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Allison-Benkart-Gao functor and the cyclicity of free alternative functors

Shikui Shang

TL;DR

The paper develops an adjoint pair of functors between nonunital alternative algebras and Lie algebras with $sl_3(k)$-actions, via the Allison-Benkart-Gao and Berman-Moody constructions, and shows the ABG Lie algebras differ from the Steinberg case in the nonunital setting. It focuses on free algebras $A(D)$, formulates homology- and dimension-based conjectures that connect $H_r(\mathcal{ABG}(A(D)))$ to the graded components $A(D)_n$ and ${\rm Inner}(A(D))_n$ through Grothendieck rings and $\lambda$-rings, and proves the cyclicity of the free alternative functor, linking multilinear structures to symmetry actions. The work provides explicit computations for small $D$ (notably $D=1,2$) and general $D$-dependent dimension formulas, together with a framework to derive consequences for homology and representation theory via cyclicity. Overall, the results aim to illuminate the structure of free alternative algebras and their $sl_3(k)$-graded Lie algebras by combining homological, representation-theoretic, and cyclicity techniques with adjunctions and Grothendieck–lambda calculus.

Abstract

Let $k$ be a field of characteristic $0$. We introduce a pair of adjoint functors, Allison-Benkart-Gao functor $\AG$ and Berman-Moody functor $\BM$, between the category of non-unital alternative algebras over $k$ and the category $\LieR$ of Lie algebras with compatible $sl_3(k)$-actions. Surprisingly, when $A$ is an alternative algebra without a unit, the Allison-Benkart-Gao Lie algebra $\AG(A)$ is not isomorphic to the more well-known Steinberg Lie algebra $st_3(A)$ in general. Let $A(D)$ be the free (non-unital) alternative algebra over $D$ generators with the inner derivation algebra $\innAD$. A conjecture on the homology $H_r(\AGAD)$ is proposed. Furthermore, consider the degree $n$ component of $A(D)_n$(resp. $\innAD_n$). The previous conjecture implies another conjecture on the dimensions on $A(D)_n$ and $\text{Inner} A(D)_n$. Some evidences are given to support these conjectures. Finally, we prove the cyclicity of the alternative structure, namely that the symmetric group $S_{1+D}$ acts on the multilinear part of $A(D)$, which plays an important role to connect the Lie algebra homology of $\AGAD$ and the character of $A(D)$.

Allison-Benkart-Gao functor and the cyclicity of free alternative functors

TL;DR

The paper develops an adjoint pair of functors between nonunital alternative algebras and Lie algebras with -actions, via the Allison-Benkart-Gao and Berman-Moody constructions, and shows the ABG Lie algebras differ from the Steinberg case in the nonunital setting. It focuses on free algebras , formulates homology- and dimension-based conjectures that connect to the graded components and through Grothendieck rings and -rings, and proves the cyclicity of the free alternative functor, linking multilinear structures to symmetry actions. The work provides explicit computations for small (notably ) and general -dependent dimension formulas, together with a framework to derive consequences for homology and representation theory via cyclicity. Overall, the results aim to illuminate the structure of free alternative algebras and their -graded Lie algebras by combining homological, representation-theoretic, and cyclicity techniques with adjunctions and Grothendieck–lambda calculus.

Abstract

Let be a field of characteristic . We introduce a pair of adjoint functors, Allison-Benkart-Gao functor and Berman-Moody functor , between the category of non-unital alternative algebras over and the category of Lie algebras with compatible -actions. Surprisingly, when is an alternative algebra without a unit, the Allison-Benkart-Gao Lie algebra is not isomorphic to the more well-known Steinberg Lie algebra in general. Let be the free (non-unital) alternative algebra over generators with the inner derivation algebra . A conjecture on the homology is proposed. Furthermore, consider the degree component of (resp. ). The previous conjecture implies another conjecture on the dimensions on and . Some evidences are given to support these conjectures. Finally, we prove the cyclicity of the alternative structure, namely that the symmetric group acts on the multilinear part of , which plays an important role to connect the Lie algebra homology of and the character of .
Paper Structure (20 sections, 201 equations)

This paper contains 20 sections, 201 equations.

Table of Contents

  1. Introduction
  2. The Allison-Benkart-Gao functor and Berman-Moody functor
  3. The Allison-Benkart-Gao construction
  4. The Berman-Moody functor
  5. The homology of Lie algebra ${\mathcal{ABG}}(A(D))$
  6. The (co)homology of Lie algebras in ${\text{\bf Lie}_{\mathfrak R}}$
  7. The homology of ${\mathcal{ABG}}(A(D))$
  8. The homogenous dimensions of $A(D)$ and $\mathcal{B}(A(D))$
  9. The Grothendieck rings and $\lambda$-operations
  10. $A(D)$ and $\mathcal{B}(A(D))$ in the Grothendieck rings
  11. The dimensions of $A(D)_n$ and $\mathcal{B}(A(D))_n$
  12. Evidences for the conjectures
  13. The case $D=1$
  14. The case $D=2$
  15. The cases $D\geq3$
  16. ...and 5 more sections