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.

[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.

[3]

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

[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.

[5]

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

[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.

[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.

[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.

[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.

[10]

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

[11]

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

[12]

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

[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.

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[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.

[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.

[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.

[22]

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

[23]

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

show all references

References:
[1]

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

[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.

[3]

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

[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.

[5]

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

[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.

[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.

[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.

[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.

[10]

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

[11]

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

[12]

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

[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.

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[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.

[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.

[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.

[22]

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

[23]

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

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