Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Free Jordan Algebras and Representations of sl_2(J)

Michael Lau, Olivier Mathieu

TL;DR

The paper develops a comprehensive representation-theoretic framework for the universal central extension $\widehat{\mathfrak{sl}}_2(J)$ of the TKK algebra associated to a unital Jordan algebra $J$. It introduces dominant $J$-spaces, Weyl and standard modules, and the associative envelopes ${\mathcal{U}}_n(J)$, linking combinatorial partition data to representation theory via Garland’s and Girard–Newton identities. A significant portion is devoted to categorical aspects: LA and LA$^+$ categories, extended $\nabla(n)$-objects, good filtrations, and generalized HW categories, culminating in Ext-vanishing theorems that generalize classical HW results. The paper also connects finiteness properties (finite generation/presentation, FP$_\infty$) and homological vanishing to conjectures on the growth of free Jordan algebras, aided by Zelmanov’s deep results on nil and Albert Jordan algebras. Finally, it discusses smooth $\widehat{\mathfrak{sl}}_2(J)$-modules, cohomology comparisons with pro-algebraic groups, and potential generalized HW structures for PSL$_2$-type categories, highlighting a rich web of algebraic and categorical interdependencies.

Abstract

Let J be a unital Jordan algebra and let sl_2(J) be the universal central extension of the Tits-Kantor-Koecher Lie algebra TKK(J). In part A, we study the category of (sl_2(J), SL_2(K))-modules. We characterize the dominant J-spaces, which are analogous to the classical notion of dominant highest weights. We also show some finitness results. For an augmented Jordan algebra J, we define in Part C the notion of smooth modules. We investigate the corresponding category in the spirit of Cline-Parshall-Scott highest weight categories. We show that the standard modules are finite dimensional if J is a finitely generated. Let J(D) be the free unital Jordan algebra over $D$ variables. The results of Part A suggest that the Lie algebra sl_2(J(D+1)) is FP_\infty. Similarly the results of part C suggest that the category of smooth sl_2(J(D))-modules is a generalized highest weight category. We show that both hypothesis implies the conjecture of [KM] about the dimensions of the homogenous components of $J(D)$. Surprisingly, the proofs of most results make use of some deep results of E. Zelmanov.

Free Jordan Algebras and Representations of sl_2(J)

TL;DR

The paper develops a comprehensive representation-theoretic framework for the universal central extension of the TKK algebra associated to a unital Jordan algebra . It introduces dominant -spaces, Weyl and standard modules, and the associative envelopes , linking combinatorial partition data to representation theory via Garland’s and Girard–Newton identities. A significant portion is devoted to categorical aspects: LA and LA categories, extended -objects, good filtrations, and generalized HW categories, culminating in Ext-vanishing theorems that generalize classical HW results. The paper also connects finiteness properties (finite generation/presentation, FP) and homological vanishing to conjectures on the growth of free Jordan algebras, aided by Zelmanov’s deep results on nil and Albert Jordan algebras. Finally, it discusses smooth -modules, cohomology comparisons with pro-algebraic groups, and potential generalized HW structures for PSL-type categories, highlighting a rich web of algebraic and categorical interdependencies.

Abstract

Let J be a unital Jordan algebra and let sl_2(J) be the universal central extension of the Tits-Kantor-Koecher Lie algebra TKK(J). In part A, we study the category of (sl_2(J), SL_2(K))-modules. We characterize the dominant J-spaces, which are analogous to the classical notion of dominant highest weights. We also show some finitness results. For an augmented Jordan algebra J, we define in Part C the notion of smooth modules. We investigate the corresponding category in the spirit of Cline-Parshall-Scott highest weight categories. We show that the standard modules are finite dimensional if J is a finitely generated. Let J(D) be the free unital Jordan algebra over variables. The results of Part A suggest that the Lie algebra sl_2(J(D+1)) is FP_\infty. Similarly the results of part C suggest that the category of smooth sl_2(J(D))-modules is a generalized highest weight category. We show that both hypothesis implies the conjecture of [KM] about the dimensions of the homogenous components of . Surprisingly, the proofs of most results make use of some deep results of E. Zelmanov.

Paper Structure

This paper contains 69 sections, 46 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. The Lie algebra $\widehat{\mathfrak{sl}}_2(J)$
  3. Conventions and notations
  4. Tits-Kantor-Koecher construction
  5. Tits-Allison-Gao construction
  6. $(\widehat{\mathfrak{sl}}_2(J),{\rm SL}_2)$-modules
  7. $J$-spaces
  8. Dominant $J$-spaces
  9. Joseph functors and Weyl modules
  10. Garland's formula
  11. The Girard-Newton formula
  12. The Dominance criterion
  13. The algebra ${\mathcal{U}}_n(J)$
  14. Examples
  15. The algebras ${\mathcal{U}}_n(J)$ for finite dimensional Jordan algebras $J$
  16. ...and 54 more sections

Key Result

Lemma 3.1

For any $(\mathfrak{B},B)$-module $V$, the Weyl module $\Delta(V)$ is a $(\mathfrak{G},{\rm SL}_2)$-module. Moreover, for any $(\mathfrak{G},{\rm SL}_2)$-module $X$, every morphism of $\mathfrak{B}$-modules $\phi:V\to X$ uniquely extends to a morphism of $\mathfrak{G}$-modules $\psi:\Delta(V)\to X$.

Theorems & Definitions (83)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • proof
  • Theorem 1
  • proof
  • ...and 73 more