有圈路代数及其商代数的上同调的研究
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摘要
本文主要研究路代数及其商代数的上同调性质。此外,还研究了路范畴和完备路代数的性质。
首先,我们研究了路范畴的性质。我们讨论了路范畴上n-微分算子的性质,并研究了相应的分次李代数。另外,我们还给出了路范畴上存在非平凡的微分分次结构的充要条件。
其次,我们刻画了有圈路代数的上同调。我们得到的结论表明,一阶上同调空间的标准基的选取依赖于箭图(作为拓扑对象)的亏格。作为准备,我们首先讨论了路代数上的分次微分算子和相应的分次李代数。
第三,我们研究了容许代数的一阶上同调,容许代数可以看做基本代数的推广。我们研究了容许代数上的微分算子空间的性质,并由此得到了容许代数的一阶上同调空间的维数公式。特别地,当箭图是平面箭图时,我们还得到了与该箭图相关的无圈完全路代数和无圈截面代数的一组基。
最后,我们研究了完备路代数的上同调的性质。完备路代数可以看成一列截面代数的逆极限。由此观点,我们可以借鉴投射有限群的上同调理论来研究完备路代数的上同调。我们得到的结论表明,完备路代数的以离散双模为系数的上同调等于一列截面代数的上同调的有向极限。
The present thesis mainly concerns the cohomology theory of path alge-bras and their quotient algebras. Besides, we investigate the properties of path categories and complete path algebras.
Firstly, we consider the graded path category associated to a quiver. We investigate all n-differentials on such a category, and also study the associated graded Lie algebra. Moreover, a sufficient and necessary condition is given that ensures the graded path category admits a DG category structure.
Secondly, we characterize the first graded Hochschild cohomology of a hered-itary algebra whose Gabriel quiver is admitted to have oriented cycles. The in-teresting conclusion we have obtained shows that the standard basis of the first graded Hochschild cohomology depends on the genus of a quiver as a topological object. As preparation, we first investigate the graded differential operators on a path algebra and the associated graded Lie algebra.
Thirdly, we study the first cohomology of admissible algebras which can be seen as a generalization of basic algebras. Differential operators on an admissible algebra are studied. Based on the discussion, the dimension formula of the fist cohomology of admissible algebras is characterized. In particular, for planar quivers, the linear bases of the first cohomology of acyclic complete monomial algebras and acyclic truncated quiver algebras are constructed over the field k of characteristic0.
At last, we study the cohomology of complete path algebras. Complete path algebras can be seen as an inverse limit of a sequence of truncated path algebras. Due to this view, we can adopt the method of studying the cohomology of profinite groups. The conclusion we get shows that the cohomology of a complete path algebra with a discrete bimodule as coefficient is a direct limit of a sequence of the cohomology of truncated path algebras.
首先,我们研究了路范畴的性质。我们讨论了路范畴上n-微分算子的性质,并研究了相应的分次李代数。另外,我们还给出了路范畴上存在非平凡的微分分次结构的充要条件。
其次,我们刻画了有圈路代数的上同调。我们得到的结论表明,一阶上同调空间的标准基的选取依赖于箭图(作为拓扑对象)的亏格。作为准备,我们首先讨论了路代数上的分次微分算子和相应的分次李代数。
第三,我们研究了容许代数的一阶上同调,容许代数可以看做基本代数的推广。我们研究了容许代数上的微分算子空间的性质,并由此得到了容许代数的一阶上同调空间的维数公式。特别地,当箭图是平面箭图时,我们还得到了与该箭图相关的无圈完全路代数和无圈截面代数的一组基。
最后,我们研究了完备路代数的上同调的性质。完备路代数可以看成一列截面代数的逆极限。由此观点,我们可以借鉴投射有限群的上同调理论来研究完备路代数的上同调。我们得到的结论表明,完备路代数的以离散双模为系数的上同调等于一列截面代数的上同调的有向极限。
The present thesis mainly concerns the cohomology theory of path alge-bras and their quotient algebras. Besides, we investigate the properties of path categories and complete path algebras.
Firstly, we consider the graded path category associated to a quiver. We investigate all n-differentials on such a category, and also study the associated graded Lie algebra. Moreover, a sufficient and necessary condition is given that ensures the graded path category admits a DG category structure.
Secondly, we characterize the first graded Hochschild cohomology of a hered-itary algebra whose Gabriel quiver is admitted to have oriented cycles. The in-teresting conclusion we have obtained shows that the standard basis of the first graded Hochschild cohomology depends on the genus of a quiver as a topological object. As preparation, we first investigate the graded differential operators on a path algebra and the associated graded Lie algebra.
Thirdly, we study the first cohomology of admissible algebras which can be seen as a generalization of basic algebras. Differential operators on an admissible algebra are studied. Based on the discussion, the dimension formula of the fist cohomology of admissible algebras is characterized. In particular, for planar quivers, the linear bases of the first cohomology of acyclic complete monomial algebras and acyclic truncated quiver algebras are constructed over the field k of characteristic0.
At last, we study the cohomology of complete path algebras. Complete path algebras can be seen as an inverse limit of a sequence of truncated path algebras. Due to this view, we can adopt the method of studying the cohomology of profinite groups. The conclusion we get shows that the cohomology of a complete path algebra with a discrete bimodule as coefficient is a direct limit of a sequence of the cohomology of truncated path algebras.
引文
[1]G. Ames, L. Cagliero and P. Tirao, Comparison morphisms and the Hochschild cohomology ring of truncated quiver algebras, J. Algebra 322 (2009):1466-1497.
[2]I. Assem, D. Simson and A. Skowronski, Elements of the Representation Theory of Associative Algebras Vol I:Techniques of Representation The-ory, London Mathematical Society Student Texts 65, Cambridge University Press, Cambridge,2006.
[3]M. Auslander, I. Reiten, S. O. Smal(?), Representation Theory of Artin Alge-bra, Cambridge University Press, Cambridge,1995.
[4]M. J. Bardzell, The alternating syzygy behavior of monomial algebras, J. Al-gebra 188 (1997), no.1,69-89.
[5]B. Bollobas, Modern Graph Theory, Graduate Texts in Mathematics 184, Springer-Verlag,1998.
[6]M.J. Bardzell, A.C. Locateli, E.N. Marcos, On the Hochschild cohomology of truncated cycle algebras, Comm in algebra,28(3),2000,1615-1639.
[7]W. Crawley-Boevey, Lectures on representations of quivers, http://wwwl.maths. leeds.ac.uk/-pmtwc/quivlecs.pdf.
[8]H. Carton, Eilenberg, Homological Algebra, Princeton:Princeton University Press,1956.
[9]C. Cibils, On the Hochschild cohomology of finite dimensional algebras, Comm. Algebra 16(1998),645-649.
[10]C. Cibils, Rigid monomial algebras, Math. Ann.289,95-109 (1991).
[11]C. Cibis, Rigidity of truncated quiver algebras, Adv. Math.79 (1990) 1842.
[12]L.Corwin, Y.Ne'eman, S.Stemnberg, Graded Lie algebras in mathematics and physics (Bose-Fermi symmetry), Rev. Modern Phys.47 (1975),573-604.
[13]C. Cibils, M. Rosso, Hopf quivers, J. Algebra 254 (2002),241-251.
[14]B. Deng, J. Du, B. Parshall, J. Wang, Finit Dimensional Algebras and Quan-tum Groups, Mathematical Surveys and Monographs,150, American Math-ematical Society, Providence, RI,2008.
[15]H. Derksen, J. Weyman, A. Zelevinsky, Quivers with potentials and their representations I:Mutations, Sel.math, New ser 14 (2008):59-119.
[16]M. Gerstenhabor, The cohomology structure of an associative ring, Ann. of Math.(2) 78 (1963),267-288.
[17]M. Gerstenhaber, On the deformation of rings and algebras, Ann. of Math. 79(1)(1964),59-103.
[18]L. Guo and F. Li, Structure of Hochschild cohomology of path algebras and differential formulation of Euler's polyhedron formula, arxiv:1010.1980v2.
[19]P. Gabriel, A.V. Roiter, Representations of finite dimensional algebras, Al-gebra VIII, Encycl. Math. Stud.,73, Springer (1992).
[20]J. L. Gross, T. W. Tucker, Topological Graph Theory, John Wiley and Sons, New York,1987.
[21]D. Happel, Hochschild cohomology of Finite Dimensional Algebras, Lecture Notes in Math., Springer-Verlag,1404 (1989):108-126.
[22]D. Happel, Triangulated Categories in the Representations Theory of Fi-nite Dimensional Algebras, London Mathematical Society lecture note series, vol.119, Cambridge University Press,1988.
[23]G. Hochschild, On the cohomology group of an associative algebra, Ann. of Math.,46(1945),58-67.
[24]J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate texts in mathematics 9, Springer-Verlag,1972.
[25]C. Kassel, A Runneth formula for the cyclic cohomology of Z/2-graded al-gebras, Math.Ann.275(1986),683-699.
[26]C. Kassel, Quantum Groups, Graduate texts in mathematics 155, Springer-Verlag,1995.
[27]C. Kassel, V. Turaev, Braid Groups, Graduate texts in mathematics 247, Springer-Verlag,2008.
[28]B. Keller, On differential graded categories, International Congress of Math-ematicians. Vol. II,151-190, Eur. Math. Soc, Zurich,2006.
[29]A.C. Locateli, Hochschild cohomology of truncated quiver algebras, Comm.Algebra,1999,27:645-664.
[30]J.L. Loday, Cyclic Homology, Grundlehren der mathematischen Wis-senschaften 301, Springer-Verlag,1992.
[31]Z.A. Lykova, Cyclic cohomology of projective limits of topological algebras, Proceedings of the Edinburgh Mathematical Society (2006) 49,173-199.
[32]F. Li, D. Tan, Graded Hochschild cohomology of a path algebra with oriented cycles,submitted.
[33]F. Li, D. Tan, Differential graded structures on path categories, accepted.
[34]F. Li, D. Tan, On the first Hochschild cohomology of admissible alge-bras,submitted.
[35]S. MacLane, Categories for the Working Mathematician, Graduate texts in mathematics 5, Springer-Verlag,1971.
[36]S. Montgomery, Hopf Algebras and Their Actions on Rings, CBMS Lecture in Math. Vol.82. Providence, RI.1993.
[37]J.C. McConnell, J.C. Robson, Noncommutative Noetherian Rings, Graduate Studies in Mathematics, Volume 30,2000.
[38]Constantin Nastasescu, F.van Oystaeyen, Methods of Graded Rings, Lecture Notes in Math.1836, Berlin:Springer-Verlag.
[39]Albert Nijenhuis, R.W. Richardson, Cohomology and deformations of alge-braic structures, Bull.Amer.Math.Soc.79(1964),406-411.
[40]A. Nijenhuis, R.W. Richardson, Cohomology and deformations in graded Lie algebras, Bull. Amer. Math. Soc.72:1-29,1966.
[41]F. Oystaeyen, P. Zhang, Quiver Hopf algebras, J. Algebra 280 (2004),577-589.
[42]J. Rotman, An Introduction to Homological Algebra, New York, San Fran-cisco, London:Academic Press,1979.
[43]S. Sanchez-Flores, The Lie module structure on the Hochschild cohomology groups of monomial algebras with radical square zero, J. Algebra,320 (2008): 4249-4269.
[44]S. Sanchez-Flores, On the semisimplicity of the outer derivations of mono-mial algebras, Comm in Algebra,39:3410-3434,2011.
[45]S. Sanchez-Flores, The Lie structure on the Hochschild cohomology of a mod-ular group algebra, J. Algebra Appl.,216(2012),718-733.
[46]G. Sardanashvily:Differential operators on a Lie and graded Lie algebras, arxiv:1004.0058vl [math-ph].
[47]S. Shatz, Profinite Groups, Arithmetic, and Geometry, Annals of Math. Study No.67.Priceton:Priceton University Press,1972.
[48]C. Strametz, The Lie algebra structure on the first Hochschild cohomology group of a monomial algebra, J. Algebra Appl.,5(3) (2006):245-270.
[49]M.E. Sweedler, Hopf Algebras, Benjamin, New York,1969.
[50]J.L. Taylor, Homology and cohomology for topological algebras, Advances in Mathematics 9,137-182 (1972).
[51]C.A. Weibel, An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics, Vol.38, Cambridge Univ. Press, Cambridge, UK, 1994.
[52]F. W.Warner, Foundations of Differentiable Manifolds and Lie Groups, Graduate texts in mathematics 94, Springer-Verlag,1983.
[53]Y.G. Xu, Y. Han and W.F. Jiang, Hochschild cohomology of truncated quiver algebras, Science in China Series A:Mathematics, Volume 50, Number 5 (2007),727-736.
[54]Y. Yao, Y. Ye, P. Zhang, Quiver Poisson algebras, J. Algebra 312 (2007), 570-589.
[55]P. Zhang, Hochschild cohomology of truncated basic cycle, Science in China Series A:Mathematics, Volume 40, Number 12 (1997),1272-1278.
[56]D.K. Zhao, Invariance of Hochschild and cyclic (co)homology of superalge-bras under graded equivalences, Comm in Algebra,38:4193-4201,2010.
[2]I. Assem, D. Simson and A. Skowronski, Elements of the Representation Theory of Associative Algebras Vol I:Techniques of Representation The-ory, London Mathematical Society Student Texts 65, Cambridge University Press, Cambridge,2006.
[3]M. Auslander, I. Reiten, S. O. Smal(?), Representation Theory of Artin Alge-bra, Cambridge University Press, Cambridge,1995.
[4]M. J. Bardzell, The alternating syzygy behavior of monomial algebras, J. Al-gebra 188 (1997), no.1,69-89.
[5]B. Bollobas, Modern Graph Theory, Graduate Texts in Mathematics 184, Springer-Verlag,1998.
[6]M.J. Bardzell, A.C. Locateli, E.N. Marcos, On the Hochschild cohomology of truncated cycle algebras, Comm in algebra,28(3),2000,1615-1639.
[7]W. Crawley-Boevey, Lectures on representations of quivers, http://wwwl.maths. leeds.ac.uk/-pmtwc/quivlecs.pdf.
[8]H. Carton, Eilenberg, Homological Algebra, Princeton:Princeton University Press,1956.
[9]C. Cibils, On the Hochschild cohomology of finite dimensional algebras, Comm. Algebra 16(1998),645-649.
[10]C. Cibils, Rigid monomial algebras, Math. Ann.289,95-109 (1991).
[11]C. Cibis, Rigidity of truncated quiver algebras, Adv. Math.79 (1990) 1842.
[12]L.Corwin, Y.Ne'eman, S.Stemnberg, Graded Lie algebras in mathematics and physics (Bose-Fermi symmetry), Rev. Modern Phys.47 (1975),573-604.
[13]C. Cibils, M. Rosso, Hopf quivers, J. Algebra 254 (2002),241-251.
[14]B. Deng, J. Du, B. Parshall, J. Wang, Finit Dimensional Algebras and Quan-tum Groups, Mathematical Surveys and Monographs,150, American Math-ematical Society, Providence, RI,2008.
[15]H. Derksen, J. Weyman, A. Zelevinsky, Quivers with potentials and their representations I:Mutations, Sel.math, New ser 14 (2008):59-119.
[16]M. Gerstenhabor, The cohomology structure of an associative ring, Ann. of Math.(2) 78 (1963),267-288.
[17]M. Gerstenhaber, On the deformation of rings and algebras, Ann. of Math. 79(1)(1964),59-103.
[18]L. Guo and F. Li, Structure of Hochschild cohomology of path algebras and differential formulation of Euler's polyhedron formula, arxiv:1010.1980v2.
[19]P. Gabriel, A.V. Roiter, Representations of finite dimensional algebras, Al-gebra VIII, Encycl. Math. Stud.,73, Springer (1992).
[20]J. L. Gross, T. W. Tucker, Topological Graph Theory, John Wiley and Sons, New York,1987.
[21]D. Happel, Hochschild cohomology of Finite Dimensional Algebras, Lecture Notes in Math., Springer-Verlag,1404 (1989):108-126.
[22]D. Happel, Triangulated Categories in the Representations Theory of Fi-nite Dimensional Algebras, London Mathematical Society lecture note series, vol.119, Cambridge University Press,1988.
[23]G. Hochschild, On the cohomology group of an associative algebra, Ann. of Math.,46(1945),58-67.
[24]J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate texts in mathematics 9, Springer-Verlag,1972.
[25]C. Kassel, A Runneth formula for the cyclic cohomology of Z/2-graded al-gebras, Math.Ann.275(1986),683-699.
[26]C. Kassel, Quantum Groups, Graduate texts in mathematics 155, Springer-Verlag,1995.
[27]C. Kassel, V. Turaev, Braid Groups, Graduate texts in mathematics 247, Springer-Verlag,2008.
[28]B. Keller, On differential graded categories, International Congress of Math-ematicians. Vol. II,151-190, Eur. Math. Soc, Zurich,2006.
[29]A.C. Locateli, Hochschild cohomology of truncated quiver algebras, Comm.Algebra,1999,27:645-664.
[30]J.L. Loday, Cyclic Homology, Grundlehren der mathematischen Wis-senschaften 301, Springer-Verlag,1992.
[31]Z.A. Lykova, Cyclic cohomology of projective limits of topological algebras, Proceedings of the Edinburgh Mathematical Society (2006) 49,173-199.
[32]F. Li, D. Tan, Graded Hochschild cohomology of a path algebra with oriented cycles,submitted.
[33]F. Li, D. Tan, Differential graded structures on path categories, accepted.
[34]F. Li, D. Tan, On the first Hochschild cohomology of admissible alge-bras,submitted.
[35]S. MacLane, Categories for the Working Mathematician, Graduate texts in mathematics 5, Springer-Verlag,1971.
[36]S. Montgomery, Hopf Algebras and Their Actions on Rings, CBMS Lecture in Math. Vol.82. Providence, RI.1993.
[37]J.C. McConnell, J.C. Robson, Noncommutative Noetherian Rings, Graduate Studies in Mathematics, Volume 30,2000.
[38]Constantin Nastasescu, F.van Oystaeyen, Methods of Graded Rings, Lecture Notes in Math.1836, Berlin:Springer-Verlag.
[39]Albert Nijenhuis, R.W. Richardson, Cohomology and deformations of alge-braic structures, Bull.Amer.Math.Soc.79(1964),406-411.
[40]A. Nijenhuis, R.W. Richardson, Cohomology and deformations in graded Lie algebras, Bull. Amer. Math. Soc.72:1-29,1966.
[41]F. Oystaeyen, P. Zhang, Quiver Hopf algebras, J. Algebra 280 (2004),577-589.
[42]J. Rotman, An Introduction to Homological Algebra, New York, San Fran-cisco, London:Academic Press,1979.
[43]S. Sanchez-Flores, The Lie module structure on the Hochschild cohomology groups of monomial algebras with radical square zero, J. Algebra,320 (2008): 4249-4269.
[44]S. Sanchez-Flores, On the semisimplicity of the outer derivations of mono-mial algebras, Comm in Algebra,39:3410-3434,2011.
[45]S. Sanchez-Flores, The Lie structure on the Hochschild cohomology of a mod-ular group algebra, J. Algebra Appl.,216(2012),718-733.
[46]G. Sardanashvily:Differential operators on a Lie and graded Lie algebras, arxiv:1004.0058vl [math-ph].
[47]S. Shatz, Profinite Groups, Arithmetic, and Geometry, Annals of Math. Study No.67.Priceton:Priceton University Press,1972.
[48]C. Strametz, The Lie algebra structure on the first Hochschild cohomology group of a monomial algebra, J. Algebra Appl.,5(3) (2006):245-270.
[49]M.E. Sweedler, Hopf Algebras, Benjamin, New York,1969.
[50]J.L. Taylor, Homology and cohomology for topological algebras, Advances in Mathematics 9,137-182 (1972).
[51]C.A. Weibel, An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics, Vol.38, Cambridge Univ. Press, Cambridge, UK, 1994.
[52]F. W.Warner, Foundations of Differentiable Manifolds and Lie Groups, Graduate texts in mathematics 94, Springer-Verlag,1983.
[53]Y.G. Xu, Y. Han and W.F. Jiang, Hochschild cohomology of truncated quiver algebras, Science in China Series A:Mathematics, Volume 50, Number 5 (2007),727-736.
[54]Y. Yao, Y. Ye, P. Zhang, Quiver Poisson algebras, J. Algebra 312 (2007), 570-589.
[55]P. Zhang, Hochschild cohomology of truncated basic cycle, Science in China Series A:Mathematics, Volume 40, Number 12 (1997),1272-1278.
[56]D.K. Zhao, Invariance of Hochschild and cyclic (co)homology of superalge-bras under graded equivalences, Comm in Algebra,38:4193-4201,2010.