The singularity theory is an important research field in differential topology. It has important applications in judging and determining the number of solutions of differential equations, giving the classification and counterexamples of the differential structure of differential manifolds, and describing the geometric properties of specific positions on differential manifolds.

It is well known that, the Laplace-Beltrami operator is an extremely important operator that acts on ${\mathbb{C}}^{\infty }$ functions on a Riemannian manifold. Over past several decades, research on the spectrum of the Laplace-Beltrami operator has always become a core issue in the study of geometry. For example, the geometry of closed minimal submanifolds in the unit sphere is closely related to the eigenvalue problem. In order to classify the isolated hypersurface singularities in complex geometry, Yau conjectured in [1] that the Laplace eigenfunctions ${\varphi }_{\lambda }$: satisfy where M is an arbitrary n-dimensional ${\mathbb{C}}^{\infty }$-smooth closed Riemannian manifold (compact and without boundary), and the symbol ${\mathcal{H}}^k$ denotes the k dimensional Hausdorff measure. Here $c,C$ depend only on the Riemannian metric on M and are independent of the eigenvalue $\lambda$.

Aiming at Yau’s conjecture, in [2], it was pointed out that the moduli algebra of an analytic function f completely determines theisolated hypersurface singularities’ complex structure. Therefore, this classification problem can be translated into classifying the moduli algebras, up to isomorphism. For further study of Yau’s conjecture, one may refer [3,4,5,6].

Let k be an algebraically closed field, and G is a simply connected algebraic group over it. Suppose that M is a finite dimensional rational G-module. Take a basis ${\left\{e_i\right\}}_{i=1}^n$ of M, and let ${\left\{x_i\right\}}_{i=1}^n\subseteq M^{\ast }$ be its dual basis. Denote by $A=S^{•}{M}^{\ast }\cong k[x_1,\dots ,x_n]$ the polynomial function ring on M (with the usual G-action), and let ${\partial }_i:=\partial /\partial x_i$ be the usual differential operators on A. If we let $A_d$ be the set of homogeneous polynomials of degree d, then for each $v\in A_d$, it is known that the Jacobian $J\left(v\right)$ of f is a subspace of $A_{d-1}$ spanned by ${\left\{{\partial }_i\left(f\right)\right\}}_{i=1}^n$. Under the notions as above, Yau conjectured that if $G=S{L}_2\left(\mathbb{C}\right)$ and $J\left(v\right)$ is G-invariant, then the highest weights of $J\left(v\right)$ is a subset of the highest weights of $A_1$ ($\cong M^{\ast }$). In [6], Xi constructed the following homomorphism of G-modules where G is an arbitrary simply connected algebraic group. Combining this with Theorem 13 [4], each invariant Jacobian $J\left(v\right)$ is a quotient of $ku\otimes M$ ($\cong M$) if the characteristic of k equals 0. As a consequence, Yau’s conjecture (see [7]) is particularly true for simply connected complex algebraic groups.

Suppose that $m\left(x\right)$ is the minimal polynomial of ${\mathsf{\Phi }}_{M^{\ast },M^{\ast }}\in End(M^{\ast }\otimes M^{\ast })$. Fixing an eigenvalue $\lambda$ of ${\mathsf{\Phi }}_{M^{\ast },M^{\ast }}$, let $m_{\lambda }\left(x\right):=m\left(x\right)/(x-\lambda )$ and $A(M,\lambda )=T\left(M^{\ast }\right)/I(m_{\lambda }\left(x\right),M^{\ast })$. In [8], using braided derivations, Chen generalized the setting in [6] as a representation of a quasi-triangular Hopf algebra H, and showed that is a homomorphism of left H modules (for details, see ([8], Theorem 3.5)).

It is known that the category of Yetter–Drinfel’d modules is an important category in the theory of Hopf algebra. Under some favourable conditions (e.g., H is a Hopf algebra with a bijective antipode), the category of Yetter–Drinfel’d modules is indeed braided monoidal through Drinfel’d double construction (see [9]). In [10], it is pointed out that symmetric Yetter–Drinfel’d categories are trivial, i.e., if H is a Hopf algebra, such that the canonical braiding of the category of Yetter–Drinfel’d modules is a symmetry, then $H=k$ in the field. Via braiding structures, the notion of the Yetter–Drinfel’d module plays an important role in the relations between knot theory and quantum groups.

It is known that the category of H modules is a spacial case in Yetter–Drinfel’d modules. So, it is natural but meaningful to ask whether the map $\varphi$, defined above, is a morphism of Yetter–Drinfel’d modules or not when M is a Yetter–Drinfel’d module. This is where the motivation for our paper comes. In this case, the results in [8] also hold in the coquasi-triangular Hopf algebra.

The paper is organized as follows. In Section 2, we mainly present some useful definitions about Yetter–Drinfel’d modules. In Section 3, we generalize the homomorphisms of the module over the groups and Lie algebras established in [6] as being morphisms in the category of (non-symmetric) Yetter–Drinfel’d modules. In Section 4, we provide a brief conclusion in this paper.

Let k be a ground field. All algebra, linear spaces, etc., will be over k; unadorned ⊗ means ${\otimes }_k$. Unless otherwise stated, H will denote a Hopf algebra with comultiplication $\mathsf{\Delta }$, counit $\epsilon$, and bijective antipode S. Then, the opposite $H^{op}$ is again a Hopf algebra with antipode $S^{-1}$. We will use the version of Sweedler’s sigma notation: $\mathsf{\Delta }\left(h\right)=h_1\otimes h_2$ for all $h\in H$. For unexplained concepts and notations about Hopf algebras, we refer to [11,12]. If M is a vector space, a left H-module (right H comodule) structure on M will be usually denoted by ${\psi }_M:H\otimes M\to M,h\otimes m\mapsto h·m$ (${\rho }_M:M\to M\otimes H,m\mapsto m_{\left(0\right)}\otimes m_{\left(1\right)}$, respectively).

We denote the category of Yetter–Drinfel’d modules by ${}_H\mathcal{Y}\mathcal{D}^H$, with the morphisms being H linear and H colinear maps. It is known that $({}_H\mathcal{Y}\mathcal{D}^H,\tilde{\otimes },k)$ forms a braided tensor category as follows:

(i) For any $M,N\in {}_H\mathcal{Y}\mathcal{D}^H$, we have $M\tilde{\otimes }N\in {}_H\mathcal{Y}\mathcal{D}^H$, where $M\tilde{\otimes }N=M\otimes N$ are the spaces, and the Yetter–Drinfel’d module structure is provided by

(ii) The braiding is defined by with inverse ${\mathsf{\Phi }}_{M,N}^{-1}\left(n\tilde{\otimes }m\right)=S\left(n_{\left(1\right)}\right)·m\tilde{\otimes }{n}_{\left(0\right)}$.

Let $M\in {}_H\mathcal{Y}\mathcal{D}^H$ be of a finite dimension. We denote the dual vector space $M^{\ast }$ by $M^{\diamond }$ as being endowed with the following Yetter–Drinfel’d module structure: for all $h\in H,f\in M^{\diamond },m\in M$.

In this paper, we mainly use the theory of the braided monoidal category to generalize the module homomorphisms for groups and Lie algebras for morphisms in the category of (non-symmetric) Yetter–Drinfel’d modules.

We then generalize the morphisms in an arbitrary category of the (non-symmetric) braided monoidal category.

Let M and N be right H comodules. We take $f\in Hom(M,N)$ and consider $\rho \left(f\right)\in Hom(M,N\otimes H)$ provided by As k is a field, $Hom(M,N)\otimes H\subseteq Hom(M,N\otimes H)$. Define If a morphism f belongs in $HOM(M,N)$, then f is said to be rational. If the Hopf algebra H is finite dimensional, then we know that all morphisms are rational. Meanwhile, it is known from [16] that $HOM(M,N)$ is a right H comodule, and that it is actually the largest H comodule contained in the pace $Hom(M,N)$. Also, recall that $\rho \left(f\right)=f_{\left(0\right)}\otimes f_{\left(1\right)}$ if and only if for all $m\in M$.

Let $M\in {}_H\mathcal{Y}\mathcal{D}^H$ be of finite dimension with a basis ${\left\{e_i\right\}}_{i=1}^n$ and let ${\left\{x_i\right\}}_{i=1}^n\subseteq M^{\ast }$ be the dual basis of M, such that $x_i\left(e_i\right)={\delta }_{i,j}$. Let $m\left(x\right)$ be the minimal polynomial of ${\mathsf{\Phi }}_{M^{\diamond },M^{\diamond }}\in End\left(M^{\diamond }\tilde{\otimes }{M}^{\diamond }\right)$. For each eigenvalue $\lambda$ of ${\mathsf{\Phi }}_{M^{\diamond },M^{\diamond }}$, set $m_{\lambda }\left(x\right):=m\left(x\right)/(x-\lambda )$ and $A(M,\lambda )=T\left(M^{\diamond }\right)/I(m_{\lambda }\left(x\right),M^{\diamond })$. From Lemma 1 it is known that $A(M,\lambda )$ is a Yetter–Drinfel’d module algebra. We define partial differential operators of ${\partial }_{\lambda ,i}(1\le i\le n)$ on $A(M,\lambda )$, and it can be seen that the algebra $A(M,\lambda )$, together with ${\partial }_{\lambda ,i}$, plays the role of the usual polynomial ring A with G-action and the usual ${\partial }_i$’s, comparable to what we can see from [8].

Set $E:=END\left(T\left(M^{\diamond }\right)\right)$. Then, it follows from Proposition 1 that E is a Yetter–Drinfel’d module algebra. Let $\psi :E\tilde{\otimes }T\left(M^{\diamond }\right)\to T\left(M^{\diamond }\right),f\tilde{\otimes }\mathbf{u}\mapsto f\left(\mathbf{u}\right)$. We first define the partial differential operators ${\partial }_{\lambda ,i}(1\le i\le n)$ on $T\left(M^{\diamond }\right)$ for each eigenvalue $\lambda$ of ${\mathsf{\Phi }}_{M^{\diamond },M^{\diamond }}$. Their action on $T^1\left(M^{\diamond }\right)=M^{\diamond }$ is provided by ${\partial }_{\lambda ,i}\left(x_j\right)={\delta }_{i,j}$, while the action on $T^d\left(M^{\diamond }\right)$ for $d>1$ is defined as follows: for all $x_{i_1}\tilde{\otimes }{x}_{i_2}\cdots \tilde{\otimes }{x}_{i_d}\in T^d\left(M^{\diamond }\right)$, we set where ${\psi }_{k,k+1}:=i{d}_{{\left(M^{\diamond }\right)}^{\tilde{\otimes }(k-1)}}\tilde{\otimes }\mathsf{\Phi }\tilde{\otimes }i{d}_{{\left(M^{\diamond }\right)}^{\tilde{\otimes }(d-k)}}$.

For each $\mathcal{A}\in End\left(M^{\diamond }\tilde{\otimes }{M}^{\diamond }\right)$, denote $\mathcal{A}\left(x_i\tilde{\otimes }{x}_j\right)={\displaystyle \sum _{1\le s,t\le n}}{\left[\mathcal{A}\right]}_{s,t}^{i,j}{x}_s\tilde{\otimes }{x}_t$.

The following proposition, which enables us to identify ${\partial }_{\lambda ,i}$ with $e_i$, is key in the discussion below.