Generalized Kac–Moody Lie Algebras and Product Quivers

REFERENCES

• Borcherds , R. ( 1988 ). Generalized Kac–Moody Lie algebras . J. Algebra 115 : 501 – 512 .
   Web of Science ®Google Scholar
• Deng , B. , Xiao , J. ( 2002 ). On double Ringel–Hall algebras . J. Algebra 255 : 110 – 149 .
   Google Scholar
• Deng , B. , Xiao , J. ( 2004 ). On Ringel–Hall algebras. Representations of Finite Dimensional Algebras and Related Topics in Lie Theory and Geometry. Fields Inst. Commun., Vol. 40. Providence, RI: Amer. Math. Soc., pp. 319–348 .
   Google Scholar
• Dlab , V. , Ringel , C. M. ( 1976 ). Indecomposable representations of graphs and algebras . Memoirs Amer. Math. Soc. 173 : 1 – 57 .
   Google Scholar
• Drinfeld , V. G. ( 1987 ). Quantum Groups. Proc. Int. Congr. Math. Berkeley 1986. Amer. Math. Soc., pp. 798–820 .
   Google Scholar
• Frenkel , I. , Malkin , A. , Vybornov , M. ( 2001 ). Affine Lie algebras and tame quivers . Selecta Math. (N.S.) 7 : 1 – 56 .
   Google Scholar
• Green , J. A. ( 1995 ). Hall algebras, hereditary algebras and quantum groups . Invent. Math. 120 : 361 – 377 .
   Web of Science ®Google Scholar
• Iyama , O. ( 2004 ). On Hall algebras of hereditary orders . Comm. Algebra 32 ( 4 ): 1571 – 1579 .
   Web of Science ®Google Scholar
• Jantzen , J. C. ( 1995 ). Lectures on Quantum Groups. Amer. Math. Society, GSM, Vol. 6 .
   Google Scholar
• Hubery , A. ( 2004 ). From triangulated categories to Lie algebras: a theorem of Peng and Xiao. Representations of Finite Dimensional Algebras and Related Topics in Lie Theory and Geometry, Fields Inst. Commun., Vol. 40. Providence, RI: Amer. Math. Soc., pp. 319–348 .
   Google Scholar
• Kac , V. ( 1990 ). Infinite Dimensional Lie Algebras. , 3rd ed. Cambridge : Cambridge University Press .
   Google Scholar
• Kang , S.-J. , Schiffmann , O. (2006). Canonical bases for quantum generalized Kac–Moody algebras. Adv. Math. 200(2):455–478, math.QA/0311089.
   Web of Science ®Google Scholar
• Kapranov , M. M. ( 1997 ). Eisenstein series and quantum affine algebras . Journal of Mathematical Science 84 ( 5 ): 1311 – 1360 .
   Google Scholar
• Kapranov , M. M. ( 1998 ). Heisenberg doubles and derived categories . J. Algebra 202 ( 2 ): 712 – 744 .
   Web of Science ®Google Scholar
• Kapranov , M. M. , Vasserot , E. ( 2000 ). Kleinian singularities, derived categories and Hall algebras . Math. Ann. 316 ( 3 ): 565 – 576 .
   Web of Science ®Google Scholar
• Li , Y. , Lin , Z. ( 2010 ). A realization of quantum groups via product valued quivers . Algebra and Representation Theory 13 ( 4 ): 427 – 444 .
   Web of Science ®Google Scholar
• Lin , Y. , Peng , L. ( 2005 ). 2-extended affine Lie algebras and tubular algebras . Adv. Math. 196 ( 2 ): 487 – 530 .
   Web of Science ®Google Scholar
• Nakajima , H. ( 2001 ). Quiver varieties and finite dimensional representations of quantum affine algebras . J. Amer. Math. Soc. 14 ( 1 ): 145 – 238 .
   Web of Science ®Google Scholar
• Peng , L. ( 1998 ). Lie algebras determined by finite Auslander–Reiten quivers . Comm. Algebra 26 ( 9 ): 2711 – 2725 .
   Web of Science ®Google Scholar
• Peng , L. , Xiao , J. ( 2000 ). Triangulated categories and Kac–Moody algebras . Invent. Math. 140 ( 3 ): 563 – 603 .
   Web of Science ®Google Scholar
• Riedtmann , Ch. ( 1994 ). Lie algebras generated by indecomposables . J. Algebra 170 : 526 – 546 .
   Web of Science ®Google Scholar
• Ringel , C. M. ( 1976 ). Representations of K-species and bimodules . J. Algebra 41 : 269 – 302 .
   Web of Science ®Google Scholar
• Ringel , C. M. ( 1990 ). Hall algebras and quantum groups . Invent. Math. 101 : 583 – 592 .
   Web of Science ®Google Scholar
• Ringel , C. M. ( 1990 ). Hall algebras . In: Topics in Algebra . Part 1 (Warsaw, 1998). Banach Center Publ., Vol. 26. Part 1, PWN , Warsaw , pp. 433 – 447 .
   Google Scholar
• Ringel , C. M. ( 1993 ). The composition algebra of a cyclic quiver . Proc. London Math. Soc. (3) 66 : 507 – 537 .
   Web of Science ®Google Scholar
• Ringel , C. M. ( 1993 ). Hall algebras revisited . Israel Math. Conf. Proc. 7 : 171 – 176 .
   Google Scholar
• Ringel , C. M. ( 1992 ). Lie algebras arising in representation theory. Representations of Algebras and Related Topics (Kyoto, 1990). London Math. Soc. Lecture Note Ser., Vol. 168. Cambridge: Cambridge Univ. Press, pp. 284–291 .
   Google Scholar
• Sevenhant , B. , Van Den Bergh , M. ( 2001 ). A relation between a conjecture of Kac and the structure of the Hall algebra . J. Pure and Appl. Algebra 160 : 319 – 332 .
   Web of Science ®Google Scholar
• Sevenhant , B. , Van Den Bergh , M. ( 1999 ). On the double of the Hall algebra of a quiver . J. Algebra 221 : 135 – 160 .
   Web of Science ®Google Scholar
• Toën , B. ( 2006 ). Derived Hall algebra . Duke Math. J. 135 ( 3 ): 587 – 615 , math.QA/ 0501343 .
   Google Scholar
• Xiao , J. ( 1997 ). Drinfeld double and Ringel–Green theory of Hall algebras . J. Algebra 190 : 100 – 144 .
   Web of Science ®Google Scholar
• Xiao , J. , Xu , F. ( 2008 ). Hall algebras associated to triangulated categories . Duke Math. J. 143 ( 2 ): 357 – 373 .
   Web of Science ®Google Scholar
• Zhang , P. ( 2000 ). PBW-bases of the composition algebra of the Kronecker algebra . J. Reine Angew. Math. 527 : 97 – 116 .
   Web of Science ®Google Scholar
• Zhang , S. ( 1997 ). Lie algebras determined by finite valued Auslander–Reiten quivers . Sci. China Ser. A 40 ( 7 ): 714 – 724 .
   Google Scholar
• Communicated by D. Nakano.
   Google Scholar