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We introduce a family of braided Hopf algebras that (in characteristic zero) generalizes the rank 1 Hopf algebras introduced by Krop and Radford and we study its cleft extensions.  相似文献   
3.
Yafit Natani 《代数通讯》2017,45(9):3872-3885
In this paper, we investigate the basis graph of the monoid algebra of a submonoid of the monoid of mappings from N = {1,…,n} to itself, defined by a nested sequence of compositions of N. Each such monoid is a left regular band (LRB), that is, a semigroup S satisfying x2 = x and xyx = xy for all x,yS. This class is su?ciently rich that every path algebra of an acyclic quiver can be embedded in such a monoid algebra. The multiplication in the monoid algebra has a particularly simple quasi-multiplicative form, allowing definition over the integers. Combining this with a formula for Ext-groups for LRBs due to Margolis et al. [6 Margolis, S., Saliola, F., Steinberg, B. (2015). Combinatorial topology and the global dimension of algebras arising in combinatorics. J. Eur. Math Soc. 17(12):30373080.[Crossref], [Web of Science ®] [Google Scholar]], we get a simple criterion for the nested composition algebras to be hereditary.  相似文献   
4.
We show that for some finite group block algebras, with nontrivial defect groups, the first Hochschild cohomology is nontrivial. Along the way we obtain methods to investigate the nontriviality of the first Hochschild cohomology of some twisted group algebras.  相似文献   
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In a recent paper [O. Bärwald, R.W. Gebert, M. Günaydin and H. Nicolai, preprint KCL-MTH-97-22, IASSNS-HEP-97/20, PSU-TH-178, AEI-029, hep-th/9703084, to appear in Commun. Math. Phys.] it was conjectured that the imaginary simple roots of the Borcherds algebra II9,1 at level 1 are its only ones. We here propose an independent test of this conjecture, establishing its validity for all roots of norm − 8. However, the conjecture fails for roots of norm −10 and beyond, as we show by computing the simple multiplicities down to norm −24, which turn out to be remarkably small in comparison with the corresponding E10 multiplicities. Our derivation is based on a modified denominator formula combining the denominator formulas for E10 and II9,1, and provides an efficient method for determining the imaginary simple roots. In addition, we compute the E10 multiplicities of all roots up to height 231, including levels up to l = 6 and norms −42.  相似文献   
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We define the BGP-reflection functors in the derived categories and the root categories. By Ringel's Hall algebra approach, the BGP-reflection functor is applicable to obtain the classical Weyl group action on the Lie algebra.  相似文献   
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The notion of sofic equivalence relation was introduced by Gabor Elek and Gabor Lippner. Their technics employ some graph theory. Here we define this notion in a more operator algebraic context, starting from Connes? Embedding Problem, and prove the equivalence of these two definitions. We introduce a notion of sofic action for an arbitrary group and prove that an amalgamated product of sofic actions over amenable groups is again sofic. We also prove that an amalgamated product of sofic groups over an amenable subgroup is again sofic.  相似文献   
10.
We introduce the notion of sofic measurable equivalence relations. Using them we prove that Connes' Embedding Conjecture as well as the Measurable Determinant Conjecture of Lück, Sauer and Wegner hold for treeable equivalence relations.  相似文献   
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