Equivariant spectral triples on the quantum SU(2) group. (English) Zbl 1028.58005
All equivariant odd spectral triples for \(SU_q(2)\) acting on its \(L_2\)-space and having a nontrivial Chern character are characterized. Applying the method used to derive this result, the ordinary Dirac operator on \(SU(2)\) is shown to be minimal in a certain sense.
To derive the result, first representation theory of \(SU_q(2)\) is reviewed in Sect. 2. Let \(t_{ij}^{(n)}\) be the \(ij\)th entry of \(t^n\), \(n\in\{0,{1\over 2},1,\dots\}\) the unique irreducible unique representation of \(SU_q(2)\) of dimension \(2n+1\). They belong to \({\mathcal A}_f\), the dense *-subalgebra of \({\mathcal A}\), the \(C^*\)-algebra of continuous functions on \(SU_q(2)\), generated by \(\alpha\) and \(\beta\), the generators of \(SU_q (2)\). Then the GNS space associated with the Haar state is spanned by \(e^{(n)}_{ij}\), the normalized \(t_{ij}^{(n)}\)’s. The generators of regular representation of \(SU_q(2)\) are denoted by \(A_0\), \(A_1\) \(({\mathbf a}\) and \({\mathbf n}\) in P. Podleś and S. L. Woronowicz, Commun. Math. Phys. 130, 381-431 (1990; Zbl 0703.22018)]. An operator \(T\) on \({\mathcal H}\) is said to be equivariant if it commutes with \(A_0\), \(A_1\) and \(A_1^*\). Since an equivaliant self-adjoint operator with discrete spectrum has the form \[ D:e_{i j}^{(n)}\to d(n,i) e^{(n)}_{ij}, \] the equivariant Dirac operator is characterized in terms of \(d(n,i)\) as follows: \(({\mathcal A},{\mathcal H},D)\) is an equivariant odd spectral triple with nontrivial Chern character if and only if \(D\) satisfies \[ d\Bigl(n+ \tfrac 12,i+ \tfrac 12\Bigr)- d(n,i)=O(1),\;d\Bigl(n +\tfrac 12,i- \tfrac 12\Bigr)- d(n,i)=O (n+i+1), \] and conditions on \[ \begin{aligned} S(m,n,r) & =\left\{ d\left({j+r \over 2},{j-r \over 2}\right): j>n\right\},\;0 \leq r\leq m,\\ T(m) & =\left\{ d\left({j+i \over 2},{j-i \over 2}\right): i>m,j \geq 0\right\}. \end{aligned} \] (Th. 4.6). Conditions on \(S(m,n,r)\) and \(T(m)\) are also described.
It is also shown that \(SU_q(2)\) admits an equivariant odd 3-summable spectral triple, but does not admit \(p\)-summable spectral triples for \(p<3\) (Th. 3.5 and 3.6). In Sect. 5, the last Section, as the adaptation of previous methods, it is shown if \((C(SU(2)\), \(L_2(SU(2)),D)\) is an equivariant spectral triple and \(D\) has nontrivial sign, then \(D\) cannot be \(p\)-summable for \(p<4\) (Th. 5.4).
To derive the result, first representation theory of \(SU_q(2)\) is reviewed in Sect. 2. Let \(t_{ij}^{(n)}\) be the \(ij\)th entry of \(t^n\), \(n\in\{0,{1\over 2},1,\dots\}\) the unique irreducible unique representation of \(SU_q(2)\) of dimension \(2n+1\). They belong to \({\mathcal A}_f\), the dense *-subalgebra of \({\mathcal A}\), the \(C^*\)-algebra of continuous functions on \(SU_q(2)\), generated by \(\alpha\) and \(\beta\), the generators of \(SU_q (2)\). Then the GNS space associated with the Haar state is spanned by \(e^{(n)}_{ij}\), the normalized \(t_{ij}^{(n)}\)’s. The generators of regular representation of \(SU_q(2)\) are denoted by \(A_0\), \(A_1\) \(({\mathbf a}\) and \({\mathbf n}\) in P. Podleś and S. L. Woronowicz, Commun. Math. Phys. 130, 381-431 (1990; Zbl 0703.22018)]. An operator \(T\) on \({\mathcal H}\) is said to be equivariant if it commutes with \(A_0\), \(A_1\) and \(A_1^*\). Since an equivaliant self-adjoint operator with discrete spectrum has the form \[ D:e_{i j}^{(n)}\to d(n,i) e^{(n)}_{ij}, \] the equivariant Dirac operator is characterized in terms of \(d(n,i)\) as follows: \(({\mathcal A},{\mathcal H},D)\) is an equivariant odd spectral triple with nontrivial Chern character if and only if \(D\) satisfies \[ d\Bigl(n+ \tfrac 12,i+ \tfrac 12\Bigr)- d(n,i)=O(1),\;d\Bigl(n +\tfrac 12,i- \tfrac 12\Bigr)- d(n,i)=O (n+i+1), \] and conditions on \[ \begin{aligned} S(m,n,r) & =\left\{ d\left({j+r \over 2},{j-r \over 2}\right): j>n\right\},\;0 \leq r\leq m,\\ T(m) & =\left\{ d\left({j+i \over 2},{j-i \over 2}\right): i>m,j \geq 0\right\}. \end{aligned} \] (Th. 4.6). Conditions on \(S(m,n,r)\) and \(T(m)\) are also described.
It is also shown that \(SU_q(2)\) admits an equivariant odd 3-summable spectral triple, but does not admit \(p\)-summable spectral triples for \(p<3\) (Th. 3.5 and 3.6). In Sect. 5, the last Section, as the adaptation of previous methods, it is shown if \((C(SU(2)\), \(L_2(SU(2)),D)\) is an equivariant spectral triple and \(D\) has nontrivial sign, then \(D\) cannot be \(p\)-summable for \(p<4\) (Th. 5.4).
Reviewer: Akira Asada (Matumoto)
MSC:
| 58B34 | Noncommutative geometry (à la Connes) |
| 46L87 | Noncommutative differential geometry |
| 47B99 | Special classes of linear operators |
