The Hidden Geometry of the Quantum Euclidean Space

The Geometry of the Quantum Euclidean Space

A detailed study is made of the noncommutative geometry of $R^3_q$, the quantum space covariant under the quantum group $SO_q(3)$. For each of its two $SO_q(3)$-covariant differential calculi we find its metric, the corresponding frame and two torsion-free covariant derivatives that are metric compatible up to a conformal factor and which yield both a vanishing linear curvature. A discussion is given of various ways of imposing reality conditions. The delicate issue of the commutative limit is discussed at the formal algebraic level. Two rather different ways of taking the limit are suggested,…

基于欧氏几何的新型量子 LDPC 码

New Quantum LDPC Codes Based on Euclidean Geometry

该研究利用欧氏几何中非原点的点、不过原点的直线及平行类，给出 QLDPC 码的构造条件并得到两类新型量子 LDPC 码。

Infinite dimensional geometry of $M_1=Diff_+(S^1)/PSL(2,R)$ and $q_R$-conformal symmetries. II. Geometric quantization and hidden symmetries of Verma modules over Virasoro algebra

Some natural hidden symmetries in the Verma modules over the Virasoro algebra are constructed in terms of geometric quantization. Their differential geometric meaning is established and their expression via $q_R$-conformal symmetries in the Verma modules over the Lie algebra $sl(2,C)$ is found. The analysis and the unraveling of the algebraic structure of these families of hidden symmetries are performed.

6j symbols for U_q(sl_2) and non-Euclidean tetrahedra

We relate the semiclassical asymptotics of the 6j symbols for the representation theory of the quantized enveloping algebra U_q(sl_2) at q a primitive root of unity, or q positive real, to the geometry of non-Euclidean tetrahedra. The formulas are motivated by the geometry of conformal blocks in the Wess-Zumino-Witten model; they generalize formulas in the case q = 1 of Wigner, Ponzano and Regge, and Schulten and Gordon, proved by J. Roberts.