Quantized function algebras at $q=0$: type $A_{n}$ case

Publication date: 28 March 2022

Abstract: We define the notion of quantised function algebras at

q = 0

for the

q

deformations of the type

A_{n}

compact Lie groups at the

C^{*}

-algebra level. The

C^{*}

-algebra

A_{n} (0)

is defined as a universal

C^{*}

-algebra given by a finite set of generators and relations. We obtain these relations by looking at the irreducible representations of the quantised function algebras for

q > 0

and taking limit as

q o 0 +

after rescaling the generating elements appropriately. We then focus on the case

n = 2

and prove that irreducible representations

A_{2} (0)

are precisely the

q o 0 +

limits of the irreducible representations of the

C^{*}

-algebras

A_{2} (q)

.

Mathematics Subject Classification ID

Quantum groups (quantized enveloping algebras) and related deformations (17B37) Quantum groups (quantized function algebras) and their representations (20G42) Quantum groups (operator algebraic aspects) (46L67)

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