In the present study, we investigate a new type of warped products on manifolds with indefinite metrics, namely, warped product lightlike submanifolds of indefinite Kaehler manifolds with a slant factor. First, we show that indefinite Kaehler manifolds do not admit any proper warped product semi-slant lightlike submanifolds of the type $N_{T}\times_{\lambda}N_{\theta}$, $N_{\theta}\times_{\lambda}N_{T}$, $N_{\perp}\times_{\lambda}N_{\theta}$ and $N_{\theta}\times_{\lambda}N_{\perp}$, where $N_{T}$ is a holomorphic submanifold, $N_{\perp}$ is a totally real submanifold and $N_{\theta}$ is a proper slant submanifold. Then, we study warped product semi-slant lightlike submanifolds of the type $B\times_{\lambda}N_{\theta}$, where $B = N_{T}\times N_{\perp}$, of an indefinite Kaehler manifold. Following this, we give one non-trivial example for this kind of warped products of indefinite Kaehler manifolds. Then, we establish a geometric estimate for the squared norm of the second fundamental form involving the Hessian of warping function $\lambda$ for this class of warped products. Finally, we present a sharp geometric inequality for the squared norm of second fundamental form of warped product semi-slant lightlike submanifolds of the type $B\times_{\lambda}N_{\theta}$.

The copointed liftings of the Fomin–Kirillov algebra $\mathcal{FK}_3$ over the algebra of functions on the symmetric group $\mathbb{S}_3$ were classified by Andruskiewitsch and the author. We demonstrate here that those associated to a generic parameter are Morita equivalent to the regular blocks of well-known Hopf algebras: the Drinfeld doubles of the Taft algebras and the small quantum groups $u_{q}(\mathfrak{sl}_2)$. The indecomposable modules over these were classified independently by Chen, Chari–Premet and Suter. Consequently, we obtain the indecomposable modules over the generic liftings of $\mathcal{FK}_3$. We decompose the tensor products between them into the direct sum of indecomposable modules. We then deduce a presentation by generators and relations of the Green ring.

We obtain a necessary and sufficient condition on the Haar coefficients of a real function $f$ defined on $\mathbb{R}^+$ for the Lipschitz $\alpha$ regularity of $f$ with respect to the ultrametric $\delta(x,y)=\inf \{|I| : x, y\in I; I\in\mathcal{D}\}$, where $\mathcal{D}$ is the family of all dyadic intervals in $\mathbb{R}^+$ and $\alpha$ is positive. Precisely, $f\in \mathrm{Lip}_\delta(\alpha)$ if and only if ${\vert\langle{f}{h^j_k}\rangle\vert}\leq C 2^{-(\alpha + 1/2)j}$ for some constant $C$, every $j\in\mathbb{Z}$ and every $k=0,1,2,\ldots$ Here, as usual, $h^j_k(x)= 2^{j/2}h(2^jx-k)$ and $h(x)=\mathcal{X}_{[0,1/2)}(x)-\mathcal{X}_{[1/2,1)}(x)$.

Flips of triangulations appear in the definition of cluster algebras by Fomin and Zelevinsky. In this article we give an interpretation of mutation in the sense of permutation using triangulations of a convex polygon. We thus establish a link between cluster variables and permutation mutations in the case of cluster algebras of type $\mathbb{A}$.