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Clifford algebra analogue of Cartan's theorem for symmetric pairs

Kieran Calvert, Karmen Grizelj, Andrey Krutov, Pavle Pandžić

TL;DR

The paper extends Kostant's Clifford-algebra framework from the absolute to the relative setting of symmetric pairs $( rak g, rak k)$, describing $K$-invariants in $Cl( rak p)$ and establishing a Clifford-analytic Cartan theorem for primary and almost primary cases. It develops a relative transgression theory via (non)commutative and quantum Weil algebras, yielding primitive invariants and Harish-Chandra projections $ ext{hc}_w$ that connect $Cl( rak p)^{ rak k}$ to $Cl( rak a)$. A central result is the factorization $Cl( rak p)^{ rak k} \, ext{ ≅ }\ Cl(P_{ ext{Cl}}( rak p), ilde B) \, ensor\ Pr(S)$, together with a concrete description of $Pr(S)$ via projection operators on the spin module and an explicit identification with $ ext{Cl}( rak a)$ in favorable cases. The paper also proves a relative version of Kostant's Clifford algebra conjecture, showing the filtrations on $ rak a$ induced by principal $ rak{sl}_2$-actions align with the grading coming from primitives, thereby extending the classical Cartan/Hopf–Koszul–Samelson picture to the relative setting. Overall, these results clarify the structure of relative Clifford-invariant algebras and provide explicit, computable tools for primitive invariants and Harish-Chandra projections in symmetric-pair contexts with potential disconnected subgroups.

Abstract

We extend Kostant's results about $\mathfrak{g}$-invariants in the Clifford algebra $Cl(\mathfrak{g})$ of a complex semisimple Lie algebra $\mathfrak{g}$ to the relative case of $\mathfrak{k}$-invariants in the Clifford algebra $Cl(\mathfrak{p})$, where $(\mathfrak{g},\mathfrak{k})$ is a classical symmetric pair and $\mathfrak{p}$ is the $(-1)$-eigenspace of the corresponding involution. In this setup we prove the Cartan theorem for Clifford algebras, a relative transgression theorem, the Harish--Chandra isomorphism for $Cl(\mathfrak{p})$, and a relative version of Kostant's Clifford algebra conjecture.

Clifford algebra analogue of Cartan's theorem for symmetric pairs

TL;DR

The paper extends Kostant's Clifford-algebra framework from the absolute to the relative setting of symmetric pairs , describing -invariants in and establishing a Clifford-analytic Cartan theorem for primary and almost primary cases. It develops a relative transgression theory via (non)commutative and quantum Weil algebras, yielding primitive invariants and Harish-Chandra projections that connect to . A central result is the factorization , together with a concrete description of via projection operators on the spin module and an explicit identification with in favorable cases. The paper also proves a relative version of Kostant's Clifford algebra conjecture, showing the filtrations on induced by principal -actions align with the grading coming from primitives, thereby extending the classical Cartan/Hopf–Koszul–Samelson picture to the relative setting. Overall, these results clarify the structure of relative Clifford-invariant algebras and provide explicit, computable tools for primitive invariants and Harish-Chandra projections in symmetric-pair contexts with potential disconnected subgroups.

Abstract

We extend Kostant's results about -invariants in the Clifford algebra of a complex semisimple Lie algebra to the relative case of -invariants in the Clifford algebra , where is a classical symmetric pair and is the -eigenspace of the corresponding involution. In this setup we prove the Cartan theorem for Clifford algebras, a relative transgression theorem, the Harish--Chandra isomorphism for , and a relative version of Kostant's Clifford algebra conjecture.
Paper Structure (24 sections, 52 theorems, 255 equations)

This paper contains 24 sections, 52 theorems, 255 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Contractions and bilinear forms
  4. Clifford algebras
  5. Symmetric pairs of Lie algebras
  6. Types of symmetric pairs
  7. Algebra structure of $\mathop{\mathrm{Cl}}\nolimits({\mathfrak p})^{\mathfrak k}$ as endomorphisms of spin modules
  8. Transgressions in Weil algebras
  9. $G$-differential algebras
  10. The commutative Weil algebra
  11. The noncommutative Weil algebra
  12. The quantum Weil algebra
  13. Transgressions in relative Weil algebras
  14. Relative noncommutative Weil algebra
  15. Transgression in the noncommutative relative Weil algebra
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $(G,K)$ be a compact symmetric pair such that $G$ is simple and connected. Assume that $({\mathfrak g},{\mathfrak k})$ is different from $(\mathfrak{e}(6),\mathfrak{sp}(8))$. (a) With the above notation, the inclusion $P_{\mathop{\mathrm{Cl}}\nolimits}(\mathfrak{p}) \hookrightarrow \mathop{\math which is an isomorphism in the primary cases, i.e., when the spin module $S$ contains only one $\ma

Theorems & Definitions (116)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8
  • proof
  • Lemma 2.9
  • ...and 106 more