Functors between representation categories. Universal modules

A.L. Agore

RECORD

Functors between representation categories. Universal modules

A.L. Agore

Rings and Algebras · Submitted 8 Jan 2023

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Under some finite dimensionality assumptions, we prove that the two functors admit left adjoints which leads to the construction of universal ${\mathcal A}$-modules and universal Lie $\mathfrak{h}$-modules as the representation theoretic counterparts of Manin-Tambara's universal coacting objects \cite{Manin, Tambara}.

how it is stated · asserted 1.00 read the record →

Comment (0) Read on arXiv ↗

Cite ▾

Claim paper

The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Open the compiled record

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,050 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.64 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.42 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; show every role.

Let $\mathfrak{g}$ and $\mathfrak{h}$ be two Lie algebras with $\mathfrak{h}$ finite dimensional and consider ${\mathcal A} = {\mathcal A} (\mathfrak{h}, \, \mathfrak{g})$ to be the corresponding universal algebra as introduced in \cite{am20}. Given an ${\mathcal A}$-module $U$ and a Lie $\mathfrak{h}$-module $V$ we show that $U \otimes V$ can be naturally endowed with a Lie $\mathfrak{g}$-module structure. This gives rise to a functor between the category of Lie $\mathfrak{h}$-modules and the category of Lie $\mathfrak{g}$-modules and, respectively, to a functor between the category of ${\mathcal A}$-modules and the category of Lie $\mathfrak{g}$-modules. Under some finite dimensionality assumptions, we prove that the two functors admit left adjoints which leads to the construction of universal ${\mathcal A}$-modules and universal Lie $\mathfrak{h}$-modules as the representation theoretic counterparts of Manin-Tambara's universal coacting objects \cite{Manin, Tambara}.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Let $\mathfrak{g}$ and $\mathfrak{h}$ be two Lie algebras with $\mathfrak{h}$ finite dimensional and consider ${\mathcal A} = {\mathcal A} (\mathfrak{h}, \, \mathfrak{g})$ to be the corresponding universal algebra as introduced in \cite{am20}.	background 0.99	0.14	0.02	asserted 0.79
s1	Given an ${\mathcal A}$-module $U$ and a Lie $\mathfrak{h}$-module $V$ we show that $U \otimes V$ can be naturally endowed with a Lie $\mathfrak{g}$-module structure.	result 0.69	0.91	0.24	asserted 1.00
s2	This gives rise to a functor between the category of Lie $\mathfrak{h}$-modules and the category of Lie $\mathfrak{g}$-modules and, respectively, to a functor between the category of ${\mathcal A}$-modules and the category of Lie $\mathfrak{g}$-modules.	result 0.69	0.86	0.14	asserted 1.00
s3	Under some finite dimensionality assumptions, we prove that the two functors admit left adjoints which leads to the construction of universal ${\mathcal A}$-modules and universal Lie $\mathfrak{h}$-modules as the representation theoretic counterparts of Manin-Tambara's universal coacting objects \cite{Manin, Tambara}.	result 0.71	0.94	0.74	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v3

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper

Original abstract

Let $\mathfrak{g}$ and $\mathfrak{h}$ be two Lie algebras with $\mathfrak{h}$ finite dimensional and consider ${\mathcal A} = {\mathcal A} (\mathfrak{h}, \, \mathfrak{g})$ to be the corresponding universal algebra as introduced in \cite{am20}. Given an ${\mathcal A}$-module $U$ and a Lie $\mathfrak{h}$-module $V$ we show that $U \otimes V$ can be naturally endowed with a Lie $\mathfrak{g}$-module structure. This gives rise to a functor between the category of Lie $\mathfrak{h}$-modules and the category of Lie $\mathfrak{g}$-modules and, respectively, to a functor between the category of ${\mathcal A}$-modules and the category of Lie $\mathfrak{g}$-modules. Under some finite dimensionality assumptions, we prove that the two functors admit left adjoints which leads to the construction of universal ${\mathcal A}$-modules and universal Lie $\mathfrak{h}$-modules as the representation theoretic counterparts of Manin-Tambara's universal coacting objects \cite{Manin, Tambara}.

Human review

How Pith checks papers → · Full audit of this paper →

How the literature uses it

1 arXiv paper cite this in 1 sentence

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Discussion

Know something about this paper? Leave a comment, or highlight any passage above to comment on it.