Yang-Baxter (YB) map systems (or set-theoretic analogs of entwining YB structures) are presented. They admit zero curvature representations with spectral parameter depended Lax triples L₁, L₂, L₃ derived from symplectic leaves of 2 × 2 binomial matrices equipped with the Sklyanin bracket. A unique factorization condition of the Lax triple implies a 3-dimensional compatibility property of these maps. In case L₁ = L₂ = L₃ this property yields the set-theoretic quantum Yang-Baxter equation, i.e. the...

By using the interplay between the Eulerian idempotent and the Dynkin idempotent, we construct explicitly a particular symmetric solution $(F,G)$ of the first equation of the Kashiwara-Vergne conjecture$$x+y-log\left({\mathrm{e}}^y{\mathrm{e}}^x\right)=(1-{\mathrm{e}}^{-\mathrm{ad}x})F(x,y)+({\mathrm{e}}^{\mathrm{ad}y}-1)G(x,y).$$Then, we explicit all the solutions of the equation in the completion of the free Lie algebra generated by two indeterminates $x$ and $y$ thanks to the kernel of the Dynkin idempotent.

We define polynomial $H$-identities for comodule algebras over a Hopf algebra $H$ and establish general properties for the corresponding $T$-ideals. In the case $H$ is a Taft algebra or the Hopf algebra $E\left(n\right)$, we exhibit a finite set of polynomial $H$-identities which distinguish the Galois objects over $H$ up to isomorphism.

In analogy with earlier work on the forward-backward case, we consider an explicit construction of the forward-forward double stochastic product integral ${\prod }^{\to \to }(1+dr)$ with generator $dr=\lambda (d{A}^{†}\otimes dA-dA\otimes d{A}^{†})$. The method of construction is to approximate the product integral by a discrete double product ${\prod }_{(j,k)\in {\mathbb{N}}_m\times \mathbb{N}ₙ}^{\to \to }\Gamma \left(R_{m,n}^{(j,k)}\right)=\Gamma \left({\prod }_{(j,k)\in {\mathbb{N}}_m\times \mathbb{N}ₙ}^{\to \to }\left(R_{m,n}^{(j,k)}\right)\right)$ of second quantised rotations $R_{m,n}^{(j,k)}$ in different planes using the embedding of ${ℂ}^m\oplus ℂⁿ$ into L²(ℝ) ⊕ L²(ℝ) in which the standard orthonormal bases of ${ℂ}^m$ and ℂⁿ are mapped to the orthonormal sets consisting of normalised indicator functions of...