Grant-in-Aid for Scientific Research (C)
|Allocation Type||Single-year Grants|
General mathematics (including Probability theory/Statistical mathematics)
|Research Institution||Osaka Prefecture University(2005)|
Osaka Women's University(2003-2004)
AIZAWA Naruhiko Osaka Prefecture University, Graduate School of Science, Assistant Professor, 理学系研究科, 助教授 (70264786)
ISHIHARA Kazuo Osaka Prefecture University, Graduate School of Science, Professor, 理学系研究科, 教授 (90090563)
IRIYE Kouyemon Osaka Prefecture University, Graduate School of Science, Professor, 理学系研究科, 教授 (40151691)
O'UCHI Moto Osaka Prefecture University, Graduate School of Science, Professor, 理学系研究科, 教授 (70127885)
KATO Kiriko Osaka Prefecture University, Graduate School of Science, Assistant Professor, 理学系研究科, 助教授 (00347478)
WATAMORI Yoko Osaka Prefecture University, Graduate School of Science, Assistant Professor, 理学系研究科, 助教授 (70240538)
|Project Period (FY)
2003 – 2005
Completed(Fiscal Year 2005)
|Budget Amount *help
¥3,200,000 (Direct Cost : ¥3,200,000)
Fiscal Year 2005 : ¥800,000 (Direct Cost : ¥800,000)
Fiscal Year 2004 : ¥1,000,000 (Direct Cost : ¥1,000,000)
Fiscal Year 2003 : ¥1,400,000 (Direct Cost : ¥1,400,000)
|Keywords||boson algebras / Hopf algebras / Representation theory / Quantum groups / Non-commutative geometry / Special functions / 国際研究者交流(インド) / リー代数の量子化 / コヒーレント状態 / weak Hopf代数|
The purposes of this research project are three fold. Firstly, seeking possibilities of generalizing the boson algebra in the framework of Hopf algebra. Secondly, investigating properties of the generalized boson algebras. Then, applying the generalized boson algebras to some mathematical or physical problems. One of the generalized boson algebras, already known, is regarded as bosonization of the quantum algebra U_q[osp(1/2)]. We, thus, started our investigation with the algebra (denoted by U) and U_q[osp(1/2)]. The main results of the present research are summarized as follows.
1.The algebra dual to U was obtained. The duality is expressed in the form of universal T-matrix.
2.The single-particle and bipartite coherent states for U were constructed. It was shown that the bipartite coherent state had entanglement which disappears in the classical limit. An analytic proof of orthogonality and completeness for the single-particle coherent state was given.
3.Irreducible representations of th
e algebra dual to U were obtained explicitly and it was shown that the representation matrices are related to little q-Jacobi polynomials.
4.Tensor product of the representations of U is decomposed into irreducible ones. It was shown that the decomposition was carried out with q-Hahn polynomials. The result No.3 and 4 imply that the algebra U gives a new algebraic background for basic hypergeometri functions.
5.A general method for constructing noncommutative space with supersymmetric nature, which means that the space is covariant under the action of quantum group OSp_q(1/2), was introduced. By the method, 3-dimensional noncommutative flat superspace and 5-dimensional noncommutative supersphere were constructed and their properties, as well as relations to boson algebras, were studied.
6.Differential geometry on noncommutative spaces has been developed by Dubois-Viollete et al. In this research, the geometry was extended to noncommutative superspaces. As an example, covariant derivative, curvature etc on 3-dimensional noncommutative superspace, which is a covariant algebra of the Jordanian quantum group OSp_h(1/2), were computed. The computation shows that the superspace is not physical, since the covariant derivative is not compatible with the metric. Less