# Generalized Yang–Baxter Operators for Dieudonné Modules

## Abstract

**:**

## 1. Introduction

## 2. Generalized Yang–Baxter Operators for Dieudonné Modules

**Definition 2.1.**A Dieudonné module ${M}_{*}$ is a graded abelian group together with a degree-preserving ${\mathbb{Z}}_{p}$-action and two homomorphisms $F:{M}_{*}\to {M}_{*}$ and $V:{M}_{*}\to {M}_{*}$, such that:

**Example 2.2.**The ring R (with grading as above defined) is a Dieudonné module, with V and F acting by means of the ring operation.

**Example 2.3.**The polynomial ring ${\mathbb{Z}}_{p}[F,V]$ is a Dieudonné module, with V and F acting by means of the ring operation (again, the grading is the one defined above).

**Example 2.4.**Fix $n\in \mathbb{N}$.

**Theorem 2.5.**([2,3]) The periodically graded Dieudonné module $D({\overline{K\left(n\right)}}_{\phantom{\rule{4pt}{0ex}}*}\left(K(\mathbb{Z}/\left({p}^{j}\right),q)\right)$ is a free $\mathbb{Z}/\left({p}^{j}\right)$ module on generators ${a}^{I}={a}_{\left(0\right)}^{{i}_{0}}\cdots {a}_{(n-1)}^{{i}_{n-1}}$, where ${i}_{k}\in \{0,1\}$ and $\sum _{k=0}^{n-1}{i}_{k}=q$, in degree $\sum _{k=0}^{n-1}{i}_{k}{p}^{k}$, with:

**Example 2.6.**Fix n and $q<n$ in $\mathbb{N}$.

**Definition 2.7.**If M is a Dieudonné module in $\mathcal{DM}$, a generalized Yang–Baxter operator for M is an invertible bilinear map $A:M{\otimes}_{\mathbb{Z}}M\to M{\otimes}_{\mathbb{Z}}M$, such that:

**Example 2.8.**For any Dieudonné module M, the identity map $A:M{\otimes}_{\mathbb{Z}}M\to M{\otimes}_{\mathbb{Z}}M$ is trivially a generalized Yang–Baxter operator.

**Example 2.9.**For any Dieudonné module M with a chosen basis, define $A:M{\otimes}_{\mathbb{Z}}M\to M{\otimes}_{\mathbb{Z}}M$ on basis elements by $A(x\otimes y)=y\otimes x$ (and expand by linearity on both arguments).

**Example 2.10.**Define $\mathsf{\alpha}:R\to R$ as the identity on all powers of V and F, except on those ${F}^{k}$ with $p\nmid k$, where $\mathsf{\alpha}\left({F}^{k}\right)={F}^{pk}$, and expand to R by linearity.

**Example 2.11.**The previous example is a particular case of a more general situation. Suppose we look for $\mathsf{\alpha}:R\to R$ and $\mathsf{\beta}:R\to R$ that map each power of F or V into another power of either (and not into a linear combination of more than one such power).

**Example 2.12.**For the Dieudonné module in Example 2.4, define $\mathsf{\beta}:M\to M$ by $\mathsf{\beta}\left({V}_{k}^{p\phantom{\rule{0.166667em}{0ex}}r}\right)={V}_{k}^{r}$ if $p\nmid r$ and $pr\le n$, and the identity elsewhere.

**Example 2.13.**If, in the setting of the previous example, we allow the behaviors of the ${V}_{k}^{r}$ to affect those for different values of k; we can put:

**Example 2.14.**For the Dieudonné module in Example 2.6. and again inspired by the Morava K-theory generators ${a}_{\left(j\right)}$, put $\mathsf{\beta}\left(I\right)=s\left(I\right)$ if $I(n-1-pr)=1$ (with $p\nmid r$ and $pr\le n$) and $I\left(j\right)=0$ for $j<n-1-pr$, and the identity elsewhere, and $\mathsf{\alpha}\left(I\right)={s}^{-1}\left(I\right)$ if $I(n-1-r(n-1\left)\right)=1$ (with $p\nmid r$ and $r<k$) and $I\left(j\right)=0$ for $j<n-1-r(n-1)$, and the identity elsewhere.

## 3. The Influence of the Dieudonné Module Yang–Baxter Operators on the Corresponding Hopf Algebras

**Definition 3.1.**The Witt polynomials ${\mathsf{\omega}}_{n}$, for $n\ge 0$, are given by:

**Theorem 3.2.**([8]) There exists a unique Hopf algebra structure on the polynomial algebra ${\mathbb{Z}}_{p}[{x}_{0},{x}_{1},\cdots ]$, such that the Witt polynomials ${\omega}_{n}$ are primitive.

**Proposition 3.3.**[8] Let $\left[p\right]:{\mathbb{Z}}_{p}[{x}_{0},{x}_{1},\cdots ]\to {\mathbb{Z}}_{p}[{x}_{0},{x}_{1},\cdots ]$ be p-times the identity map in the abelian group of Hopf algebra maps ${\mathbb{Z}}_{p}[{x}_{0},{x}_{1},\cdots ]\to {\mathbb{Z}}_{p}[{x}_{0},{x}_{1},\cdots ]$. Then, $\left[p\right]\left({x}_{i}\right)\cong {x}_{i-1}^{p}\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}p)$.

**Definition 3.4.**For a Hopf algebra H over ${\mathbb{F}}_{p}$, the Frobenius is the homomorphism $F:H\to H$ taking an element x to the element ${x}^{p}$. The Verschiebung $V:H\to H$ is the dual to the Frobenius in the dual algebra.

**Definition 3.5.**The Dieudonné module for a Hopf algebra $H\in {\mathcal{H}A}_{*}$ is the graded abelian group:

**Theorem 3.6.**([5,6]) The above functor D has a right adjoint $U:{\mathcal{D}M}_{*}\to {\mathcal{H}A}_{*}$, and the pair $(D,U)$ forms an equivalence of categories.

## 4. Yang–Baxter Operators on Dieudonné Modules, Hopf Ring and Hopf Coring Structures

**Lemma 4.1.**Any bilinear pairing ${\circ}_{ij}:D{H}_{i}\otimes D{H}_{j}\to D{H}_{i+j}$ induces a bilinear pairing ${\circ}_{ij}^{\prime}:\phantom{\rule{3.33333pt}{0ex}}{H}_{i}\otimes {H}_{j}\to {H}_{i+j}$.

**Proof.**Suppose first that the characteristic of the base field is zero.

**Example 4.2.**R, viewed as a Dieudonné module (as in Example 2.2), is equivalent to $DH\left(\infty \right)$, since clearly $R\simeq {\mathrm{Hom}}_{{\mathcal{H}A}_{*}}(H\left(n\right),H\left(\infty \right))$.

**Example 4.3.**For Example 2.3, we have ${\mathrm{Hom}}_{{\mathcal{H}A}_{*}}(H\left(n\right),CW\left(\infty \right))\simeq {\mathbb{Z}}_{p}[F,V]$, and so, $DCW\left(\infty \right)\simeq {\mathbb{Z}}_{p}[F,V]$.

**Example 4.4.**The Dieudonné module from Example 2.4 was suggested by the one for the Hopf algebra $K{\left(n\right)}_{*}\left({\mathbf{K}}_{1}\right)$, where ${\mathbf{K}}_{1}$ is the first Eilenberg–MacLane space $K(\mathbb{Z}/(p),1)$. This and the Hopf ring for further Eilenberg–MacLane spaces are completely described in [1]. By analogy, in our example, we get that the Hopf algebra corresponding to M is a truncated polynomial algebra generated by the ${a}_{\left(i\right)}^{k}$, where the p-th algebra power of each of these generators is zero (for $K{\left(n\right)}_{*}\left({\mathbf{K}}_{1}\right)$, the algebra relations depend on elements ${v}_{n}$ that we are not considering in this example). The coalgebra structure is given by $\psi \left({a}_{\left(i\right)}^{k}\right)={\sum}_{j=0}^{i}\phantom{\rule{0.166667em}{0ex}}{a}_{\left(j\right)}^{k}\otimes {a}_{(i-j)}^{k}$.

**Example 4.5.**For the Hopf algebra corresponding to the Dieudonné module in Example 2.6, we again adapt the periodically-graded situation from [1]. Each map $I:{\mathbb{N}}_{0}\to \{0,1\}$ in the conditions of Example 2.6 (namely, non-zero, except eventually on the n consecutive integers $i,\cdots ,i+n-1$) will correspond to an element of the form ${a}_{\left(i\right)}^{I\left(i\right)}\circ \cdots \circ {a}_{(i+n-1)}^{I(i+n-1)}$, where the ∘ notation is inspired by the subjacent Hopf ring structure (which is not dealt with here). The algebra in question will be free on these elements (over ${\mathbb{F}}_{p}$), with the algebra product of an ${a}_{\left(i\right)}^{I\left(i\right)}\circ \cdots \circ {a}_{(i+n-1)}^{I(i+n-1)}$ and an ${a}_{\left(j\right)}^{I\left(j\right)}\circ \cdots \circ {a}_{(j+n-1)}^{I(j+n-1)}$ given by an element ${a}_{\left(k\right)}^{I\left(k\right)}\circ \cdots \circ {a}_{(k+n-1)}^{I(k+n-1)}$ obtained by rearranging the ${a}_{\left(i\right)}$ and ${a}_{\left(j\right)}$ in increasing order of indexes, summing (mod two) the superscripts $I\left(i\right)$ and $I\left(j\right)$ for the same indexes and multiplying the result by the index of the permutation obtained from $(i,\cdots ,i+n-1)$ and $(j,\cdots ,j+n-1)$ by concatenation and by eliminating any repetitions of indexes that may appear in both of these sub-permutations. We determine also that this product should be zero if the resulting element is not of the form of those I in the definition of the original Dieudonné module (this has to do with I being nonzero only for q elements in a range of n consecutive natural numbers.)

**Example 4.6.**Consider first the switch operator from Example 2.9.

**Example 4.7.**Continuing Example 2.10, identify ${V}^{k}$ with ${\widehat{\omega}}_{k}$, in the notation of Lemma 4.1, and ${F}^{k}$ with $1\circ {f}^{k}$, where f is the map mentioned just before Definition 3.5.

**Example 4.8.**In Example 2.12, we again had a Yang–Baxter operator of the form $A=\alpha \otimes \beta $. The Hopf algebra in Example 4.4 will then have two induced coalgebra structures, coming from ${A}_{1}(x,y)=\alpha \left(x\right)$ and ${A}_{2}(x,y)=\beta \left(y\right)$ (where x and y are in the Dieudonné module).

**Example 4.9.**For the same Hopf algebra and the Yang–Baxter operator from Example 2.13, we get structures similar to those in the previous example, the difference being in the range of indexes where the generators of the algebra exist.

**Example 4.10.**For Example 2.14 and the Hopf algebra in Example 4.5, we have also ${A}_{1}(I,J)=\mathsf{\alpha}\left(I\right)$ and ${A}_{2}(I,J)=\mathsf{\beta}\left(J\right)$, and so, the two new ring structures will be projections on the first and second factors, except if $I(n-1-r(n-1\left)\right)=1$ and $I\left(j\right)=0$ for $j<n-1-r(n-1)$ (with $p\nmid r$ and $r<k$), for which ${A}_{1}(I,J)={s}^{-1}\left(I\right)$, and if $J(n-1-pr)=1$ and $J\left(j\right)=0$ for $j<n-1-pr$ (with $p\nmid r$ and $pr\le n$), for which ${A}_{2}(I,J)=s\left(J\right)$.

**Lemma 4.11.**[12] Any cobilinear map $g:DH\to D{H}_{1}\otimes D{H}_{2}$, where H, ${H}_{1}$ and ${H}_{2}$ are connected Hopf algebras in $\mathcal{HA}$, induces a cobilinear map ${g}^{\prime}:H\to {H}_{1}\otimes {H}_{2}$.

**Proof.**Since H is connected, it is enough to define ${g}^{\prime}$ on primitives and induced primitives [12].

**Example 4.12.**For the switch operator A from Example 2.9 on any $DH$, we get ${A}_{1}\left(x\right)=A\circ {i}_{1}\left(x\right)=A(x\otimes \widehat{1})=\widehat{1}\otimes x$ and ${A}_{2}\left(x\right)=A\circ {i}_{2}\left(x\right)=A(\widehat{1}\otimes x)=x\otimes \widehat{1}$ for any $x\in DH$.

**Example 4.13.**Continuing Example 2.10, we get ${A}_{1}\left(x\right)=A\circ {i}_{1}\left(x\right)=A(x\otimes \widehat{1})=\mathsf{\alpha}\left(\widehat{1}\right)\otimes \mathsf{\beta}\left(x\right)=\widehat{1}\otimes \mathsf{\beta}\left(x\right)$.

**Example 4.14.**The Yang–Baxter operator from Example 2.12 is also of the form $A=\mathsf{\alpha}\otimes \mathsf{\beta}$. This means that the deductions in the previous example are also at hand.

**Example 4.15.**Example 2.13 gives the same definitions for the induced products as the previous example. Nonetheless, in this case, the elements ${a}_{\left(i\right)}^{k}$ for different values of k are not independent, which means that the restrictions on the range of values that r and k may assume make for structures that differ from those in that example.

**Example 4.16.**Continuing Example 2.14, for the Hopf algebra in Example 4.5, we again have $A=\mathsf{\alpha}\otimes \mathsf{\beta}$, and so, ${A}_{1}\left(x\right)=\widehat{1}\otimes \mathsf{\beta}\left(x\right)$ and ${A}_{2}\left(x\right)=\mathsf{\alpha}\left(x\right)\otimes \widehat{1}$. Reading the definitions of α and β, on the generators I, we get then, as structures:

**Example 4.17.**The switch operator from Example 2.9 induces the same ${A}_{3}$ as the identity operator, which is simply the diagonal: ${A}_{3}\left(x\right)=sw\circ \mathsf{\Delta}\left(x\right)=sw(x\otimes x)=x\otimes x$ for $x\in DH$.

**Example 4.18.**For Example 2.10 and the Hopf algebra $H\left(\infty \right)$, ${A}_{3}\left(x\right)=\alpha \left(x\right)\otimes \beta \left(x\right)$.

**Example 4.19.**For Example 2.12, we again have ${A}_{3}\left(x\right)=\mathsf{\alpha}\left(x\right)\otimes \mathsf{\beta}\left(x\right)$ for x in the Dieudonné module. The considerations in Example 4.14 imply that the induced ${A}_{3}^{\prime}$ on the generators ${a}_{\left(i\right)}^{k}$ will be the diagonal, except if either $p\nmid (n-1-i)$ (and $n-1-i\le k$) or $n-1-i=pr$ for some r, such that $p\nmid r$ (and $pr\le n$). Both conditions cannot be satisfied simultaneously. This means that, because of the behavior of these particular α and β, one of the components in the image by ${A}_{3}^{\prime}$ of any element in the Hopf algebra will always be the identity.

**Example 4.20.**The previous example works also for the Yang–Baxter operator in Example 2.13 if we take into account the restrictions discussed in Example 4.15.

**Example 4.21.**From Example 2.14, we get that ${A}_{3}$ of a generator I can be the identity, $1\otimes I$, $I\otimes 1$, $1\otimes s\left(I\right)$, ${s}^{-1}\left(I\right)\otimes 1$ or ${s}^{-1}\left(I\right)\otimes s\left(I\right)$. These values will depend, as before, on the range of the indexes at play. In particular, the last value, which corresponds to a $\mathsf{\alpha}\left(I\right)\otimes \mathsf{\beta}\left(I\right)$ where neither α nor β are the identity, occurs whenever $n-1-pr=n-1-r(n-1)$, that is if $n-1$ is the prime p. Since we must also have that $pr\le n$, this implies that $r=1$.

## Conflicts of Interest

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Saramago, R.M.
Generalized Yang–Baxter Operators for Dieudonné Modules. *Axioms* **2015**, *4*, 177-193.
https://doi.org/10.3390/axioms4020177

**AMA Style**

Saramago RM.
Generalized Yang–Baxter Operators for Dieudonné Modules. *Axioms*. 2015; 4(2):177-193.
https://doi.org/10.3390/axioms4020177

**Chicago/Turabian Style**

Saramago, Rui Miguel.
2015. "Generalized Yang–Baxter Operators for Dieudonné Modules" *Axioms* 4, no. 2: 177-193.
https://doi.org/10.3390/axioms4020177