Noncommutative cohomology and electromagnetism on $C_q[SL_2]$ at roots of unity

Noncommutative Ricci curvature and Dirac operator on $C_q[SL_2]$ at roots of unity

We find a unique torsion free Riemannian spin connection for the natural Killing metric on the quantum group $C_q[SL_2]$, using a recent frame bundle formulation. We find that its covariant Ricci curvature is essentially proportional to the metric (i.e. an Einstein space). We compute the Dirac operator and find for $q$ an odd $r$'th root of unity that its eigenvalues are given by $q$-integers $[m]_q$ for $m=0,1,...,r-1$ offset by the constant background curvature. We fully solve the Dirac equation for $r=3$.

Electromagnetism and Gauge Theory on the Permutation Group $S_3$

Using noncommutative geometry we do U(1) gauge theory on the permutation group $S_3$. Unlike usual lattice gauge theories the use of a nonAbelian group here as spacetime corresponds to a background Riemannian curvature. In this background we solve spin 0, 1/2 and spin 1 equations of motion, including the spin 1 or `photon' case in the presence of sources, i.e. a theory of classical electromagnetism. Moreover, we solve the U(1) Yang-Mills theory (this differs from the U(1) Maxwell theory in noncommutative geometry), including the moduli spaces of flat connections. We show that the Yang-Mills ac…

Aspects of a new class of braid matrices: roots of unity and hyperelliptic $q$ for triangularity, L-algebra,link-invariants, noncommutative spaces

Various properties of a class of braid matrices, presented before, are studied considering $N^2 \times N^2 (N=3,4,...)$ vector representations for two subclasses. For $q=1$ the matrices are nontrivial. Triangularity $(\hat R^2 =I)$ corresponds to polynomial equations for $q$, the solutions ranging from roots of unity to hyperelliptic functions. The algebras of $L-$ operators are studied. As a crucial feature one obtains $2N$ central, group-like, homogenous quadratic functions of $L_{ij}$ constrained to equality among themselves by the $RLL$ equations. They are studied in detail for $N =3$ and …

Noncommutative discriminants via Poisson primes

We present a general method for computing discriminants of noncommutative algebras. It builds a connection with Poisson geometry and expresses the discriminants as products of Poisson primes. The method is applicable to algebras obtained by specialization from families, such as quantum algebras at roots of unity. It is illustrated with the specializations of the algebras of quantum matrices at roots of unity and more generally all quantum Schubert cell algebras.

On $q$-commutative power and Laurent series rings at roots of unity

We continue the first and second authors' study of $q$-commutative power series rings $R=k_q[[x_1,\ldots,x_n]]$ and Laurent series rings $L=k_q[[x^{\pm 1}_1,\ldots,x^{\pm 1}_n]]$, specializing to the case in which the commutation parameters $q_{ij}$ are all roots of unity. In this setting, $R$ is a PI algebra, and we can apply results of De Concini, Kac, and Procesi to show that $L$ is an Azumaya algebra whose degree can be inferred from the $q_{ij}$. Our main result establishes an exact criterion (dependent on the $q_{ij}$) for determining when the centers of $L$ and $R$ are commutative Laure…

单位根处的量子格拉斯曼流形及其相关的量子舒伯特簇

Quantum Grassmannians and their Associated Quantum Schubert Varieties at roots of unity

本文研究了单位根处量子代数的 PI degree，通过证明相关矩阵不变因子为 2 的幂来显式计算分划子代数的 PI degree，并利用非交换去齐次化方法推导出量子舒伯特簇和量子格拉斯曼流形的 PI degree。