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There exist a large family of noncommutative spaces that arise from the quantum matrices. These purely algebraic objects q-deform the coordinate rings of certain complex varieties. For example, take the quantum SL(2), this is the algebra $\mathbb{C} < a,b,c,d\ >$ quotiented by the ideal generated by $$ab - qba, ~~ ac - qca, ~~ bc - cb, ~~ bd - qdb, ~~ cd - qdc, ~~ ad - da - (q - q^{-1})bc\$$ and the "q-det" relation $$ad -qbc - 1.$$ where $q$ is some complex number. Clearly, when $q=1$ we get back the coordinate ring of $SU(2)$. Other members of this family include quantum $SU(n)$, for $n >2$, quantum $U(2)$, quantum $\mathbb{CP}^n$, quantum Grassamnnian space, quantum spheres .....

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The basic quantum matrices originally arose from the work of Lenigrad physicists on the inverse scattering problem, and so has a completely different origin to operator algebraic structures. While it is possiple to put norms and involutions on all these onjects algebraic objects and complete them to $C^*$-algebras, but they are still very interesting and well studied objects in their own right. For more information see Shahn Majid's book: A quantum groups primer, or the paper link text.

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There exist a large family of noncommutative spaces that arise from the quantum matrices. These purely algebraic objects q-deform the coordinate rings of certain complex varieties. For example, take the quantum SL(2), this is the algebra $\mathbb{C} < a,b,c,d\ >$ quotiented by the ideal generated by $$ab - qba, ~~ ac - qca, ~~ bc - cb, ~~ bd - qdb, ~~ cd - qdc, ~~ ad - da - (q - q^{-1})bc\$$ and the "q-det" relation $$ad -qbc - 1.$$ where $q$ is some complex number. Clearly, when $q=1$ we get back the coordinate ring of $SU(2)$. Other members of this family include quantum $SU(n)$, for $n >2$, quantum $U(2)$, quantum $\mathbb{CP}^n$, quantum Grassamnnian space, quantum spheres .....

It is possiple to put norms and involutions on these onjects and complete them to $C^*$-algebras, but they are very interesting and well studied objects in their own right. For more information see Shahn Majid's book: A quantum groups primer, or the paper link text.