Universal coverings of Steinberg Lie algebras of small characteristic

Second homology groups and universal coverings of Steinberg Leibniz algebras of small characteristic

It is known that the second Leibniz homology group $HL_2(stl_n(R))$ of the Steinberg Leibniz algebra $stl_n(R)$ is trivial for $n\geq 5$. In this paper, we determine $HL_2(stl_n(R))$ explicitly (which are shown to be not necessarily trivial) for $n=3, 4$ without any assumption on the base ring.

Quantum supergroups VI. Roots of $1$

A quantum covering group is an algebra with parameters $q$ and $π$ subject to $π^2=1$ and it admits an integral form; it specializes to the usual quantum group at $π=1$ and to a quantum supergroup of anisotropic type at $π=-1$. In this paper we establish the Frobenius-Lusztig homomorphism and Lusztig-Steinberg tensor product theorem in the setting of quantum covering groups at roots of 1. The specialization of these constructions at $π=1$ recovers Lusztig's constructions for quantum groups at roots of 1.

Universal coverings of Lie tori (A finite presentation)

Using the well-known recognition and structural theorem(s) for root-graded Lie algebras and their universal coverings, we give a finite presentation for the universal covering algebra of a centerless Lie torus of type $X\not=A,C,BC$. We follow a unified approach for the types under consideration.

Universal Vassiliev invariants of links in coverings of 3-manifolds

We study Vassiliev invariants of links in a 3-manifold $M$ by using chord diagrams labeled by elements of the fundamental group of $M$. We construct universal Vassiliev invariants of links in $M$, where $M=P^2\times [0,1]$ is a cylinder over the real projective plane $P^2$, $M=Σ\times [0,1]$ is a cylinder over a surface $Σ$ with boundary, and $M=S^1\times S^2$. A finite covering $p:N\longrightarrow M$ induces a map $π_1(p)^*$ between labeled chord diagrams that corresponds to taking the preimage $p^{-1}(L)\subset N$ of a link $L\subset M$. The maps $p^{-1}$ and $π_1(p)^*$ intertwine the constr…

Coverings of graded pointed Hopf algebras

We introduce the concept of a covering of a graded pointed Hopf algebra. The theory developed shows that coverings of a bosonized Nichols algebra can be concretely expressed by biproducts using a quotient of the universal coalgebra covering group of the Nichols algebra. If there are enough quadratic relations, the universal coalgebra covering is given by the bosonization by the enveloping group of the underlying rack.

量子化覆盖的非交换几何

Noncommutative Geometry of Quantized Coverings

本文提出一个统一的公理化理论，涵盖多种$C^*$-代数覆盖，并推广万有覆盖、基本群等几何构造，给出纯代数证明。