Sweedler 4维Hopf代数的Rota-Baxter代数结构
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摘要
设H_4是Sweedler4维Hopf代数.本文根据Rota-Baxter算子的定义和性质,建立H_4的权为λ的Rota-Baxter算子在选定基下的矩阵元素满足的二次方程组.通过求解权λ=0时的二次齐次方程组和权λ=1时的二次非齐次方程组,给出了Rota-Baxter算子相应的矩阵形式.
Let H_4 be the Sweedler's 4-dimensional Hopf algebra. In this paper by the definition and property of Rota-Baxter operator, we establish the system of quadratic equations of the matrix elements of Rota-Baxter operators of H_4 with weight A for a given base. By solving the system of the equations with weight λ = 0 and λ=1 the matrix forms of Rota-Baxter operators are given.
Let H_4 be the Sweedler's 4-dimensional Hopf algebra. In this paper by the definition and property of Rota-Baxter operator, we establish the system of quadratic equations of the matrix elements of Rota-Baxter operators of H_4 with weight A for a given base. By solving the system of the equations with weight λ = 0 and λ=1 the matrix forms of Rota-Baxter operators are given.
引文
[1]Aguiar M., Pre-Poisson algebras, Lett. Math. Phys., 2000, 54(4):263-277.
[2]Aguiar M., Infinitesimal Hopf algebras, Contemp. Math. Amer. Math. Soc., 2011:1-29.
[3]Aguiar M., On the associative analog of Lie bialgebras, J. Algebra, 2001, 244(2):492-532.
[4]An H., Bai C., From Rota-Baxter algebras to pre-Lie algebras, J. Phys. A:Math. Theor., 2007, 41(1):19-25.
[5]Andrews G. E., Guo L., Keigher W., et al., Baxter lgebras and Hopf algebras, Trans. Amer. Math. Soc.,2003, 355(11):4639-4656.
[6]Baxter G., An analytic problem whose solution follows from a simple algebraic identity, Pacific J. Math.,1960, 10(3):731-742.
[7]Connes A., Kreimer D., Hopf algebras, renormalization and noncommutative geometry, Commun. Math.Phys., 1998, 199(1):203-242.
[8]Connes A., Kreimer D., Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem, Commun. Math. Phys., 2000, 210:249-273.
[9]Connes A., Kreimer D., Renormalization in quantum field theory and the Riemann-Hilbert problem. II. The L-function, diffeomorphisms and the renormalization group, Commun. Math. Phys., 2001, 216(1):215-241.
[10]Rota G., Baxter operators and Combinatorial identities, I. Bull. Amer. Math. Soc., 1969, 75(2):325-329.
[11]Rota G. C., Baxter Operators, an Introduction, in:Joseph P. S. Kung(Ed.), Gian-Carlo Rota on Combinar torics, Introductory Papers and Commentaries, Birkhauser, Boston, 1995.
[12]Ebrahimi-Fard K., Guo L., On the products and dual of binary, quadratic, regular operads, J. Pure Appl.Algebra, 2005, 200(3):293-317.
[13]Ebrahimi-Fard K., Guo L., Quasi-shuffles, mixable shuffles and Hopf algebras, J. Algebraic Comb., 2006, 24:83-101.
[14]Guo L., Baxter algebras and the umbral calculus, Adv. Appl. Math., 2001, 27(2-3):405-426.
[15]Guo L., Baxter algebras, Stirling numbers and partitions, J. Algebra Appl., 2005, 4(02):153-164.
[16]Guo L., An Introduction to Rota-Baxter Algebra, Higher Education Press, Beijing, 2012.
[17]Guo L., Keigher W., Baxter algebras and shuffle product, Adv. Math., 2000, 150(1):117-149.
[18]Guo L., Keigher W., On free Baxter algebras:completions and the internal construction, Adv. Math., 2000,151(1):101-127..
[19]Guo L., Rosenkranz M., Zheng S. H., Classification of Rota-Baxter operators on semigroup algebras of order two and three, https://arxiv.org/abs/1402.3702.
[20]Jian R. Q., From quantum quasi-shuffle algebras to braided Rota-Baxter algebras, Lett. Math. Phys., 2013,103(8):851-863.
[21]Jian R. Q., Rosso M., Braided cofree Hopf algebras and quantum multi-brace algebras, J. Reine Angew.Math., 2012, 2012(667):193-220.
[22]Li X., Hou D., Bai C., Rota-Baxter operators on pre-Lie algebras, J. Nonlinear Math. Phys., 2007, 14(2):269-289.
[23]Leroux P., Ennea-algebras, J. Algebra, 2003, 281(1):287-302.
[24]Leroux P., Construction of Nijenhuis operators and dendriform trialgebras, J. Math. Sci., 2007, 2004(49):2595-2615.
[25]Manchon D., Paycha S., Shuffle relations for regularized integrals of symbols, Commun. Math. Phys., 2007,270(1):13-51.
[26]Tang X.,Zhang Y., Sun Q., Rota-Baxter operators on 4-dimensional complex simple associative algebras,Appl. Math. Comput., 2014, 229(C):173-186.
[27]Williams R. E., Finite dimensional Hopf Algebras, Thesis(Ph.D.)-The Florida State University, Tallahassee,Florida, USA, 1988.
[2]Aguiar M., Infinitesimal Hopf algebras, Contemp. Math. Amer. Math. Soc., 2011:1-29.
[3]Aguiar M., On the associative analog of Lie bialgebras, J. Algebra, 2001, 244(2):492-532.
[4]An H., Bai C., From Rota-Baxter algebras to pre-Lie algebras, J. Phys. A:Math. Theor., 2007, 41(1):19-25.
[5]Andrews G. E., Guo L., Keigher W., et al., Baxter lgebras and Hopf algebras, Trans. Amer. Math. Soc.,2003, 355(11):4639-4656.
[6]Baxter G., An analytic problem whose solution follows from a simple algebraic identity, Pacific J. Math.,1960, 10(3):731-742.
[7]Connes A., Kreimer D., Hopf algebras, renormalization and noncommutative geometry, Commun. Math.Phys., 1998, 199(1):203-242.
[8]Connes A., Kreimer D., Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem, Commun. Math. Phys., 2000, 210:249-273.
[9]Connes A., Kreimer D., Renormalization in quantum field theory and the Riemann-Hilbert problem. II. The L-function, diffeomorphisms and the renormalization group, Commun. Math. Phys., 2001, 216(1):215-241.
[10]Rota G., Baxter operators and Combinatorial identities, I. Bull. Amer. Math. Soc., 1969, 75(2):325-329.
[11]Rota G. C., Baxter Operators, an Introduction, in:Joseph P. S. Kung(Ed.), Gian-Carlo Rota on Combinar torics, Introductory Papers and Commentaries, Birkhauser, Boston, 1995.
[12]Ebrahimi-Fard K., Guo L., On the products and dual of binary, quadratic, regular operads, J. Pure Appl.Algebra, 2005, 200(3):293-317.
[13]Ebrahimi-Fard K., Guo L., Quasi-shuffles, mixable shuffles and Hopf algebras, J. Algebraic Comb., 2006, 24:83-101.
[14]Guo L., Baxter algebras and the umbral calculus, Adv. Appl. Math., 2001, 27(2-3):405-426.
[15]Guo L., Baxter algebras, Stirling numbers and partitions, J. Algebra Appl., 2005, 4(02):153-164.
[16]Guo L., An Introduction to Rota-Baxter Algebra, Higher Education Press, Beijing, 2012.
[17]Guo L., Keigher W., Baxter algebras and shuffle product, Adv. Math., 2000, 150(1):117-149.
[18]Guo L., Keigher W., On free Baxter algebras:completions and the internal construction, Adv. Math., 2000,151(1):101-127..
[19]Guo L., Rosenkranz M., Zheng S. H., Classification of Rota-Baxter operators on semigroup algebras of order two and three, https://arxiv.org/abs/1402.3702.
[20]Jian R. Q., From quantum quasi-shuffle algebras to braided Rota-Baxter algebras, Lett. Math. Phys., 2013,103(8):851-863.
[21]Jian R. Q., Rosso M., Braided cofree Hopf algebras and quantum multi-brace algebras, J. Reine Angew.Math., 2012, 2012(667):193-220.
[22]Li X., Hou D., Bai C., Rota-Baxter operators on pre-Lie algebras, J. Nonlinear Math. Phys., 2007, 14(2):269-289.
[23]Leroux P., Ennea-algebras, J. Algebra, 2003, 281(1):287-302.
[24]Leroux P., Construction of Nijenhuis operators and dendriform trialgebras, J. Math. Sci., 2007, 2004(49):2595-2615.
[25]Manchon D., Paycha S., Shuffle relations for regularized integrals of symbols, Commun. Math. Phys., 2007,270(1):13-51.
[26]Tang X.,Zhang Y., Sun Q., Rota-Baxter operators on 4-dimensional complex simple associative algebras,Appl. Math. Comput., 2014, 229(C):173-186.
[27]Williams R. E., Finite dimensional Hopf Algebras, Thesis(Ph.D.)-The Florida State University, Tallahassee,Florida, USA, 1988.