Automorphisms and Ideals of the Weyl Algebra TeX Export

@misc{citeulike:486110, abstract = {Let \$A\_1\$ be the (first) Weyl algebra, and let \$G\$ be its automorphism group. We study the natural action of \$G\$ on the space of isomorphism classes of right ideals of \$A\_1\$ (equivalently, of finitely generated rank 1 torsion-free right \$A\_1\$-modules). We show that this space breaks up into a countable number of orbits each of which is a finite dimensional algebraic variety. Our results are strikingly similar to those for the commutative algebra of polynomials in two variables; however, we do not know of any general principle that would allow us to predict this in advance. As a key step in the proof, we obtain a new description of the bispectral involution of \cite{W1}. We also make some comments on the group \$G\$ from the viewpoint of Shafaravich's theory of infinite dimensional algebraic groups.}, archivePrefix = {arXiv}, author = {Berest, Yuri and Wilson, George}, citeulike-article-id = {486110}, citeulike-linkout-0 = {http://arxiv.org/abs/math.QA/0102190}, citeulike-linkout-1 = {http://arxiv.org/pdf/math.QA/0102190}, day = {25}, eprint = {math.QA/0102190}, keywords = {geometry, noncommutative, weylalgebra}, month = {Feb}, posted-at = {2006-01-30 19:59:38}, priority = {0}, title = {Automorphisms and Ideals of the Weyl Algebra}, url = {http://arxiv.org/abs/math.QA/0102190}, year = {2001} }

TY - ELEC ID - citeulike:486110 L3 - citeulike-article-id:486110 N2 - Let $A_1$ be the (first) Weyl algebra, and let $G$ be its automorphism group. We study the natural action of $G$ on the space of isomorphism classes of right ideals of $A_1$ (equivalently, of finitely generated rank 1 torsion-free right $A_1$-modules). We show that this space breaks up into a countable number of orbits each of which is a finite dimensional algebraic variety. Our results are strikingly similar to those for the commutative algebra of polynomials in two variables; however, we do not know of any general principle that would allow us to predict this in advance. As a key step in the proof, we obtain a new description of the bispectral involution of \cite{W1}. We also make some comments on the group $G$ from the viewpoint of Shafaravich's theory of infinite dimensional algebraic groups. TI - Automorphisms and Ideals of the Weyl Algebra KW - geometry KW - noncommutative KW - weylalgebra AU - Berest, Yuri AU - Wilson, George PY - 2001/Feb/25/ UR - http://arxiv.org/abs/math.QA/0102190 ER -