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Triangular Decomposition of the Crystal Lattice of Quantized Function Algebras: Revisited

Ayan Dey

Abstract

Let $\g$ be a simple complex Lie algebra of type $G_2$, $F_4$, or $E_8$, and let $G$ be the unique connected simply connected Lie group with $\mathrm{Lie}(G)=\g$ with compact real form $K$. We prove a triangular decomposition theorem for the lower crystal lattice $\OAztG$ of the quantized function algebra $\OtG$, establishing that $\OAztG = \RAzm \cdot \RAzp$. This extends the triangular decomposition recently obtained for types $A_n, B_n, C_n, D_n, E_6$, and $E_7$ in~\cite{DDPa} to all complex simple Lie algebras. As a consequence, we obtain: (i) the inclusion $\OAztG\subseteq\OAztK$ conjectured by Matassa-Yuncken and (ii) the crystal limit $\CpKo$ is a compact quantum semigroup for all connected, simply connected, compact simple Lie groups $K$.

Triangular Decomposition of the Crystal Lattice of Quantized Function Algebras: Revisited

Abstract

Let be a simple complex Lie algebra of type , , or , and let be the unique connected simply connected Lie group with with compact real form . We prove a triangular decomposition theorem for the lower crystal lattice of the quantized function algebra , establishing that . This extends the triangular decomposition recently obtained for types , and in~\cite{DDPa} to all complex simple Lie algebras. As a consequence, we obtain: (i) the inclusion conjectured by Matassa-Yuncken and (ii) the crystal limit is a compact quantum semigroup for all connected, simply connected, compact simple Lie groups .
Paper Structure (16 sections, 6 theorems, 24 equations, 2 figures)

This paper contains 16 sections, 6 theorems, 24 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Prerequisites
  3. Lie-theoretic conventions
  4. The quantized enveloping algebra
  5. Crystal bases and global bases
  6. The quantized function algebra and its crystal lattice
  7. The algebras $R^{A_0}_+$ and $R^{A_0}_-$
  8. The compact real form $O^{A_0}_t(K)$
  9. The quasi-minuscule representations
  10. Obstruction in the quasi-minuscule case
  11. A Crystal Lemma for the Quasi-Minuscule Representation
  12. The G2 case: crystal graphs
  13. The Triangular Decomposition
  14. Consequences
  15. The inclusion $O^{A_0}_t(G) \;\subseteq\; O^{A_0}_t(K)$.
  16. ...and 1 more sections

Key Result

Lemma 3.1

Let $\mathfrak{g}$ be a complex semi-simple Lie algebra and $\varpi_i$ be a fundamental weight. Assume that $V(\varpi_i)$ appears as a direct summand of $V(\varpi_i)\otimes V(\varpi_i)$ into irreducibles. Let $x = b_{\varpi_i}\otimes c$ be a highest weight node of weight $\varpi_i$ in the tensor pro

Figures (2)

  • Figure 1: The crystal graph $B(\varpi_1)$ for $G_2$. Blue arrows are $\tilde{f}_1$-edges, red arrows are $\tilde{f}_2$-edges. The weight-zero node is $b_4$. (See Proposition 2.1, KM94)
  • Figure 2: The subcrystal $\mathcal{C}\cong B(\varpi_1)$ inside $B(\varpi_1)\otimes B(\varpi_1)$ for $G_2$. The yellow node $x_4=b_2\otimes b_6$ is the unique weight-zero node; its left factor $b_2$ has weight $\varpi_1-\alpha_1>0$ and its right factor $b_6$ has weight $-(\varpi_1-\alpha_1)<0$. The right tensor factor is $b_4$ (weight $0$) only at the top node $x_1$; from $x_2$ onward every right factor has strictly negative weight. Blue arrows are $\tilde{f}_1$-edges, red arrows are $\tilde{f}_2$-edges.

Theorems & Definitions (11)

  • Lemma 3.1
  • proof
  • Theorem 4.1
  • proof
  • Corollary 4.2
  • proof
  • Proposition 4.3: DDPa
  • Corollary 5.1
  • Remark 5.2
  • Remark 5.3
  • ...and 1 more