Quantum superalgebras at roots of unity and topological invariants of three-manifolds
Quantum superalgebras at roots of unity and topological invariants of three-manifolds
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DOI：
10.1017/s0004972700035498
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发表时间：
2006-01
影响因子：
0.7
通讯作者：
Sacha C. Blumen
中科院分区：
文献类型：
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作者：
Sacha C. Blumen
遵循Reshetikhin和Turaev的一般方法，从一类新的称为伪模Hopf代数的代数出发，发展了闭的、连通的、可定向的三维流形的拓扑不变量。伪模Hopf代数是一类推广了模Hopf代数概念的Z_2-分次带状Hopf代数。C上的量子超代数U_q(OSP(1|2n))被认为是所有整数N>；=3的本原N^次单位根。对于这样的q，U_q(OSP(1|2n))的某个左理想I也是双边Hopf理想，商代数U_q^(N)(OSP(1|2n))=U_q(OSP(1|2n))/I是Z_2-分次带状Hopf代数。对于所有n和所有N>；=3，定义了U_q^(N)(OSP(1|2n))的有限维表示的有限集合。U_q^(N)(OSP(1|2n))的每一个这样的表示都用属于截断占优Weyl腔的积分占优权重来标记。讨论了这些表示的性质：计算了每个表示的量子超维，证明了每个表示都是自对偶的，更重要的是，得到了任意多个这样的表示的张量积对于偶数N的分解.证明了当N>=6是两个奇数时，商代数U_q^(N)(OSP(1|2n))与上面讨论的有限维表示集一起构成了伪模Hopf代数.利用这个伪模Hopf代数，我们构造了一个3-流形的拓扑不变量。该不变量不同于量子SO(2n+1)在单位根处产生的3-流形的拓扑不变量。
The general method of Reshetikhin and Turaev is followed to develop topological invariants of closed, connected, orientable 3-manifolds from a new class of algebras called pseudo-modular Hopf algebras. Pseudo-modular Hopf algebras are a class of Z_2-graded ribbon Hopf algebras that generalise the concept of a modular Hopf algebra. The quantum superalgebra U_q(osp(1|2n)) over C is considered with q a primitive N^th root of unity for all integers N >= 3. For such a q, a certain left ideal I of U_q(osp(1|2n)) is also a two-sided Hopf ideal, and the quotient algebra U_q^(N)(osp(1|2n)) = U_q(osp(1|2n)) / I is a Z_2-graded ribbon Hopf algebra. For all n and all N >= 3, a finite collection of finite dimensional representations of U_q^(N)(osp(1|2n)) is defined. Each such representation of U_q^(N)(osp(1|2n)) is labelled by an integral dominant weight belonging to the truncated dominant Weyl chamber. Properties of these representations are considered: the quantum superdimension of each representation is calculated, each representation is shown to be self-dual, and more importantly, the decomposition of the tensor product of an arbitrary number of such representations is obtained for even N. It is proved that the quotient algebra U_q^(N)(osp(1|2n)), together with the set of finite dimensional representations discussed above, form a pseudo-modular Hopf algebra when N >= 6 is twice an odd number. Using this pseudo-modular Hopf algebra, we construct a topological invariant of 3-manifolds. This invariant is shown to be different to the topological invariants of 3-manifolds arising from quantum so(2n+1) at roots of unity.