m-循环复形范畴的Bridgeland--Hall代数的余代数结构

张海诚

数学学报 ›› 2015, Vol. 58 ›› Issue (6) : 881-896.

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数学学报 ›› 2015, Vol. 58 ›› Issue (6) : 881-896. DOI: 10.12386/A2015sxxb0088
论文

m-循环复形范畴的Bridgeland--Hall代数的余代数结构

    张海诚1,2
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Coalgebra Structure on the Bridgeland–Hall Algebra of Categories of m-Cyclic Complexes

    Hai Cheng ZHANG1,2
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摘要

设 A 是一个遗传 Abel 范畴且P 是 A 的投射对象构成的满子范畴.本文主要研究 m-循环复形范畴 Cm(P) 的 Bridgeland--Hall 代数的余代数结构 (其中 m≥2). 受 Yanagida 工作的启发, 我们在 Cm(P) 上定义一个新的正合结构, 由此得到了其 Bridgeland--Hall 代数的余代数结构.同时, 证明了存在 A 的扩展 Ringel--Hall 代数到m-循环复形范畴 Cm(P) 的 Bridgeland--Hall代数的余代数嵌入.

Abstract

Let A be a hereditary abelian category and P the full subcategory consisting of projective objects of A . We mainly study the coalgebra structure on the Bridgeland–Hall algebra of the category Cm(P) of m-cyclic complexes for any positive integer m ≥ 2. Inspired by the work of Yanagida, we define a new exact structure on Cm(P) and then obtain a coalgebra structure on the Bridgeland–Hall algebra of Cm(P). Moreover, we prove that there exists a coalgebra embedding from the extended Ringel–Hall algebra of A to the Bridgeland–Hall algebra of Cm(P).

关键词

循环复形 / Bridgeland--Hall 代数 / 余代数

Key words

cyclic complexes / Bridgeland–Hall algebra / coalgebra

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张海诚. m-循环复形范畴的Bridgeland--Hall代数的余代数结构. 数学学报, 2015, 58(6): 881-896 https://doi.org/10.12386/A2015sxxb0088
Hai Cheng ZHANG. Coalgebra Structure on the Bridgeland–Hall Algebra of Categories of m-Cyclic Complexes. Acta Mathematica Sinica, Chinese Series, 2015, 58(6): 881-896 https://doi.org/10.12386/A2015sxxb0088

参考文献

[1] Bridgeland T., Quantum groups via Hall algebras of complexes, Ann. Math., 2013, 177: 1–21.

[2] Chen Q., Deng B., Cyclic complexes, Hall polynomials and simple Lie algebras, J. Algebra, 2015, 440: 1–32.

[3] Drinfeld V. G., Quantum groups, Proceedings of the International Congress of Mathematicians (Berkeley, 1986), 798–820, Amer. Math. Soc., 1987.

[4] Green J. A., Hall algebras, hereditary algebras and quantum groups, Invent. Math., 1995, 120: 361–377.

[5] Hall P., The algebra of partitions, Proceedings of the 4th Canadian Mathematical Congress, Banff 1957, University of Toronto Press, Toronto, 1959: 147–159.

[6] Hubery A., From triangulated categories to Lie algebras: A theorem of Peng and Xiao, Trends in representation theory of algebras and related topics, Contemp. Math., 2006, 406: 51–66.

[7] Jimbo M., A q-difference analogue of U(g) and the Yang–Baxter equation, Lett. Math. Phy., 1985, 10: 63–69.

[8] Ringel C. M., Hall algebras and quantum groups, Invent. Math., 1990, 101: 583–592.

[9] Ringel C. M., Hall algebras, Topics in Algebra, Part 1, S. Balcerzyk, et al. (eds.), Banach Center Publications, 1988, 26: 433–447.

[10] Ringel C. M., The Hall algebra approach to quantum groups, Aportaciones Matemáticas Comunicaciones, 1995, 15: 85–114.

[11] Ringel C. M., Green's Theorem on Hall Algebras, Representation Theory of Algebras and Related Topics, Canad. Math. Soc. Conf. Proc. 19, Amer. Math. Soc., Providence, RI, 1996: 185–245.

[12] Ringel C. M., Zhang P., Representations of quivers over the algebra of dual numbers, arXiv: 1112.1924v2.

[13] Schiffmann O., Lectures on Hall algebras, arXiv: 0611617.

[14] Steinitz E., Zur Theorie der Abel'schen Gruppen, Jahrsber. Deutsch. Math-Verein., 1901, 9: 80–85.

[15] Töen B., Derived Hall algebras, Duke Math. J., 2006, 135: 587–615.

[16] Xiao J., Drinfeld double and Ringel–Green theory of Hall algebras, J. Algebra, 1997, 190: 100–144.

[17] Xiao J., Xu F., Hall algebras associated to triangulated categories, Duke Math. J., 2008, 143: 357–373.

[18] Yanagida S., A note on Bridgeland's Hall algebra of two-periodic complexes, arXiv: 1207.0905.

[19] Yanagida S., Bialgebra structure on Bridgeland's Hall algebra of two-periodic complexes, arXiv: 1304.6970.
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