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In this paper we show that a certain algebra being a comodule algebra over the Taft Hopf algebra of dimension $n^2$ is equivalent to a set of identities related to the $q$-binomial coefficient, when $q$ is a primitive $n^{th}$ root of 1. We then give a direct combinatorial proof of these identities.
Let A denote the ring of differential operators on the affine line with its two usual generators t and d/dt given degrees +1 and -1 respectively. Let X be the stack having coarse moduli space the affine line Spec k[z] and isotropy groups Z/2 at each
We introduce a factorized difference operator L(u) annihilated by the Frenkel-Reshetikhin screening operator for the quantum affine algebra U_q(C^{(1)}_n). We identify the coefficients of L(u) with the fundamental q-characters, and establish a number
In terms of the $q$-Saalsch{u}tz identity and the Chinese remainder theorem for coprime polynomials, we establish some $q$-supercongruences modulo the third power of a cyclotomic polynomial. In particular, we give a $q$-analogue of a formula due to L
Let $E_n$ be the $n$-th Euler number and $(a)_n=a(a+1)cdots (a+n-1)$ the rising factorial. Let $p>3$ be a prime. In 2012, Sun proved the that $$ sum^{(p-1)/2}_{k=0}(-1)^k(4k+1)frac{(frac{1}{2})_k^3}{k!^3} equiv p(-1)^{(p-1)/2}+p^3E_{p-3} pmod{p^4}, $
This paper is a fundamental study of comodules and contramodules over a comonoid in a closed monoidal category. We study both algebraic and homotopical aspects of them. Algebraically, we enrich the comodule and contramodule categories over the origin