Social Inequality SOC 360
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Date Created: 10/28/15
An introduction to the invariant theory of noncommutative rings Part II Ellen Kirkman Septem ber 16 2008 Based on joint work with James Kuzmanovich and James Zhang University of Washington Seattle Outline Review of First Lecture Shephard Todd Chevalley Theorem Quasi reflections Outline Review of First Lecture Shephard Todd Chevalley Theorem Quasi reflections New Reflection Groups Outline Review of First Lecture Shephard Todd Chevalley Theorem Quasi reflections New Reflection Groups Gorenstein Invariant Subrings Outline Review of First Lecture Shephard Todd Chevalley Theorem Quasi reflections New Reflection Groups Gorenstein Invariant Subrings Questions Outline Review of First Lecture Shephard Todd Chevalley Theorem Quasi reflections New Reflection Groups Gorenstein Invariant Subrings Questions Invariants under Hopf Algebra Actions Shephard Todd Chevalley Theorem Theorem The ring of invariants kx17 XnG under a finite group G is a polynomial ring if and only if G is generated by reflections A linear map g on V is called a reflection of V if all but one of the eigenvalues of g are 1 or dim Vg dim V71 Noncommutative Generalization Replace kx17 7Xn with some nice polynomial like noncommutative algebra A and consider groups G of linear automorphisms acting on A Artin Schelter regular algebras eg skew polynomial rings Proper question NOT when is A6 E A but under what conditions on G is A6 regular The trace of a graded automorphism gt The trace of a graded automorphism g of a graded ring A with An the elements of degree n is the formal power series TrAg7 t Z traceglAnt ni0 gt A graded automorphism of an AS regular algebra A of GKdim n is called a quasi reflection of A if its trace has a pole of order n 71 at t1 mgr r where p1q1 y o G is called a reflection group on A if A6 is regular New Reflections Groups Let A C1X1X2 and G be the group generated by the mystic reflections i 0 1 d i 0 139 g1 71 0 an 2 7 l 0 The fixed ring is A6 CX1X27Xil l X3 a commutative polynomial ring G is a reflection group 71 0 gf 17 gfg l 0 ill 17 17 0 71 ngng igl 1 O G is the Quaternion Group of order 8 which is not a classical reflection group Infinite Set of New Reflection Groups Take 7 O g2 7 ikil 0 for a primitive 2m th root of unity G ghgg has fixed ring is A6 CX1X2X12m l Xgm gt G has order 4m gt m 1 G is cyclic of order 4 gt m 2 G is quaternion of order 8 gt G is non abelian for m 2 2 gt G has a unique element of order 2 G is different from the infinite families of the cyclic groups the symmetric groups and the groups Cm7 p7 n so for In gtgt O is not a classical reflection group Noncommutative Watanabe s Theorem Watanabe s Theorem If G is a finite subgroup of SLV acting naturally on the commutative polynomial algebra kV then the fixed subring kVG is Gorenstein 71 0 klxiylg kltX27Xy7y2gt Jorgensen Zhang s Theorem If G is a finite subgroup of GLV acting linearly on an Artin Schelter regular algebra A if the homological determinant of each g in G is trivial then the fixed subring kVG is Gorenstein Example Let g l 31 act on kx7 y Example Let g act on k1x7 y 0 l l 0 llt1xyg is generated by 01 X l y and 02 X3 l y3 Noncommutative Stanley s Theorem Jorgensen Zhang s Theorem Let A be AS regular and some technical conditions and G be a finite group of graded automorphisms of A Then A6 is AS Gorenstein if and only its Hilbert series satisfies the functional equation HAGtT1 ithAGU for some integer m Example Let g T01 31 act on kx7 y Then 1 t2 HAGf 7 SO 7 1 r t41 r t4 t2 1 7 7 2 HAGt 717t722 uff tzi 1y 1 HAG Example Letg l 0 1 act on k1xy Then 1 17tt2 HM Quasi Bi reflections Characterize C so that AC is a complete intersection In the commutative case it is necessary that G be generated by bi reflections g with all but 2 eigenvalues equal 1 For regular algebras of dimension n is it necessary that Tng7 1 has a pole of order n72 at t1 Example A k1xyz and 0710 g100 0071 which has eigenvalues 71777139 so is not a bi reflection of A1 but klxi Viz W Ag g W2 7 X2 4Y2Z is a commutative hypersurface Find the Quasi reflections of other Regular Algebras then the quasi reflections of A are either If HAt aft classical reflections or mystic reflections A quasi reflection g is a mystic reflection if the order of g is 4 and there is a basis b17 7bn of A1 such that gb1ib1 gb2 7172 and gbj bj forj 2 3 Many regular algebras have NO quasi reflections Question Are there regular algebras with other kinds of quasi reflections Noether s Bound Noether 1916 lGl is an upper bound on the degrees of the generators of the algebra of invariants Example b Let g 1 1 act on A k1xy Then 2 and invariants are generated by 01 X l y and 02 X3 l y3 So Noether bound fails Noether39s bound fails for kx17 7Xn in characteristic p and finding a reasonable bound is of interest Question Find an analogue of Noether39s bound say for skew polynomial rings Invariant Theory for Hopf Algebra Actions In a group action g ab g ag b There are algebras H7 A e 5 called Hopf algebras where the co multiplication A is used to define the action h ab The invariants of H on A are AHaeAlhaehaforaheH One can get invariant subrings A that are not isomorphic to A6 for any finite group We have a versions of Molien39s Theorem Watanabe39s Theorem and Stanley39s Theorem for A when H is finite dimensional and semi simple Question What conditions on H force A to be regular even when A kx17 7Xr7 When is H a reflection quantum group References Commutative Invariant Theory 1 DJ Benson Polynomial invariants of finite groups London Mathematical Society Lecture Note Series 190 Cambridge University Press Cambridge 1993 2 H Derksen G Kemper Computational lnvariant Theory Encyclopedia of Mathematical Sciences 130 lnvariant Theoy and Algebraic Transformation Groups I Springer Verlag Berlin 2002 3 M D Neusel lnvariant Theory American Mathematical Society Providence RI 2007 Invariant Theory of Regular Algebras 1 N Jing and JJ Zhang On the trace of graded automorphisms J Algebra 189 1997 no 2 353 376 2 N Jing and JJ Zhang Gorensteinness of invariant subrings of quantum algebras J Algebra 221 1999 no 2 669 691 3 P J rgensen and JJ Zhang Gourmet39s guide to Gorensteinness Adv Math 151 2000 no 2 313 345 4 E Kirkman J Kuzmanovich and JJ Zhang Rigidity of graded regular algebras Trans Amer Math Soc 360 2008 6331 6369 5 E Kirkman J Kuzmanovich and JJ Zhang Shephard Todd Chevalley Theorem for Skew Polynomial Rings to appear Algebr Represent Theory Hopf Algebras 1 S Montgomery Hopf algebras and their actions on rings CBMS Regional Conference Series inMathematics 82 Providence RI 1993
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