A Quantum Analogue of the First Fundamental Theorem of Classical Invariant Theory

Autor:	Ruibin Zhang, Hechun Zhang, Gustav I. Lehrer
Rok vydání:	2010
Předmět:	Pure mathematics Fundamental theorem Mathematics::Operator Algebras Quantum group Associative algebra Subalgebra Lie algebra Statistical and Nonlinear Physics Noncommutative geometry Mathematical Physics Invariant theory Vector space Mathematics
Zdroj:	Communications in Mathematical Physics. 301:131-174
ISSN:	1432-0916 0010-3616
Popis:	We establish a noncommutative analogue of the first fundamental theorem of classical invariant theory. For each quantum group associated with a classical Lie algebra, we construct a noncommutative associative algebra whose underlying vector space forms a module for the quantum group and whose algebraic structure is preserved by the quantum group action. The subspace of invariants is shown to form a subalgebra, which is finitely generated. We determine generators of this subalgebra of invariants and determine their commutation relations. In each case considered, the noncommutative modules we construct are flat deformations of their classical commutative analogues. Our results are therefore noncommutative generalisations of the first fundamental theorem of classical invariant theory, which follows from our results by taking the limit as q → 1. Our method similarly leads to a definition of quantum spheres, which is a noncommutative generalisation of the classical case with orthogonal quantum group symmetry.
Databáze:	OpenAIRE
Externí odkaz:	https://explore.openaire.eu/search/publication?articleId=doi_________::c38ee4263dac491afa487157bd8089be https://doi.org/10.1007/s00220-010-1143-3 Zobrazit plný text záznamu Full text from SpringerLink