2011, 18: 12-21. doi: 10.3934/era.2011.18.12

On subgroups of the Dixmier group and Calogero-Moser spaces

1. 

Department of Mathematics, Cornell University, Ithaca, NY 14853-4201, United States

2. 

Department of Mathematics, University of Arizona, Tucson, AZ 85721-0089, United States

3. 

Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, United States

Received  August 2010 Revised  February 2011 Published  March 2011

We describe the structure of the automorphism groups of algebras Morita equivalent to the first Weyl algebra $ A_1(k) $. In particular, we give a geometric presentation for these groups in terms of amalgamated products, using the Bass-Serre theory of groups acting on graphs. A key rôle in our approach is played by a transitive action of the automorphism group of the free algebra $ k< x, y>$ on the Calogero-Moser varieties $ \CC_n $ defined in [5]. In the end, we propose a natural extension of the Dixmier Conjecture for $ A_1(k) $ to the class of Morita equivalent algebras.
Citation: Yuri Berest, Alimjon Eshmatov, Farkhod Eshmatov. On subgroups of the Dixmier group and Calogero-Moser spaces. Electronic Research Announcements, 2011, 18: 12-21. doi: 10.3934/era.2011.18.12
References:
[1]

J. Alev, Action de groupes sur $A_1(\c)$, Lecture Notes in Math. 1197, Springer, Berlin, 1986, 1-9.  Google Scholar

[2]

R. C. Alperin, Homology of the group of automorphisms of $ k[x,y] $, J. Pure Appl. Algebra, 15 (1979), 109-115. doi: 10.1016/0022-4049(79)90027-6.  Google Scholar

[3]

H. Bass, "Algebraic $K$-Theory," W. A. Benjamin Inc., New York-Amsterdam, 1968.  Google Scholar

[4]

Yu. Berest and O. Chalykh, $A_{\infty}$-modules and Calogero-Moser spaces, J. reine angew Math., 607 (2007), 69-112. doi: 10.1515/CRELLE.2007.046.  Google Scholar

[5]

Yu. Berest and G. Wilson, Automorphisms and ideals of the Weyl algebra, Math. Ann., 318 (2000), 127-147. doi: 10.1007/s002080000115.  Google Scholar

[6]

Yu. Berest and G. Wilson, Classification of rings of differential operators on affine curves, Internat. Math. Res. Notices, 2 (1999), 105-109. doi: 10.1155/S1073792899000057.  Google Scholar

[7]

Yu. Berest and G. Wilson, Ideal classes of the Weyl algebra and noncommutative projective geometry (with an Appendix by M. Van den Bergh), Internat. Math. Res. Notices, 26 (2002), 1347-1396. doi: 10.1155/S1073792802108051.  Google Scholar

[8]

Yu. Berest and G. Wilson, Mad subalgebras of rings of differential operators on curves, Adv. Math., 212 (2007), 163-190. doi: 10.1016/j.aim.2006.09.018.  Google Scholar

[9]

Yu. Berest and G. Wilson, Differential isomorphism and equivalence of algebraic varieties in "Topology, Geometry and Quantum Field Theory" (Ed. U. Tillmann), London Math. Soc. Lecture Note Ser. 308, Cambridge Univ. Press. Cambridge, 2004, pp. 98-126.  Google Scholar

[10]

P. M. Cohn, The automorphism group of the free algebras of rank two, Serdica Math. J., 28 (2002), 255-266.  Google Scholar

[11]

J. Dixmier, Sur les alg\`ebres de Weyl, Bull. Soc. Math. France, 96 (1968), 209-242.  Google Scholar

[12]

V. Ginzburg, Non-commutative symplectic geometry, quiver varieties, and operads, Math. Res. Lett., 8 (2001), 377-400.  Google Scholar

[13]

M. H. Gizatullin and V. I. Danilov, Automorphisms of affine surfaces. I, II, Math. USSR Izv., 9 (1975), 493-534; Math. USSR Izv., 11 (1977), 51-98. doi: 10.1070/IM1977v011n01ABEH001695.  Google Scholar

[14]

K. M. Kouakou, "Isomorphismes Entre Algèbres d'opérateurs Différentielles sur les Courbes Algébriques Affines," Thèse de Doctorat, Universite Claude Bernard-Lyon I, 1994. Google Scholar

[15]

L. Makar-Limanov, Automorphisms of a free algebra with two generators, Funct. Anal. Appl., 4 (1970), 262-264. doi: 10.1007/BF01075252.  Google Scholar

[16]

L. Makar-Limanov, On automorphisms of the Weyl algebra, Bull. Soc. Math. France, 112 (1984), 359-363.  Google Scholar

[17]

J.-P. Serre, "Trees," Springer-Verlag, Berlin, 1980.  Google Scholar

[18]

I. R. Shafarevich, "Collected Mathematical Papers," Springer, Berlin, 1989, pp. 430, 607.  Google Scholar

[19]

J. T. Stafford, Endomorphisms of right ideals of the Weyl algebra, Trans. Amer. Math. Soc., 299 (1987), 623-639. doi: 10.1090/S0002-9947-1987-0869225-3.  Google Scholar

[20]

J. T. Stafford and M. Van den Bergh, Noncommutative curves and noncommutative surfaces, Bull. Amer. Math. Soc., 38 (2001), 171-216. doi: 10.1090/S0273-0979-01-00894-1.  Google Scholar

[21]

G. Wilson, Collisions of Calogero-Moser particles and an adelic Grassmannian (with an Appendix by I. G. Macdonald), Invent. Math., 133 (1998), 1-41. doi: 10.1007/s002220050237.  Google Scholar

[22]

G. Wilson, Bispectral commutative ordinary differential operators, J. reine angew. Math., 442 (1993), 177-204. doi: 10.1515/crll.1993.442.177.  Google Scholar

[23]

D. Wright, Two-dimensional Cremona groups acting on simplicial complexes, Trans. Amer. Math. Soc., 331 (1992), 281-300. doi: 10.2307/2154009.  Google Scholar

show all references

References:
[1]

J. Alev, Action de groupes sur $A_1(\c)$, Lecture Notes in Math. 1197, Springer, Berlin, 1986, 1-9.  Google Scholar

[2]

R. C. Alperin, Homology of the group of automorphisms of $ k[x,y] $, J. Pure Appl. Algebra, 15 (1979), 109-115. doi: 10.1016/0022-4049(79)90027-6.  Google Scholar

[3]

H. Bass, "Algebraic $K$-Theory," W. A. Benjamin Inc., New York-Amsterdam, 1968.  Google Scholar

[4]

Yu. Berest and O. Chalykh, $A_{\infty}$-modules and Calogero-Moser spaces, J. reine angew Math., 607 (2007), 69-112. doi: 10.1515/CRELLE.2007.046.  Google Scholar

[5]

Yu. Berest and G. Wilson, Automorphisms and ideals of the Weyl algebra, Math. Ann., 318 (2000), 127-147. doi: 10.1007/s002080000115.  Google Scholar

[6]

Yu. Berest and G. Wilson, Classification of rings of differential operators on affine curves, Internat. Math. Res. Notices, 2 (1999), 105-109. doi: 10.1155/S1073792899000057.  Google Scholar

[7]

Yu. Berest and G. Wilson, Ideal classes of the Weyl algebra and noncommutative projective geometry (with an Appendix by M. Van den Bergh), Internat. Math. Res. Notices, 26 (2002), 1347-1396. doi: 10.1155/S1073792802108051.  Google Scholar

[8]

Yu. Berest and G. Wilson, Mad subalgebras of rings of differential operators on curves, Adv. Math., 212 (2007), 163-190. doi: 10.1016/j.aim.2006.09.018.  Google Scholar

[9]

Yu. Berest and G. Wilson, Differential isomorphism and equivalence of algebraic varieties in "Topology, Geometry and Quantum Field Theory" (Ed. U. Tillmann), London Math. Soc. Lecture Note Ser. 308, Cambridge Univ. Press. Cambridge, 2004, pp. 98-126.  Google Scholar

[10]

P. M. Cohn, The automorphism group of the free algebras of rank two, Serdica Math. J., 28 (2002), 255-266.  Google Scholar

[11]

J. Dixmier, Sur les alg\`ebres de Weyl, Bull. Soc. Math. France, 96 (1968), 209-242.  Google Scholar

[12]

V. Ginzburg, Non-commutative symplectic geometry, quiver varieties, and operads, Math. Res. Lett., 8 (2001), 377-400.  Google Scholar

[13]

M. H. Gizatullin and V. I. Danilov, Automorphisms of affine surfaces. I, II, Math. USSR Izv., 9 (1975), 493-534; Math. USSR Izv., 11 (1977), 51-98. doi: 10.1070/IM1977v011n01ABEH001695.  Google Scholar

[14]

K. M. Kouakou, "Isomorphismes Entre Algèbres d'opérateurs Différentielles sur les Courbes Algébriques Affines," Thèse de Doctorat, Universite Claude Bernard-Lyon I, 1994. Google Scholar

[15]

L. Makar-Limanov, Automorphisms of a free algebra with two generators, Funct. Anal. Appl., 4 (1970), 262-264. doi: 10.1007/BF01075252.  Google Scholar

[16]

L. Makar-Limanov, On automorphisms of the Weyl algebra, Bull. Soc. Math. France, 112 (1984), 359-363.  Google Scholar

[17]

J.-P. Serre, "Trees," Springer-Verlag, Berlin, 1980.  Google Scholar

[18]

I. R. Shafarevich, "Collected Mathematical Papers," Springer, Berlin, 1989, pp. 430, 607.  Google Scholar

[19]

J. T. Stafford, Endomorphisms of right ideals of the Weyl algebra, Trans. Amer. Math. Soc., 299 (1987), 623-639. doi: 10.1090/S0002-9947-1987-0869225-3.  Google Scholar

[20]

J. T. Stafford and M. Van den Bergh, Noncommutative curves and noncommutative surfaces, Bull. Amer. Math. Soc., 38 (2001), 171-216. doi: 10.1090/S0273-0979-01-00894-1.  Google Scholar

[21]

G. Wilson, Collisions of Calogero-Moser particles and an adelic Grassmannian (with an Appendix by I. G. Macdonald), Invent. Math., 133 (1998), 1-41. doi: 10.1007/s002220050237.  Google Scholar

[22]

G. Wilson, Bispectral commutative ordinary differential operators, J. reine angew. Math., 442 (1993), 177-204. doi: 10.1515/crll.1993.442.177.  Google Scholar

[23]

D. Wright, Two-dimensional Cremona groups acting on simplicial complexes, Trans. Amer. Math. Soc., 331 (1992), 281-300. doi: 10.2307/2154009.  Google Scholar

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