LECTURES ON QUANTUM GROUPS (Graduate Studies in Mathematics 6) By Jens Carsten Jantzen: 266 pp., US$44.00, ISBN 0 8218 0478 2 (American Mathematical Society, 1996).

We propose several constructions of commutative or cocommutative Hopf algebras based on various combinatorial structures, and investigate the relations between them. A commutative Hopf algebra of permutations is obtained by a general construction based on graphs, and its non-commutative dual is realized in three different ways, in particular as the Grossman-Larson algebra of heap ordered trees. 
Extensions to endofunctions, parking functions, set compositions, set partitions, planar binary trees and rooted forests are discussed. Finally, we introduce one-parameter families interpolating between different structures constructed on the same combinatorial objects.

Commutative combinatorial Hopf algebras

In this paper, we study the Green ring (or the representation ring) of Drinfeld quantum double $D(H_4)$ of Sweedler's 4-dimensional Hopf algebra $H_4$. We first give the decompositions of the tensor products of finite dimensional indecomposable modules into the direct sum of indecomposable modules over $D(H_4)$. Then we describe the structure of the Green ring $r(D(H_4))$ of $D(H_4)$ and show that $r(D(H_4))$ is generated, as a ring, by infinitely many elements subject to a family of relations.

The Green Ring of Drinfeld Double $D(H_4)$

Bergeron and Li have introduced a set of axioms which guarantee that the Grothendieck groups of a tower of algebras $\bigoplus_{n\ge0}A_n$
 can be a pair of graded dual Hopf algebras. Hivert and Nzeutchap, and independently Lam and Shimozono constructed dual graded graphs from primitive elements in Hopf algebras. In this paper we apply the composition of these constructions to towers of algebras. We show that if a tower $\bigoplus_{n\ge0}A_n$
 gives rise to a pair of graded dual Hopf algebras, then $\dim(A_n)=r^nn!$
 where $r = \dim(A_1)$
 . In the case of r = 1 we give a conjectural classification. We then investigate a quantum version of the main theorem. We conclude with some open problems and a categorification of these constructions.

Combinatorial Hopf Algebras and Towers of Algebras—Dimension, Quantization and Functorality

In this paper, a generalization of the class of semicommutative rings is inves- tigated. A ring R is called central semicommutative if for any a;b 2 R, ab = 0 implies arb is a central element of R for each r 2 R. We prove that some results on semicommutative rings can be extended to central semicommutative rings for this general settings.

On a Class of Semicommutative Rings

A ring R is defined to be quasi-normal if ae = 0 implies eaRe = 0 for a ∈ N(R) and e ∈ E(R), where E(R) and N(R) stand, respectively, for the set of idempotents and the set of nilpotents of R. It is proved that R is quasi-normal if and only if eR(1 − e)Re = 0 for each e ∈ E(R) if and only if T n (R, R) is quasi-normal for any positive integer n. And it follows that for a quasi-normal ring R, R is Π-regular if and only if N(R) is an ideal of R and R/N(R) is regular. Also, using quasi-normal ring, we proved the following: (1) R is an abelian ring if and only if R is a quasi-normal left idempotent reflexive ring; (2) R is a strongly regular ring if and only if R is a von Neumann regular quasi-normal ring; (3) Let R be a quasi-normal ring. Then R is a clean ring if and only if R is an exchange ring; (4) Let R be a quasi-normal Π-regular ring. Then R is a (S,2)-ring if and only if ℤ/2ℤ is not a homomorphic image of R.

Quasi-Normal Rings

We study integral almost square-free modular categories; i.e., integral modular categories of Frobenius–Perron dimension pnm, where p is a prime number, m is a square-free natural number and gcd(p,m) = 1. We prove that, if n ≤ 5 or m is prime with m < p, then they are group-theoretical. This generalizes several results in the literature and gives a partial answer to the question posed by the first author and Tucker. As an application, we prove that an integral modular category whose Frobenius–Perron dimension is odd and less than 1125 is group-theoretical.

Integral almost square-free modular categories

Let k be an arbitrary field of characteristic zero, and U be the quantized enveloping algebra U q (sl(2)) over k. The aim of this present paper is to study the ideals of U at q not a root of unity. It turns out that every non-zero ideal of U can be generated by at most two highest weight vectors under the adjoint action, and by a sum of two highest weight vectors. This weight property make it possible to give a complete list of all prime (primitive, maximal) ideals of U according to their generators.

WEIGHT PROPERTY FOR IDEALS OF U q (sl(2))

The first author was funded by China Postdoctoral Science Foundation (Grant No. 2017M610316), the second author was funded by the NSF of China (Grant No. 11471282).

/pdf/a-criterion-for-the-jacobson-semisimplicity-of-the-green-5gt3lay72m.pdf

A criterion for the jacobson semisimplicity of the green ring of a finite tensor category

We introduce in this paper the concepts of rings characterized by minimal one-sided ideals and concern ourselves with rings containing an injective maximal left ideal. Some known results for idempotent reflexive rings and left HI rings can be extended to left MC2 rings. As applications, we are able to give some new characterizations of regular left self-injective rings with non-zero socle and extend some known results for strongly regular rings.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/C201EB8C35356F445137C4FCB122F716/S0017089509990103a.pdf/div-class-title-mc2-rings-and-wqd-rings-a-href-fn01-ref-type-fn-a-div.pdf

Mc2 rings and wqd rings

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