Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The three graces in the Tits--Kantor--Koecher category

Vladimir Dotsenko, Iryna Kashuba

TL;DR

This work examines algebras in the non-symmetric Tits--Kantor--Koecher category $\mathfrak{T}$ of completely reducible $\mathfrak{sl}_2$-modules, probing how far operadic intuition survives without symmetry. It proves that the main conjecture holds for free associative algebras in $\mathfrak{T}$ but fails for free commutative associative algebras, and identifies the Lie subalgebra generated by the generators of a free associative algebra with the TKC construction of the free special Jordan algebra. Through Gröbner–Shirshov theory, Anick resolutions, and homology computations, the paper relates vanishing patterns to Koszul properties and shows how PBW-type phenomena may fail in this non-symmetric setting. The results highlight deep connections to Jordan algebras, the Tits functor, and related functorial constructions, while outlining limitations and directions for braided and superalgebra generalizations.

Abstract

A metaphor of Loday describes Lie, associative, and commutative associative algebras as ``the three graces'' of the operad theory. In this article, we study the three graces in the category of $\mathfrak{sl}_2$-modules that are sums of copies of the trivial and the adjoint representation. That category is not symmetric monoidal, and so one cannot apply the wealth of results available for algebras over operads. Motivated by a recent conjecture of the second author and Mathieu, we embark on the exploration of the extent to which that category ``pretends'' to be symmetric monoidal. To that end, we examine various homological properties of free associative algebras and free associative commutative algebras, and study the Lie subalgebra generated by the generators of the free associative algebra.

The three graces in the Tits--Kantor--Koecher category

TL;DR

This work examines algebras in the non-symmetric Tits--Kantor--Koecher category of completely reducible -modules, probing how far operadic intuition survives without symmetry. It proves that the main conjecture holds for free associative algebras in but fails for free commutative associative algebras, and identifies the Lie subalgebra generated by the generators of a free associative algebra with the TKC construction of the free special Jordan algebra. Through Gröbner–Shirshov theory, Anick resolutions, and homology computations, the paper relates vanishing patterns to Koszul properties and shows how PBW-type phenomena may fail in this non-symmetric setting. The results highlight deep connections to Jordan algebras, the Tits functor, and related functorial constructions, while outlining limitations and directions for braided and superalgebra generalizations.

Abstract

A metaphor of Loday describes Lie, associative, and commutative associative algebras as ``the three graces'' of the operad theory. In this article, we study the three graces in the category of -modules that are sums of copies of the trivial and the adjoint representation. That category is not symmetric monoidal, and so one cannot apply the wealth of results available for algebras over operads. Motivated by a recent conjecture of the second author and Mathieu, we embark on the exploration of the extent to which that category ``pretends'' to be symmetric monoidal. To that end, we examine various homological properties of free associative algebras and free associative commutative algebras, and study the Lie subalgebra generated by the generators of the free associative algebra.
Paper Structure (14 sections, 11 theorems, 50 equations)

This paper contains 14 sections, 11 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and recollections
  3. The Tits--Kantor--Koecher category
  4. Some recollections on Jordan algebras
  5. Gröbner--Shirshov bases
  6. Free commutative associative algebras
  7. Free associative algebras
  8. Free Lie algebras
  9. Universal enveloping algebras: the Poincaré--Birkhoff--Witt non-theorem
  10. Some other observations
  11. Computational evidence
  12. Braided categories of the same size
  13. Koszul duality
  14. The case of superalgebras

Key Result

Proposition 2.3

The free algebra $\pazocal{P}^{\mathop{\mathrm{\mathfrak{T}}}\nolimits}(U)$ is isomorphic to the quotient of the free algebra $\pazocal{P}(U)$ by the ideal generated by all "wrong" isotypic components for the $\mathfrak{sl}_2$-action (components of the form $L(k)$ for $k\ne 0,2$).

Theorems & Definitions (27)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Remark 2.4
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Corollary 3.3
  • proof
  • ...and 17 more