2025/12/04 by Hironori Oya, Oya, Hironori, Qin Fan +3 · 1 citation
Mathematics · #13F60 #17B37 #20G15 #20G42 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2512.05228
openalex publication_date 2025/12/04 · openalex created_date 2025/12/09 · openalex updated_date 2026/07/28
We develop (quantum) cluster algebra structures over arbitrary commutative unital rings \Bbbk and prove that the (quantized) coordinate rings of connected simply-connected complex simple algebraic groups G over \Bbbk admit such structures. We first show that the integral form of the quantized coordinate ring of G admits an upper quantum cluster algebra structure over \mathbbA=ℤ[q±(1)/(2)] by using a combination of tools from quantum groups, canonical bases and cluster algebras and a previous result of the second and third authors over ℚ(q(1)/(2)). We then obtain (integral) quantum versions of recent results of the first author: when G is not of type F4, the quantized coordinate ring of G admits a quantum cluster algebra structure over \mathbbA', where \mathbbA'=\mathbbA when G is not of types G2, E8, and F4; \mathbbA'=\mathbbA[(q2+1)-1] when G is of type G2, and \mathbbA'=ℚ(q(1)/(2)) when G is of type E8. We furthermore prove that the classical versions of these results hold over \mathbbA' (where \mathbbA'=ℤ if G is not of type F4 or G2 and \mathbbA'=ℤ[(1)/(2)] if G is of type G2) and that the integral form of the coordinate ring of G of type F4 is an upper cluster algebra. Finally, by using common triangular bases of (quantum) cluster algebras, we prove that the above results also hold under specializations of \mathbbA and \mathbbA' to commutative unital rings \Bbbk.