The crystal limit <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$C(K_{0})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of the q-family of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$C^{}$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mi>C</mi> <mrow /> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> </math></EquationSource> </InlineEquation>-algebras <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$C(K_{q})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mi>q</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> was introduced by Giri and Pal for all <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$K=SU(n+1),\, n\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>=</mo> <mi>S</mi> <mi>U</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="0.166667em" /> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. This article aims to prove that the crystal limit <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$C(K_{0})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> has the property that the representations of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$C(K_{q})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mi>q</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> give rise to a representation of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$C(K_{0})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> by sending generators of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$C(K_0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> to the limit of (scaled) generators of <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$C(K_q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mi>q</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and every representation of <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$C(K_0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> occurs in this way. This work addresses a question raised by Giri and Pal (J. Noncommut. Geom. 20 (2026), no. 2, 657–703). As a consequence, one can realize <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$C(K_{0})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> as the <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$C^{}$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mi>C</mi> <mrow /> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> </math></EquationSource> </InlineEquation>-algebra generated by the limit operators of faithful representations of <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$C(K_{q})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mi>q</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.

On representations of the crystallization of the quantized function algebra $\varvec{C(SU_{q}(n+1))}$

期刊论文
发表日期：

2026年7月15日

查看全文

Saikat Das,
Ayan Dey

摘要

The crystal limit $C(K_{0})$ $C (K_{0})$ of the q-family of $C^{*}$ $C^{*}$ -algebras $C(K_{q})$ $C (K_{q})$ was introduced by Giri and Pal for all $K=SU(n+1),\, n\ge 2$ $K = S U (n + 1), n \geq 2$ . This article aims to prove that the crystal limit $C(K_{0})$ $C (K_{0})$ has the property that the representations of $C(K_{q})$ $C (K_{q})$ give rise to a representation of $C(K_{0})$ $C (K_{0})$ by sending generators of $C(K_0)$ $C (K_{0})$ to the limit of (scaled) generators of $C(K_q)$ $C (K_{q})$ and every representation of $C(K_0)$ $C (K_{0})$ occurs in this way. This work addresses a question raised by Giri and Pal (J. Noncommut. Geom. 20 (2026), no. 2, 657–703). As a consequence, one can realize $C(K_{0})$ $C (K_{0})$ as the $C^{*}$ $C^{*}$ -algebra generated by the limit operators of faithful representations of $C(K_{q})$ $C (K_{q})$ .

On representations of the crystallization of the quantized function algebra \(\varvec{C(SU_{q}(n+1))}\)

摘要