we announced a construction of a cohomology controlling deformations of quasi-coassociative (or Drinfel'd) bialgebras. The full…
The connection between simple Lie algebras and their Yangian algebras has a long history. In this work, we construct finite-dimensional representations of Yangian algebras $\mathsf{Y}(\mathfrak{sl}_{n})$ using the quiver approach. Starting…
The Reynolds Transport Theorem, colloquially known as 'differentiation under the integral sign', is a central tool of applied mathematics, finding application in a variety of disciplines such as fluid dynamics, quantum mechanics, and…
We present new min-max relations in digraphs between the number of paths satisfying certain conditions and the order of the corresponding cuts. We define these objects in order to capture, in the context of solving the half-integral linkage…
We introduce and study certain deformations of Drinfeld quasi-modular forms by using rigid analytic trivialisations of corresponding Anderson's t-motives. We show that a sub-algebra of these deformations has a natural graduation by the…
Drinfel'd double of Lie bialgebroids plays an important role in T-duality of string theories. In the presence of $H$ and $R$ fluxes, Lie bialgebroids should be extended to proto Lie bialgebroids. For both cases, the pair is given by two…
For a diagram automorphism of an affine Kac-Moody algebra such that the folded diagram is still an affine Dynkin diagram, we show that the associated Drinfeld-Sokolov hierarchy also admits an induced automorphism. Then we show how to obtain…
Kupavskii, Volostnov, and Yarovikov have recently shown that any set of $n$ points in general position in the plane has at least as many (partial) triangulations as the convex $n$-gon. We generalize this in two directions: we show that…
We present a short proof of the Alexandrov-Fenchel inequalities for mixed volumes of convex bodies.
This two-page note gives a non-computational derivation of the dual Steenrod algebra as the automorphisms of the formal additive group. Instead of relying on computational tools like spectral sequences and Steenrod operations, the argument…
A Drinfel'd algebra gives the systematic construction of generalized parallelizable spaces and this allows us to study an extended T-duality, known as the Poisson-Lie T-duality. Recently, in order to find a generalized U-duality, an…
We provide a unifying interpretation of various optimal transport problems as a minimisation of a linear functional over the set of all Choquet representations of a given pair of probability measures ordered with respect to a certain convex…
The famous Drinfeld-Kohno theorem for simple Lie algebras states that the monodromy representation of the Knizhnik-Zamolodchikov equations for these Lie algebras expresses explicitly via R-matrices of the corresponding Drinfeld-Jimbo…
We express parallel transport for several common matrix Lie groups with a family of pseudo-Riemannian metrics in terms of matrix exponential and exponential actions. The metrics are constructed from a deformation of a bi-invariant metric…
We consider the bi-Hamiltonian representation of the two-component coupled KdV equations discovered by Drinfel'd and Sokolov and rediscovered by Sakovich and Foursov. Connection of this equation with the supersymmetric…
The (parallel linear) transports in tensor spaces generated by derivations of the tensor algebra along paths are axiomatically described. Certain their properties are investigated. Transports along paths defined by derivations of the tensor…
Quasi-Lie bialgebras are natural extensions of Lie-bialgebras, where the cobracket satisfies the co-Jacobi relation up to some natural obstruction controlled by a skew-symmetric 3-tensor $\phi$. This structure was introduced by Drinfeld…
We revisit the derivation of Knizhnik-Zamolodchikov equations in the case of nonsemisimple categories of modules of a superalgebra in the case of the generic affne level and representations parameters. A proof of existence of asymptotic…
We consider the WDVV associativity equations in the four dimensional case. These nonlinear equations of third order can be written as a pair of six component commuting two-dimensional non-diagonalizable hydrodynamic type systems. We prove…