# American Institute of Mathematical Sciences

January  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. \textbf{1197}, 1197 (1986), 1. Google Scholar [2] R. C. Alperin, Homology of the group of automorphisms of $k[x,y]$,, J. Pure Appl. Algebra, 15 (1979), 109. doi: 10.1016/0022-4049(79)90027-6. Google Scholar [3] H. Bass, "Algebraic $K$-Theory,", W. A. Benjamin Inc., (1968). Google Scholar [4] Yu. Berest and O. Chalykh, $A_{\infty}$-modules and Calogero-Moser spaces,, J. reine angew Math., 607 (2007), 69. 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. 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. 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), 26 (2002), 1347. 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. doi: 10.1016/j.aim.2006.09.018. Google Scholar [9] Yu. Berest and G. Wilson, Differential isomorphism and equivalence of algebraic varieties, in, 308 (2004), 98. Google Scholar [10] P. M. Cohn, The automorphism group of the free algebras of rank two,, Serdica Math. J., 28 (2002), 255. Google Scholar [11] J. Dixmier, Sur les alg\ebres de Weyl,, Bull. Soc. Math. France, 96 (1968), 209. Google Scholar [12] V. Ginzburg, Non-commutative symplectic geometry, quiver varieties, and operads,, Math. Res. Lett., 8 (2001), 377. Google Scholar [13] M. H. Gizatullin and V. I. Danilov, Automorphisms of affine surfaces. I, II,, Math. USSR Izv., 9 (1975), 493. doi: 10.1070/IM1977v011n01ABEH001695. Google Scholar [14] K. M. Kouakou, "Isomorphismes Entre Algèbres d'opérateurs Différentielles sur les Courbes Algébriques Affines,", Thèse de Doctorat, (1994). Google Scholar [15] L. Makar-Limanov, Automorphisms of a free algebra with two generators,, Funct. Anal. Appl., 4 (1970), 262. doi: 10.1007/BF01075252. Google Scholar [16] L. Makar-Limanov, On automorphisms of the Weyl algebra,, Bull. Soc. Math. France, 112 (1984), 359. Google Scholar [17] J.-P. Serre, "Trees,", Springer-Verlag, (1980). Google Scholar [18] I. R. Shafarevich, "Collected Mathematical Papers,", Springer, (1989). Google Scholar [19] J. T. Stafford, Endomorphisms of right ideals of the Weyl algebra,, Trans. Amer. Math. Soc., 299 (1987), 623. 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. 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), 133 (1998), 1. doi: 10.1007/s002220050237. Google Scholar [22] G. Wilson, Bispectral commutative ordinary differential operators,, J. reine angew. Math., 442 (1993), 177. 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. doi: 10.2307/2154009. Google Scholar

show all references

##### References:
 [1] J. Alev, Action de groupes sur $A_1(\c)$,, Lecture Notes in Math. \textbf{1197}, 1197 (1986), 1. Google Scholar [2] R. C. Alperin, Homology of the group of automorphisms of $k[x,y]$,, J. Pure Appl. Algebra, 15 (1979), 109. doi: 10.1016/0022-4049(79)90027-6. Google Scholar [3] H. Bass, "Algebraic $K$-Theory,", W. A. Benjamin Inc., (1968). Google Scholar [4] Yu. Berest and O. Chalykh, $A_{\infty}$-modules and Calogero-Moser spaces,, J. reine angew Math., 607 (2007), 69. 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. 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. 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), 26 (2002), 1347. 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. doi: 10.1016/j.aim.2006.09.018. Google Scholar [9] Yu. Berest and G. Wilson, Differential isomorphism and equivalence of algebraic varieties, in, 308 (2004), 98. Google Scholar [10] P. M. Cohn, The automorphism group of the free algebras of rank two,, Serdica Math. J., 28 (2002), 255. Google Scholar [11] J. Dixmier, Sur les alg\ebres de Weyl,, Bull. Soc. Math. France, 96 (1968), 209. Google Scholar [12] V. Ginzburg, Non-commutative symplectic geometry, quiver varieties, and operads,, Math. Res. Lett., 8 (2001), 377. Google Scholar [13] M. H. Gizatullin and V. I. Danilov, Automorphisms of affine surfaces. I, II,, Math. USSR Izv., 9 (1975), 493. doi: 10.1070/IM1977v011n01ABEH001695. Google Scholar [14] K. M. Kouakou, "Isomorphismes Entre Algèbres d'opérateurs Différentielles sur les Courbes Algébriques Affines,", Thèse de Doctorat, (1994). Google Scholar [15] L. Makar-Limanov, Automorphisms of a free algebra with two generators,, Funct. Anal. Appl., 4 (1970), 262. doi: 10.1007/BF01075252. Google Scholar [16] L. Makar-Limanov, On automorphisms of the Weyl algebra,, Bull. Soc. Math. France, 112 (1984), 359. Google Scholar [17] J.-P. Serre, "Trees,", Springer-Verlag, (1980). Google Scholar [18] I. R. Shafarevich, "Collected Mathematical Papers,", Springer, (1989). Google Scholar [19] J. T. Stafford, Endomorphisms of right ideals of the Weyl algebra,, Trans. Amer. Math. Soc., 299 (1987), 623. 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. 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), 133 (1998), 1. doi: 10.1007/s002220050237. Google Scholar [22] G. Wilson, Bispectral commutative ordinary differential operators,, J. reine angew. Math., 442 (1993), 177. 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. doi: 10.2307/2154009. Google Scholar
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