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Let $m$, $n$ be two positive integers, $\Bbbk$ be an algebraically closed field with char($\Bbbk)\nmid mn$. Radford constructed an $mn^{2}$-dimensional Hopf algebra $R_{mn}(q)$ such that its Jacobson radical is not a Hopf ideal. We show that the Drinfeld double $D(R_{mn}(q))$ of Radford Hopf algebra $R_{mn}(q)$ has ribbon elements if and only if $n$ is odd. Moreover, if $m$ is even and $n$ is odd, then $D(R_{mn}(q))$ has two ribbon elements, if both $m$ and $n$ are odd, then $D(R_{mn}(q))$ has only one ribbon element. Finally, we compute explicitly all ribbon elements of $D(R_{mn}(q))$.

Traffic prediction has long been a focal and pivotal area in research, witnessing both significant strides from city-level to road-level predictions in recent years. With the advancement of Vehicle-to-Everything (V2X) technologies, autonomous driving, and large-scale models in the traffic domain, lane-level traffic prediction has emerged as an indispensable direction. However, further progress in this field is hindered by the absence of comprehensive and unified evaluation standards, coupled with limited public availability of data and code. This paper extensively analyzes and categorizes existing research in lane-level traffic prediction, establishes a unified spatial topology structure and prediction tasks, and introduces a simple baseline model, GraphMLP, based on graph structure and MLP networks. We have replicated codes not publicly available in existing studies and, based on this, thoroughly and fairly assessed various models in terms of effectiveness, efficiency, and applicability, providing insights for practical applications. Additionally, we have released three new datasets and corresponding codes to accelerate progress in this field, all of which can be found on https://github.com/ShuhaoLii/TITS24LaneLevel-Traffic-Benchmark.

There exist bipartite entangled states whose violations of Clauser-Horne-Shimony-Holt (CHSH) Bell inequality can be observed by a single Alice and arbitrarily many sequential Bobs [Phys. Rev. Lett. 125, 090401 (2020)]. Here we consider its analogues for tripartite systems: a tripartite entangled state is shared among Alice, Bob and multiple Charlies. The first Charlie measures his qubit and then passes his qubit to the next Charlie who measures again with other measurements and so on. The goal is to maximize the number of Charlies that can observe some kind of nonlocality with the single Alice and Bob. It has been shown that at most two Charlies could share genuine nonlocality of the Greenberger-Horne-Zeilinger (GHZ) state via the violation of Svetlichny inequality with Alice and Bob [Quantum Inf. Process. 18, 42 (2019) and Phys. Rev. A 103, 032216 (2021)]. In this work, we show that arbitrarily many Charlies can have standard nonlocality (via violations of Mermin inequality) and some other kind of genuine nonlocality (which is known as genuinely nonsignal nonlocality) with the single Alice and single Bob., Comment: 12 pages, 2figure

Let $H$ be a semisimple Hopf algebra over an algebraically closed field $\mathbbm{k}$ of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$ and $p\nmid 2\dim_{\mathbbm{k}}(H)$. In this paper, we consider the smash product semisimple Hopf algebra $H\#\mathbbm{k}G$, where $G$ is a cyclic group of order $n:=2\dim_{\mathbbm{k}}(H)$. Using irreducible representations of $H$ and those of $\mathbbm{k}G$, we determine all non-isomorphic irreducible representations of $H\#\mathbbm{k}G$. There is a close relationship between the Grothendieck algebra $(G_0(H\#\mathbbm{k}G)\otimes_{\mathbb{Z}}\mathbbm{k},*)$ of $H\#\mathbbm{k}G$ and the Grothendieck algebra $(G_0(H)\otimes_{\mathbb{Z}}\mathbbm{k},*)$ of $H$. To establish this connection, we endow with a new multiplication operator $\star$ on $G_0(H)\otimes_{\mathbb{Z}}\mathbbm{k}$ and show that the Grothendieck algebra $(G_0(H\#\mathbbm{k}G)\otimes_{\mathbb{Z}}\mathbbm{k},\ast)$ is isomorphic to the direct sum of $(G_0(H)\otimes_{\mathbb{Z}}\mathbbm{k},*)^{\oplus\frac{n}{2}}$ and $(G_0(H)\otimes_{\mathbb{Z}}\mathbbm{k},\star)^{\oplus\frac{n}{2}}$., Comment: 14 pages. Any comments and suggestions will be appreciated

For a finite-dimensional Hopf algebra $A$ with a nonzero left integral $\Lambda$, we investigate a relationship between $P_n(\Lambda)$ and $P_n^J(\Lambda)$, where $P_n$ and $P_n^J$ are respectively the $n$-th Sweedler power maps of $A$ and the twisted Hopf algebra $A^J$. We use this relation to give several invariants of the representation category Rep$(A)$ considered as a tensor category. As applications, we distinguish the representation categories of 12-dimensional pointed nonsemisimple Hopf algebras. Also, these invariants are sufficient to distinguish the representation categories Rep$(K_8)$, Rep$(\kk Q_8)$ and Rep$(\kk D_4)$, although they have been completely distinguished by their Frobenius-Schur indicators. We further reveal a relationship between the right integrals $\lambda$ in $A^*$ and $\lambda^J$ in $(A^J)^*$. This can be used to give a uniform proof of the remarkable result which says that the $n$-th indicator $\nu_n(A)$ is a gauge invariant of $A$ for any $n\in \mathbb{Z}$. We also use the expression for $\lambda^J$ to give an alternative proof of the known result that the Killing form of the Hopf algebra $A$ is invariant under twisting. As a result, the dimension of the Killing radical of $A$ is a gauge invariant of $A$., Comment: 19 pages,comments are welcome

Let $H$ be a semisimple Hopf algebra over an algebraically closed field $\mathbbm{k}$ of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$. We show that the antipode $S$ of $H$ satisfies the equality $S^2(h)=\mathbf{u}h\mathbf{u}^{-1}$, where $h\in H$, $\mathbf{u}=S(\Lambda_{(2)})\Lambda_{(1)}$ and $\Lambda$ is a nonzero integral of $H$. The formula of $S^2$ enables us to define higher Frobenius-Schur indicators for the Hopf algebra $H$. This generalizes the notions of higher Frobenius-Schur indicators from the case of characteristic 0 to the case of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$. These indicators defined here share some properties with the ones defined over a field of characteristic 0. Especially, all these indicators are gauge invariants for the tensor category Rep$(H)$ of finite dimensional representations of $H$.

Let $\frak{g}$ be a finite dimensional simple complex Lie algebra and $U=U_q(\frak{g})$ the quantized enveloping algebra (in the sense of Jantzen) with $q$ being generic. In this paper, we show that the center $Z(U_q(\frak{g}))$ of the quantum group $U_q(\frak{g})$ is isomorphic to a monoid algebra, and that $Z(U_q(\frak{g}))$ is a polynomial algebra if and only if $\frak{g}$ is of type $A_1, B_n, C_n, D_{2k+2}, E_7, E_8, F_4$ or $G_2.$ Moreover, in case $\frak{g}$ is of type $D_{n}$ with $n$ odd, then $Z(U_q(\frak{g}))$ is isomorphic to a quotient algebra of a polynomial algebra in $n+1$ variables with one relation; in case $\frak{g}$ is of type $E_6$, then $Z(U_q(\frak{g}))$ is isomorphic to a quotient algebra of a polynomial algebra in fourteen variables with eight relations; in case $\frak{g}$ is of type $A_{n}$, then $Z(U_q(\frak{g}))$ is isomorphic to a quotient algebra of a polynomial algebra described by $n$-sequences., Comment: Section 4 is added

We study integral almost square-free modular categories; i.e., integral modular categories of Frobenius-Perron dimension $p^nm$, where $p$ is a prime number, $m$ is a square-free natural number and ${\rm gcd}(p,m)=1$. We prove that if $n\leq 5$ or $m$ is prime with $m

In this paper, we continue our study of the Green rings of finite dimensional pointed Hopf algebras of rank one initiated in \cite{WLZ}, but focus on those Hopf algebras of non-nilpotent type. Let $H$ be a finite dimensional pointed rank one Hopf algebra of non-nilpotent type. We first determine all non-isomorphic indecomposable $H$-modules and describe the Clebsch-Gordan formulas for them. We then study the structures of both the Green ring $r(H)$ and the Grothendieck ring $G_0(H)$ of $H$ and establish the precise relation between the two rings. We use the Cartan map of $H$ to study the Jacobson radical and the idempotents of $r(H)$. It turns out that the Jacobson radical of $r(H)$ is exactly the kernel of the Cartan map, a principal ideal of $r(H)$, and $r(H)$ has no non-trivial idempotents. Besides, we show that the stable Green ring of $H$ is a transitive fusion ring. This enables us to calculate Frobenius-Perron dimensions of objects of the stable category of $H$. Finally, as an example, we present both the Green ring and the Grothendieck ring of the Radford Hopf algebra., Comment: 29 pages, no figures

Let $H$ be a finite dimensional pointed rank one Hopf algebra of nilpotent type. We first determine all finite dimensional indecomposable $H$-modules up to isomorphism, and then establish the Clebsch-Gordan formulas for the decompositions of the tensor products of indecomposable $H$-modules by virtue of almost split sequences. The Green ring $r(H)$ of $H$ will be presented in terms of generators and relations. It turns out that the Green ring $r(H)$ is commutative and is generated by one variable over the Grothendieck ring $G_0(H)$ of $H$ modulo one relation. Moreover, $r(H)$ is Frobenius and symmetric with dual bases associated to almost split sequences, and its Jacobson radical is a principal ideal. Finally, we show that the stable Green ring, the Green ring of the stable module category, is isomorphic to the quotient ring of $r(H)$ modulo the principal ideal generated by the projective cover of the trivial module. It turns out that the complexified stable Green algebra is a group-like algebra and hence a bi-Frobenius algebra., Comment: small modified

In this paper, we investigate the Green ring $r(H_{n,d})$ of the generalized Taft algebra $H_{n,d}$, extending the results of Chen, Van Oystaeyen and Zhang in \cite{Coz}. We shall determine all nilpotent elements of the Green ring $r(H_{n,d})$. It turns out that each nilpotent element in $r(H_{n,d})$ can be written as a sum of indecomposable projective representations. The Jacobson radical $J(r(H_{n,d}))$ of $r(H_{n,d})$ is generated by one element, and its rank is $n-n/d$. Moreover, we will present all the finite dimensional indecomposable representations over the complexified Green ring $R(H_{n,d})$ of $H_{n,d}.$ Our analysis is based on the decomposition of the tensor product of indecomposable representations and the observation of the solutions for the system of equations associated to the generating relations of the Green ring $r(H_{n,d})$., Comment: To appear in Contemporary Math. AMS, Proceeding of Hopf Algebra conference in Almeria, 2011