# Supporting Qualitative Analysis with Large Language Models: Combining
Codebook with GPT-3 for Deductive Coding

Ziang Xiao<EMAIL_ADDRESS>Microsoft ResearchMontrealQuebecCanada ,
Xingdi Yuan<EMAIL_ADDRESS>Microsoft ResearchMontrealQuebecCanada ,
Q. Vera Liao<EMAIL_ADDRESS>Microsoft ResearchMontrealQuebecCanada ,
Rania Abdelghani<EMAIL_ADDRESS>InriaFrance and  Pierre-Yves
Oudeyer<EMAIL_ADDRESS>InriaFrance

(2023)

###### Abstract.

Qualitative analysis of textual contents unpacks rich and valuable information
by assigning labels to the data. However, this process is often labor-
intensive, particularly when working with large datasets. While recent AI-
based tools demonstrate utility, researchers may not have readily available AI
resources and expertise, let alone be challenged by the limited
generalizability of those task-specific models. In this study, we explored the
use of large language models (LLMs) in supporting deductive coding, a major
category of qualitative analysis where researchers use pre-determined
codebooks to label the data into a fixed set of codes. Instead of training
task-specific models, a pre-trained LLM could be used directly for various
tasks without fine-tuning through prompt learning. Using a curiosity-driven
questions coding task as a case study, we found, by combining GPT-3 with
expert-drafted codebooks, our proposed approach achieved fair to substantial
agreements with expert-coded results. We lay out challenges and opportunities
in using LLMs to support qualitative coding and beyond.

Qualitative Analysis, Deductive Coding, Large Language Model, GPT-3

††journalyear: 2023††copyright: rightsretained††conference: 28th International
Conference on Intelligent User Interfaces; March 27–31, 2023; Sydney, NSW,
Australia††booktitle: 28th International Conference on Intelligent User
Interfaces (IUI ’23 Companion), March 27–31, 2023, Sydney, NSW,
Australia††doi: 10.1145/3581754.3584136††isbn: 979-8-4007-0107-8/23/03††ccs:
Human-centered computing HCI design and evaluation methods††ccs: Computing
methodologies Natural language processing

##  1\. Introduction

Qualitative coding is a method of qualitative analysis that is used to
identify patterns and categories in qualitative data, such as social media
posts, open-ended survey responses, and field notes. Often, scientists will
use coded data to derive theory or build models to understand the observed
phenomenon further (Hsieh and Shannon, 2005). Qualitative coding can be a
challenging process because it requires the researcher to understand and
analyze data that is often complex, nuanced, and open to multiple
interpretations. Multiple researchers need to spend a significant amount of
time and effort to review the data, develop a codebook that can accurately
describe each label, and iteratively code the data until reaching a reasonable
inter-rater agreement. It is particularly hard when working with large
datasets. In this study, we explore a novel approach that leverages large
language models (LLMs) to facilitate qualitative coding.

Researchers have built AI-based tools to assist qualitative analysis (Rietz
and Maedche, 2021; Liew et al., 2014; Muller et al., 2016; Paredes et al.,
2017). These tools use natural language processing (NLP) and machine learning
(ML) algorithms to help researchers identify patterns and themes in
qualitative data. Two categories of models are often used, 1) unsupervised
models, e.g., topic models, to help researchers discover themes, or 2)
supervised models, e.g., logistic regression, to classify data into labels.
However, both methods face their own challenges. Unsupervised models are
difficult to steer. It is difficult for researchers to use those models for
their complex or nuanced research questions. And supervised models often
require high-quality large datasets or computing resources to achieve
reasonable performance. Therefore, most of today’s qualitative coding still
relies on manual effort.

The recent advent of LLMs (e.g., GPT-3 (Brown et al., 2020), PaLM (Chowdhery
et al., 2022), OPT (Zhang et al., 2022)) offers new capabilities, including
creative writing (Yuan et al., 2022a), programming
111https://github.com/features/copilot, etc. Unlike traditional task-specific
models, LLMs accept natural language prompts as input to perform various tasks
(Liu et al., 2021). For example, LLMs could perform classification tasks if
the prompt specifies a set of labels as the output space (Gao et al., 2021).
Compared to unsupervised models, the prompt could include specific
instructions and examples (e.g., a codebook) to increase LLMs’ performance in
a new task with unseen data in a zero-shot/few-shot fashion. Prior work shows
that LLMs can be prompted to boost an NLG system’s performance by selecting
better outputs from sampled candidates, according to some pre-defined metrics
(Yuan et al., 2022b). Since LLMs operate effectively with natural language
input and do not require training datasets, they lower the barriers for
researchers without extensive AI expertise or resources to leverage AI
capabilities in data analysis (although many of today’s LLMs are proprietary).

Although recent studies have demonstrated LLMs utility in many domains,
studies have shown LLMs are error-prone and have limited capability in
capturing language structure and nuanced meanings (Korngiebel and Mooney,
2021), which are crucial in qualitative analysis. Therefore, in this
preliminary study, we asked two research questions,

  * •

RQ1: To what extent does our LLM-based approach agree with experts in a
deductive coding task?

  * •

RQ2: How do different prompt designs affect the coding results?

We examined LLMs’ capability in facilitating a deductive coding task where we
combined GPT-3 with expert-developed codebooks to analyze children’s
curiosity-driven questions in terms of question complexity and syntactic
structure. We found our proposed approach achieved fair to substantial
agreements with experts (Question complexity: Cohen’s $\kappa=0.61$; Syntactic
Structure: Cohen’s $\kappa=0.38$). Based on our preliminary findings, we
present challenges and opportunities for utilizing LLMs in qualitative
analysis.

##  2\. Deductive Coding Task

We selected a deductive coding task as our starting point. The goal of
deductive coding is to label the data based on a codebook. It is a top-down
process where the researchers start by developing a codebook with an initial
set of codes along with descriptions and examples based on the research focus
or theory (Hsieh and Shannon, 2005). Note that a coding process often involves
both inductive coding (e.g. developing the code book) and deductive coding
(coding all the data according to a code book). Our approach can be used to
complete the second part or combined with the first part to help coders
rapidly iterate on their codebook.

###  2.1. Case Study: Curiosity-driven questions Analysis

We examined our approach in analyzing children’s curiosity-driven questions.
Understanding children’s ability to ask curiosity-driven questions informs
psychologists of a child’s learning stage. We looked at two dimensions of a
curiosity-driven question, question complexity and syntactic structure. The
question complexity looks at if the answer to a question is a simple fact
(e.g., “How big is a dinosaur?”) or requires explaining a mechanism, a
relationship, etc.(e.g., “Why were dinosaurs so big?”) (Wilen, 1991). The
syntactic structure has four categories, 1) ‘closed’ or declarative questions
(e.g., “Dinosaurs were big?”), 2) questions with questioning words in the
middle of the sentence (e.g., “The dinosaurs were how big?”), 3) questions
without an interrogative formulation (e.g., “Why the dinosaurs are big?”), and
4) questions with a questioning word at the beginning of the sentence that has
interrogative syntax (e.g., “Why are dinosaurs big?”).

We collected a dataset with 668 children’s questions in French (Abdelghani et
al., 2022) 222For question complexity, we first used GPT-3 to translate
questions into English. For the syntactic structure, we kept questions in
French to preserve its syntactic structure.. A team of psychologists has
developed a codebook and coded each question on the dimension of question
complexity and syntactic structure. Our chosen dimensions cover two main
categories of deductive coding, binary coding, and multi-level coding, with
one focusing on the semantic meaning and the other looking at the syntactic
structure. Additionally, the dataset and codebook have never been published
online which is unseen by LLMs.

##  3\. GPT-3 Setup and Prompt Design

We chose GPT-3 (davinci-text-002) with a temperature of 0.0 during the
prompting process, because it was the most advanced version of GPT-3 that was
publicly available when we conducted the experiments and 0.0 temperature
(greedy decoding) guarantees the reproducibility of this study.

We explored two design dimensions of the prompt,

  * •

Codebook-centered vs. Example-centered: This dimension regards the structure
of a prompt. In the codebook-centered prompt, we designed the prompt similar
to how researchers read a codebook. The prompt follows the structure of [Code/
Description/ Examples]. For example,

(a) Cohen’s $\kappa$ for Question Complexity

(b) Cohen’s $\kappa$ for Syntactic Structure

Figure 1. The Cohen’s $\kappa$ between GPT-3 and experts shows substantial
agreement in Question Complexity coding and fair agreement in Syntactic
Structure coding. In general, Codebook-centered prompts with examples achieves
the highest agreement.

For all prompt variants, we included an identity modifier, “I am a
developmental psychologist who has expertise in linguistics.” and an
instruction to constrain output space, “Choose from the following candidates:
[Code Set]”.

##  4\. Results

We measured the performance of our GPT-3 based approach with Cohen’s Kappa
(McHugh, 2012). Cohen’s Kappa measures inter-rater reliability, which
indicates how two coders agree with each other. We computed two sets of
Cohen’s Kappa, between GPT-3 with the expert’s final coding results and
between two experts who originally coded the dataset using the same codebook.

For RQ1, our results suggest that it is feasible to use GPT-3 with an expert-
developed codebook for deductive coding. When analyzing curiosity-driven
questions, our GPT-3-based approach achieved fair (Syntactic Structure:
Cohen’s $\kappa=0.38$) to substantial (Question complexity: Cohen’s
$\kappa=0.61$) agreement with expert rating (Cohen’s $\kappa$’s interpretation
is based on (McHugh, 2012)), see Fig. 1. However, there is a gap between
experts’ agreement with our approach and the agreement among experts (Question
complexity: Cohen’s $\kappa=0.88$; Syntactic Structure: Cohen’s
$\kappa=0.90$).

For RQ2, we examined the different prompt designs. We found the codebook-
centered design performs better than the example-centered designs, see Fig. 1.
And examples play an important role. We observed the largest performance gain
when shifting from a zero-shot to a one-shot setting. However, the performance
between one-shot and few-shot settings did not differ much.

##  5\. Opportunities and Challenges

Our preliminary findings indicate the feasibility of using LLMs for
qualitative analysis. In a deductive coding task, by combining GPT-3 and a
codebook, our LLM-based approach achieved fair to substantial agreement with
experts. Considering the accessibility and flexibility of LLMs, we believe
this approach has the potential to effectively help researchers to analyze
qualitative data, especially for increasingly large datasets. We lay out a few
challenges and opportunities for our future studies,

  * •

Interrogate Model Capability: In this preliminary exploration, we only
measured the level of agreement. To better understand model capability, we
ought to conduct more detailed error analyses on disagreed items. Also, it is
unclear if the LLM-based method could be extended to different contexts and
different coding schemes where the coder needs to capture more nuanced
signals.

  * •

Design for Appropriate Reliance: Although our preliminary results show fair to
substantial agreement with expert rating, the model produces incorrect labels.
When deploying such an imperfect AI system, we must design for appropriate
reliance to prevent over-trusting and misuse. For example, when designing the
interface, we could consider explainable AI methods to calibrate user trust
over time.

  * •

Design Codebook for LLMs: We constructed our prompts using the codebook for
experts. Although it provides transparency and explicit control, it may limit
the model’s performance. Future study is required to understand how to design
a more effective codebook for task performance and model understanding.

  * •

Support Inductive Coding: We demonstrated the feasibility of using LLMs in
deductive coding. However, for inductive coding, where the process is more
bottom-up and exploratory, novel interaction and Human-AI collaboration
diagrams are required. We should study interaction techniques and controls to
let researchers use LLMs more effectively in the qualitative analysis.

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$\delta_{H}^{n}\left(P_{n}(H)\right)\subset\left(H^{\otimes
n}\right)^{\Sigma_{n}}$, i.e. the subspace
$\delta_{H}^{n}\left(P_{n}(H)\right)$ is invariant under the permutation
action of $\Sigma_{n}$.

Moreover, if the characteristic of $\mathds{k}$ is zero, then the above
properties are equivalent.

###### Proof.

Suppose that $H$ is good. Let $n\in\mathds{N}$. We prove that, for $a\in
P_{n}(H)$, $\delta_{H}^{n}(a)\in H^{\otimes n}$ is fixed under the
$\Sigma_{n}$-action. The hypothesis implies that $g_{n}$ is surjective. We
choose $b\in P_{1}(H)^{\otimes n}$ such that $a=g_{n}(b)$. Recall the
notations in Lemma 10.7. The idempotent $e^{\otimes n}_{H}$ on
$P_{1}(H)^{\otimes n}$ induces a direct decomposition $P_{1}(H)^{\otimes
n}\cong\mathrm{Prim}(H)^{\otimes n}\oplus W$. Let
$c\in\mathrm{Prim}(H)^{\otimes n}$ and $c^{\prime}\in W$ be the components of
$b$ with respect to this decomposition. By Lemma 10.7, we have
$\delta_{H}^{n}(g_{n}(c^{\prime}))=0$. Hence, we obtain
$\delta_{H}^{n}(g_{n}(b))=\delta_{H}^{n}(g_{n}(c))=f_{n}(h_{n}(c))$ which
leads to $\delta_{H}^{n}(a)=s_{n}(c)$, by Lemma 10.6. Thus
$\delta_{H}^{n}(a)\in H^{\otimes n}$ is fixed under the $\Sigma_{n}$-action.

We now consider $\mathds{k}$ a field of characteristic zero. We start from the
second part of our statement and prove that $P_{1}(H)^{n}=P_{n}(H)$ by
induction on $n$. It is obvious for $n=1$. Suppose that
$P_{1}(H)^{n-1}=P_{n-1}(H)$ for $n\geq 2$. It suffices to prove that
$P_{n}(H)\subset P_{1}(H)^{n}$ by Proposition 10.1. Let $a\in P_{n}(H)$. By
the hypothesis $\delta_{H}^{n}\left(P_{n}(H)\right)\subset\left(H^{\otimes
n}\right)^{\Sigma_{n}}$, we have

$s_{n}(\delta_{H}^{n}(a))=\sum_{\sigma\in\Sigma_{n}}\sigma(\delta_{H}^{n}(a))=n!~{}\delta_{H}^{n}(a),$

or equivalently, $s_{n}\circ f_{n}=n!~{}f_{n}$ . By Lemma 10.6, we obtain

(10) $\displaystyle f_{n}\circ h_{n}\circ f_{n}=n!~{}f_{n}.$

Let $\pi_{n}:P_{n}(H)\to P_{n}(H)/P_{n-1}(H)$ be the projection. Note that
$f_{n}$ induces an injection
$\overline{f_{n}}:P_{n}(H)/P_{n-1}(H)\to\mathrm{Prim}(H)^{\otimes n}$ such
that $\overline{f_{n}}\circ\pi_{n}=f_{n}$ since $P_{n-1}(H)$ is the kernel of
$\delta^{n}$. By (10), we have $\overline{f_{n}}\circ\pi_{n}\circ h_{n}\circ
f_{n}=n!~{}\overline{f_{n}}\circ\pi_{n}$ which gives $\pi_{n}\circ h_{n}\circ
f_{n}=n!~{}\pi_{n}$ since $\overline{f_{n}}$ is injective. Hence, we have
$a-(n!)^{-1}\cdot h_{n}(f_{n}(a))\in P_{n-1}(H)$ (here, we use the assumption
on the characteristic). By the hypothesis of the induction, we have
$a-(n!)^{-1}\cdot h_{n}(f_{n}(a))\in P_{1}(H)^{n-1}$. Note that
$(n!)^{-1}\cdot h_{n}(f_{n}(a))\in P_{1}(H)^{n}$ by the definition of $h_{n}$.
Therefore, we have $a\in P_{1}(H)^{n}$, since $P_{1}(H)^{n-1}\subset
P_{1}(H)^{n}$. ∎

######  Corollary 10.9.

Let $\mathds{k}$ be a field of characteristic zero. Every cocommutative
bialgebra over $\mathds{k}$ is good. In particular, a primitive bialgebra over
$\mathds{k}$ is good.

###### Proof.

It is obvious that $\delta^{2}_{H}:H\to H\otimes H$ is fixed under the
$\Sigma_{2}$-action, since $H$ is cocommutative. Moreover, $\delta_{H}^{n}$ is
equal to the $n$-times iterated application of $\delta^{2}_{H}$, and thus,
$\delta^{n}_{H}$ is fixed under the $\Sigma_{n}$-action. Hence, the converse
in Theorem 10.8 implies the claim. The last claim follows from the fact that a
primitive bialgebra is cocommutative. ∎

######  Remark 10.10.

If $\mathds{k}$ is a field of positive characteristic, there exist
cocommutative bialgebras which are not good. For example, the shuffle
bialgebra $Sh(\mathds{k})$ is cocommutative but not good (see Example 10.3).

######  Corollary 10.11.

Let $\mathds{k}$ be a field of characteristic zero. A bialgebra over
$\mathds{k}$ is primitive if and only if it is conilpotent and good.

###### Proof.

A primitive bialgebra $H$ is conilpotent since
$H=\bigcup_{n\in\mathds{N}}P_{1}(H)^{n}\subset\bigcup_{n\in\mathds{N}}P_{n}(H)\subset
H$. By Corollary 10.9, the only if part is proved. Conversely, for a
conilpotent and good bialgebra $H$, we have
$\bigcup_{n\in\mathds{N}}P_{1}(H)^{n}=\bigcup_{n\in\mathds{N}}P_{n}(H)$ since
it is good. The conilpotency of $H$ implies
$\bigcup_{n\in\mathds{N}}P_{1}(H)^{n}=H$. ∎

######  Remark 10.12.

Consider $\mathds{k}$ a field of characteristic zero. One can deduce from a
classical structure theorem for bialgebras (e.g. see [11, Theorem 1.3.4]) that
conilpotency is equivalent to primitivity for a cocommutative bialgebra. We
remark that Corollary 10.9 and 10.11 lead to the same conclusion without
applying the structure theorem.

####  10.2. Primitive exponential $\mathsf{gr}^{\mathsf{op}}$-modules

In this section, we give a relation between primitive cocommutative Hopf
algebras and primitive exponential $\mathsf{gr}^{\mathsf{op}}$-modules. In
particular, we prove an equivalence of them if the ground ring $\mathds{k}$ is
a field.

Let $\mathds{k}$ be a commutative unital ring. Recall the functor
$\mathrm{ev}_{1}:\mathcal{F}^{\mathsf{exp}}(\mathsf{gr^{\mathsf{op}}};\mathds{k})\to\mathsf{Hopf}^{\mathsf{cc}}_{\mathds{k}}$
in Remark 8.1. In the following statements, let $H=\mathrm{ev}_{1}(F)$ be the
induced cocommutative Hopf algebra for
$F\in\mathcal{F}^{\mathsf{exp}}(\mathsf{gr}^{\mathsf{op}};\mathds{k})$. Note
that the exponentiality of $F$ induces the isomorphism $H^{\otimes m}\cong
F(m)$.

For a Lie algebra $\mathfrak{g}$, the assignment of $\mathfrak{g}^{\otimes n}$
to $n\in\mathds{N}$ extends to a left $\mathsf{Cat}_{\mathfrak{Lie}}$-module
which we denote by $\mathfrak{g}^{\otimes}$. This construction is natural so
that if we have a Lie algebra homomorphism $f:\mathfrak{g}\to\mathfrak{h}$,
then it induces a natural transformation
$\mathfrak{g}^{\otimes}\to\mathfrak{h}^{\otimes}$. For a Hopf algebra $H$, we
are mainly interested in the Lie algebras $\mathrm{Prim}(H)$ and $H$ itself,
which are Lie algebras equipped with the commutator.

######  Lemma 10.13.

Let $H$ be a cocommutative Hopf algebra over $\mathds{k}$. The inclusion
$\mathrm{Prim}(H)\hookrightarrow H$ induces the natural transformation

$\mathrm{Prim}(H)^{\otimes}\to\mathcal{B}^{\ast}\mathcal{E}^{\ast}\left(\alpha(H)\right)$

where
$\mathsf{Cat}_{\mathfrak{Lie}}\stackrel{{\scriptstyle\mathcal{B}}}{{\to}}\mathsf{Cat}_{\mathfrak{Ass}^{u}}\stackrel{{\scriptstyle\mathcal{E}}}{{\to}}\mathds{k}\mathsf{gr}^{\mathsf{op}}$
are the canonical functors introduced in section 6.3.

###### Proof.

The inclusion $\mathrm{Prim}(H)\to H$ is a Lie algebra homomorphism. Hence,
this induces a natural transformation $\mathrm{Prim}(H)^{\otimes}\to
H^{\otimes}$ where
$H^{\otimes}:\mathsf{Cat}_{\mathfrak{Lie}}\to\mathsf{Mod}_{\mathds{k}};n\mapsto
H^{\otimes n}$ is the functor induced by the Lie algebra structure on $H$.
Note that we have
$H^{\otimes}=\mathcal{B}^{\ast}\mathcal{E}^{\ast}\left(\alpha(H)\right)$ by
definitions. ∎

Let $F\in\mathcal{F}^{\mathsf{exp}}(\mathsf{gr}^{\mathsf{op}};\mathds{k})$.
Applying Lemma 10.13 to $\mathrm{ev}_{1}(F)$ gives a natural transformation
$\mathrm{Prim}(\mathrm{ev}_{1}(F))^{\otimes}\to\mathcal{B}^{\ast}\mathcal{E}^{\ast}\left(\alpha(\mathrm{ev}_{1}(F))\right)\cong\mathcal{B}^{\ast}\mathcal{E}^{\ast}(F)$.

Recall that we have a natural inclusion
$\mathcal{P}(F)\hookrightarrow\mathcal{B}^{\ast}\mathcal{E}^{\ast}(F)$ of
$\mathsf{Cat}_{\mathfrak{Lie}}$-modules by Proposition 9.12.

######  Lemma 10.14.

Let $F\in\mathcal{F}^{\mathsf{exp}}(\mathsf{gr}^{\mathsf{op}};\mathds{k})$.
The natural transformation
$\mathrm{Prim}(\mathrm{ev}_{1}(F))^{\otimes}\to\mathcal{B}^{\ast}\mathcal{E}^{\ast}(F)$
in Lemma 10.13 factors through
$\mathcal{P}(F)\hookrightarrow\mathcal{B}^{\ast}\mathcal{E}^{\ast}(F)$:

${\mathrm{Prim}(\mathrm{ev}_{1}(F))^{\otimes}}$${\mathcal{B}^{\ast}\mathcal{E}^{\ast}(F)}$${\mathcal{P}(F).}$$\scriptstyle{\exists\beta}$

If $\mathds{k}$ is a field, then the factorized natural transformation $\beta$
is an isomorphism.

###### Proof.

Let $f_{n}:\mathrm{Prim}(\mathrm{ev}_{1}(F))^{\otimes
n}\to\mathrm{ev}_{1}(F)^{\otimes n}=F(1)^{\otimes n}\cong F(n)$ be the
$n$-fold tensor product of the inclusions. It suffices to show the objectwise
factorization below since $\mathcal{P}(F)$ is a sub
$\mathsf{Cat}_{\mathfrak{Lie}}$-module of
$\mathcal{B}^{\ast}\mathcal{E}^{\ast}(F)$:

${F(n)}$${F(n)}$${\mathrm{Prim}(\mathrm{ev}_{1}(F))^{\otimes
n}}$${(\mathcal{P}(F))(n).}$$\scriptstyle{\exists\beta_{n}}$$\scriptstyle{f_{n}}$

By Proposition 9.5, it suffices to prove that the image of $f_{n}$ lies in
$L_{i}=\mathrm{Ker}\left(F(\mathrm{id}_{i-1}\ast\theta\ast\mathrm{id}_{n-i})\right)$
for every $1\leq i\leq n$. In fact, we have
$\mathrm{Prim}(\mathrm{ev}_{1}(F))=\mathrm{Ker}(F(\theta))$ by definition, so
that we have $F(\mathrm{id}_{i-1}\ast\theta\ast\mathrm{id}_{n-i})\circ
f_{n}=0$.

${0}$${L_{i}}$${F(n)}$${F(n+1)}$${\mathrm{Ker}(F(\theta))^{\otimes
n}.}$$\scriptstyle{F(\mathrm{id}_{i-1}\ast\theta\ast\mathrm{id}_{n-i})}$$\scriptstyle{f_{n}}$$\scriptstyle{0}$

We now assume that $\mathds{k}$ is a field, By applying Lemma 4.9, we see that
$L_{i}=F(1)^{\otimes(i-1)}\otimes\mathrm{Ker}(F(\theta))\otimes
F(1)^{\otimes(n-i)}$. In particular, Proposition 9.5 implies that
$\left(\mathcal{P}(F)\right)(n)=\bigcap^{n}_{i=1}L_{i}=\mathrm{Ker}(F(\theta))^{\otimes
n}$ so that $\beta_{n}$ gives an isomorphism. ∎

Let $\mathsf{Hopf}^{\mathsf{cc,prim}}_{\mathds{k}}$ be the full subcategory of
$\mathsf{Hopf}^{\mathsf{cc}}_{\mathds{k}}$ consisting of primitive Hopf
algebras.

Let
$\mathcal{F}^{\mathsf{exp}}_{\mathsf{prim}}(\mathsf{gr^{\mathsf{op}}};\mathds{k})$
be the full subcategory of
$\mathcal{F}^{\mathsf{exp}}(\mathsf{gr^{\mathsf{op}}};\mathds{k})$ consisting
of primitive $\mathsf{gr^{\mathsf{op}}}$-modules.

In the following, recall the functor
$\alpha:\mathsf{Hopf}^{\mathsf{cc}}_{\mathds{k}}\to\mathcal{F}^{\mathsf{exp}}(\mathsf{gr^{\mathsf{op}}};\mathds{k})$
in Remark 8.1, which gives an equivalence of categories.

######  Theorem 10.15.

The functor $\alpha$ induces a fully faithful functor:

$\mathsf{Hopf}^{\mathsf{cc,prim}}_{\mathds{k}}\to\mathcal{F}^{\mathsf{exp}}_{\mathsf{prim}}(\mathsf{gr^{\mathsf{op}}};\mathds{k}).$

If $\mathds{k}$ is a field, then it gives an equivalence of categories.

###### Proof.

We first prove that, if $H$ is a cocommutative primitive Hopf algebra, then
$F=\alpha(H)$ is primitive. Let $m\in\mathds{N}$. Recall the notations in
Theorem 9.6. For $k\in\mathds{N}$, consider the following diagram:

(11) ${H^{\otimes m}}$${F(m)}$${\mathsf{Cat}_{\mathfrak{Ass}^{u}}(k,m)\otimes
H^{\otimes k}}$${\mathsf{Cat}_{\mathfrak{Ass}^{u}}(k,m)\otimes
F(k)}$${{}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(k,m)\otimes\mathrm{Prim}(H)^{\otimes
k}}$${{}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(k,m)\otimes\left(\mathcal{P}(F)\right)(k).}$$\scriptstyle{\cong}$$\scriptstyle{T_{k}}$$\scriptstyle{\cong}$$\scriptstyle{J_{k}}$$\scriptstyle{\mathrm{id}\otimes\beta_{k}}$

The map $T_{k}$ is induced by the underlying algebra structure of $H$. Note
that
${}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(k,m)=\mathsf{Cat}_{\mathfrak{Ass}^{u}}(k,m)$.
The top square commutes due to the definition of $J_{k}$: it is induced by the
$\mathsf{Cat}_{\mathfrak{Ass}^{u}}$-module structure on $\mathcal{E}^{\ast}F$.
The square below commutes due to the definition of $\beta$ in Lemma 10.14. Let
$\widetilde{T}_{k}$ be the composition of the left vertical maps. By
definition, if $H$ is primitive, then $H^{\otimes m}$ is equal to the image of
$\Sigma_{k\in\mathds{N}}\widetilde{T}_{k}$. Hence, if $H$ is primitive, then
$F(m)$ is equal to the image of $\Sigma_{k\in\mathds{N}}\widetilde{J}_{k}$,
i.e. the $\mathsf{gr}^{\mathsf{op}}$-module $F=\alpha(H)$ is primitive.
Therefore, the composition
$\mathsf{Hopf}^{\mathsf{cc,prim}}_{\mathds{k}}\hookrightarrow\mathsf{Hopf}^{\mathsf{cc}}_{\mathds{k}}\stackrel{{\scriptstyle\mathrm{ev}_{1}}}{{\to}}\mathcal{F}^{\mathsf{exp}}(\mathsf{gr^{\mathsf{op}}};\mathds{k})$
factors through
$\mathcal{F}^{\mathsf{exp}}_{\mathsf{prim}}(\mathsf{gr^{\mathsf{op}}};\mathds{k})$.
Moreover, the factorized functor is faithfully full since so is
$\mathrm{ev}_{1}$.

We now suppose that $\mathds{k}$ is a field. It suffices to prove that an
exponential $\mathsf{gr}^{\mathsf{op}}$-module $F$ is primitive only if the
Hopf algebra $H=\mathrm{ev}_{1}(F)$ is primitive. By the last statement in
Lemma 10.14, the horizontal map $\mathrm{id}\otimes\beta_{k}$ in (11) is an
isomorphism for any $k$. Thus, if $F$ is primitive, then $H^{\otimes m}$ is
equal to the image of $\Sigma_{k\in\mathds{N}}\widetilde{T}_{k}$, by the
commutative diagram (11). By applying this to $m=1$, it proved that $H$ is
primitive. ∎

######  Remark 10.16.

Every primitive bialgebra is conilpotent for any ground ring $\mathds{k}$ (see
Proposition 10.1). For a cocommutative bialgebra over a field of
characteristic zero, its conilpotency is equivalent to primitivity (see Remark
10.12). Hence, if $\mathds{k}$ is a field of characteristic zero, the
equivalences of categories in Theorem 8.2 and 10.15 are the same. In positive
characteristic, however,
$\mathsf{Hopf}^{\mathsf{cc,prim}}_{\mathds{k}}\subset\mathsf{Hopf}^{\mathsf{cc,conil}}_{\mathds{k}}$
is an (essentially) proper inclusion. For example, the shuffle bialgebra
$\mathrm{Sh}(\mathds{k})$ associated with the one-dimensional vector space is
conilpotent but non-primitive (see section 4.3.2). In particular, the
inclusion
$\mathcal{F}^{\mathsf{exp}}_{\mathsf{prim}}(\mathsf{gr}^{\mathsf{op}};\mathds{k})\hookrightarrow\mathcal{F}^{\mathsf{exp}}_{\omega}(\mathsf{gr}^{\mathsf{op}};\mathds{k})$
(by Proposition 9.9) is (essentially) proper.

####  10.3. Equivalence with restricted Lie algebras

In this section, we suppose that the ground ring $\mathds{k}$ is a field of
positive characteristic $p$. Then the functors
$P:\mathsf{Hopf}^{\mathsf{cc,conil}}_{\mathds{k}}\to\mathsf{Lie}_{\mathds{k}}$
and
$U:\mathsf{Lie}_{\mathds{k}}\to\mathsf{Hopf}^{\mathsf{cc,conil}}_{\mathds{k}}$
in section 8.3 do not induce an equivalence of categories. We give some
counterexamples below.

######  Example 10.17.

The functor $P$ is not full. Consider the conilpotent Hopf algebra
$\mathds{k}[t]$ in section 4.3.1. By Theorem 4.13, $P_{1}(\mathds{k}[t])$ is
infinite-dimensional since $L_{p}(i)=1$ if and only if $i=p^{k}$ for some
$k\in\mathds{N}$. Hence, $P(\mathds{k}[t])$ is an infinite-dimensional abelian
Lie algebra, and thus, the map
$P:\mathrm{End}_{\mathsf{Hopf}_{\mathds{k}}}(\mathds{k}[t])\to\mathrm{End}_{\mathsf{Lie}_{\mathds{k}}}(P(\mathds{k}[t]))$
is not surjective.

######  Example 10.18.

The functor $U$ is not essentially surjective. For example, the conilpotent
Hopf algebra $H$ in Example 4.20 is not isomorphic to any universal enveloping
algebra.

There is a well-known modification of (7) in positive characteristic (see [12,
Theorem 6.11]). In fact, we have a well-behaved subcategory
$\mathsf{Hopf}^{\mathsf{cc,prim}}_{\mathds{k}}\subset\mathsf{Hopf}^{\mathsf{cc,conil}}_{\mathds{k}}$
which is (essentially) proper by Remark 10.16. The assignment of the induced
restricted Lie algebra $\bar{P}(H)$ to a primitive cocommutative Hopf algebra
$H$ gives an equivalence of categories:

$\bar{P}:\mathsf{Hopf}^{\mathsf{cc,prim}}_{\mathds{k}}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\mathsf{ResLie}_{\mathds{k}}$

where $\mathsf{ResLie}_{\mathds{k}}$ is the category of restricted Lie
algebras.

In the following statement, we denote by
$\mathsf{ResLie}^{\mathsf{ab}}_{\mathds{k}}$ the category of abelian
restricted Lie algebras.

######  Corollary 10.19.

If $\mathds{k}$ is a field with positive characteristic, then the composition
of the functor $\mathrm{ev}_{1}$ with $\bar{P}$ induces equivalences of
categories:

  1. (1)

$\mathcal{F}^{\mathsf{exp}}_{\mathsf{prim}}(\mathsf{gr^{\mathsf{op}}};\mathds{k})\simeq\mathsf{ResLie}_{\mathds{k}}$.

  2. (2)

$\mathcal{F}^{\mathsf{exp}}_{\mathsf{prim},\mathsf{out}}(\mathsf{gr^{\mathsf{op}}};\mathds{k})\simeq\mathsf{ResLie}^{\mathsf{ab}}_{\mathds{k}}$.

###### Proof.

The first claim is immediate from Theorem 10.15. The second one is proved by
combining Theorem 8.3 and Theorem 10.15. ∎

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