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Let $X=(x_{ij})_{mn}$ be a quantum matrix with the commutation relations between entries: \begin{alignat*}{2} & x_{ij} x_{il} = q x_{il} x_{ij}, && j < l, \\ & x_{ij} x_{kj} = q x_{kj} x_{ij}, && …

I am trying to figure out what should the following definition correspond to in the setting of quantum affine algebra: $$ X \circ Y = Ind_{\beta, \gamma}^{\beta+\gamma} X \boxtimes Y \quad (1) $$ This …

A coproduct $\varphi: \mathbb{C}_q[U] \to \mathbb{C}_q[U] \otimes \mathbb{C}_q[U]$ is given by: $x \mapsto 1 \otimes x + x \otimes 1$, where $x$ is a generator of $\mathbb{C}_q[U]$. There is a coprodu …

Many properties of quantum toroidal algebras are similar to quantum affine algebras. Every simple module of a quantum affine algebra of rank $n$ corresponds to an $n$-tuple of Drinfeld polynomials. …

Let $H$ be a Hopf algebra and $A$ an algebra. Let $H^*$ be the dual Hopf algebra of $H$. Then by Proposition 1.6.11 in the book Foundations of Quantum Group Theory by Shahn Majid, $A$ is a right $H$-c …

Lie $g$ be a finite dimensional complex simple Lie algebra and $U_q(g)$ the corresponding quantum group, where $q$ is not a root of unity. Every simple finite dimensional $g$-module is of the form $V( …

Are there some references of the proof of the following result? Let $(V, c)$ be a braided vector space over a field $k$ with a basis $x_1, \ldots, x_n$, where $c$ is a diagonal braiding given by \beg …

Let $H$ be a braided Hopf algebra. The multiplication on $H \otimes H$ is defined by $(a \otimes b)(c \otimes d) = a \Psi(b \otimes c) d$, $a,b,c,d \in H$. Let $H = T(V)$. There is a algebra map $\D …