Ellen Kirkman and James Kuzmanovich Wake Forest University James Zhang University of Washington
Shephard-Todd-Chevalley Theorem Theorem. The ring of invariants C[x,, x n ] G under a finite group G is a polynomial ring if and only if G is generated by reflections. Question. For an Artin-Schelter regular algebra A, when is A G isomorphic to A for a finite group of graded automorphisms G? Question. For an Artin-Schelter regular algebra A, when is A G Artin-Schelter regular?
Results in Another Direction Theorem. (S. P. Smith, 989) The Weyl algebra A (k) is not the fixed subring S G under a finite solvable group of automorphisms of a domain S. Theorem. (J. Alev and P. Polo, 995). Let g and g be two semisimple Lie algebras. Let G be a finite group of algebra automorphisms of U(g) such that U(g) G = U(g ). Then G is trivial and g = g. 2. If G is a finite group of algebra automorphisms of A n (k), then the fixed subring A n (k) G is isomorphic to A n (k) only when G is trivial.
Questions in this Other Direction Let A be Artin-Schelter regular. Question. When is it the case that A G is never isomorphic to A for a finite group G of graded automorphisms? Call such algebras rigid. Question. When is it the case that A G is never Artin-Schelter regular for a finite group G of graded automorphisms?
Example Let A = C [x, y] and let g where g = nth root of unity ξ. [ ξ 0 0 ] for a primitive If n is odd, then A g = C [x n, y] which is isomorphic to A. If n is even, then A g = C[x n, y], a commutative polynomial ring, which is not isomorphic to A.
Hilbert Series of Regular Algebras Let B be a graded algebra. The Hilbert series of B is defined by H B (t) = dimb k t k. k=0 Proposition. (Stephenson-Zhang, Jing-Zhang, ATV) Let B be an Artin-Schelter regular algebra and let H B (t) = ( t) n p(t) where p() 0. Furhtermore n = GKdim(B) and p(t) is a product of cyclotomic polynomials.
Traces of Graded Automorphisms Let g be a graded automorphism of a graded algebra A. The trace of g is defined by Note H A (t) = Tr(Id, t). Tr(g, t) = tr(g An )t n. n=0
Molien s Theorem Theorem. (Jørgensen-Zhang) Let B be a connected graded K-algebra and let G be a finite group of graded automorphisms of B with G K. Then H B G (t) = Tr B (g, t). G g G
Quasi-Reflections Let G be an automorphism of an AS-regular algebra A with GKdim(A) = n. We call g a quasi-reflection if with p() 0. Tr(g, t) = ( t) n p(t)
You Need Quasi-Reflections Theorem. Let G be a finite group of graded automorphisms of a Noetherian AS-regular algebra A. If A G is AS-regular, then G must contain a quasi-reflection. Lemma. Let f (t) = a 0 + a t + + a n t n be a palindrome polynomial; that is, a n i = a i for all i. Then f () = n f () 2.
A Sketch of the Proof Proof. Assume that G does not contain a quasi-reflection. Let H A (t) = ( t) n p(t). Suppose that A G is regular. Then H A G (t) = ( t) n q(t), where q(t) is a product of cyclotomic polynomials. Since G is nontrivial, l = deg(q(t)) > deg(p(t)) = k.
Expand H A (t) and H A G (t) into a Laurent series about t =. H A (t) = H A G (t) = ( t) n p() + p () ( t) n p() 2 + ( t) n q() + ( t) n q () q() 2 +
If we expand H A G (t) = Tr(g, t) into a Laurent series G g G around t =, the first terms come entirely from the trace of the identity. H A G (t) = G [ ( t) n p() + p () ( t) n p() 2 + H A G (t) = ( t) n q() + ( t) n q () q() 2 + Equating coefficients q() = G p(), and q () q() 2 = p () G p() 2.
Since p(t) and q(t) are products of cyclotomic polynomials, they are palindrome polynomials, and hence by the Lemma q () = l q() 2 and p () = k p() 2. Substituting in q () q() 2 = G p () l we have p() 2 2q() = G Since q() = G p(), it follows that l = k, which is a contradiction. k 2p().
Jordan Plane The Jordan Plane C J [x, y] is defined by yx xy = x 2. All graded automorphisms are of the form g = ξid with trace Tr(g, t) = Hence the Jordan Plane is rigid. ( ξt) 2.
Graded Down-Up Algebras Let α, β C with β 0. Then the down-up algebra A(α, β, 0) is the algebra generated by two elements u, d subject to the relations d 2 u = αdud + βud 2 du 2 = αudu + βu 2 d. Then A is a Noetherian Artin-Schelter domain with gldim(a) = 3. Benkart and Roby have shown that A has a vector space basis consisting of all monomials of the form u i (du) j d k. Hence H A (t) = ( t) 2 ( t 2 ).
Proposition. The graded automorphisms of a graded down-up algebra A = A(α, β, 0) are given by [ ] w 0. for any A(α, β, 0). 0 z [ ] 0 x 2. when A = A(0,, 0) or A(α,, 0) for any α. y 0 [ ] w x 3., when A = A(0,, 0) or A(2,, 0). y z Furthermore if the eigenvalues of the defining matrix for g are λ and µ, then Tr(g, t) = ( λt)( µt)( λµt 2 ).
Graded Down-Up Algebras are Rigid If g is a graded automorphism, then Tr(g, t) = Hence if g is a quasi-reflection Tr(g, t) = ( λt)( µt)( λµt 2 ). ( t)( µt)( µt 2 ). Thus µ = and we have a pole of order 3, not 2.
A More General Result Theorem. Let A be Noetherian regular with gldim(a) = 3. If A is generated by two elements of degree, then A is rigid. Nonproof. If g is a quasi-reflection, then Tr(g, t) = for roots of unity ξ, ξ 2. ( t) 2 ( ξ t)( ξ 2 t)
Quantum Polynomial Rings A quantum polynomial ring of dimension n is a Noetherian AS-regular domain of global dimension n with Hilbert series H A (t) = ( t) n. Hence quasi-reflections have traces of the form Tr A (g, t) = ( t) n ( ξt).
The Quasi-Reflections Theorem. The quasi-reflections of a quantum polynomial ring A are of two types. If g is a quasi-reflection, then either (Reflections) there is a basis {b,, b n } of A, such that g(b ) = ξb and g(b j ) = b j for j 2, or (Mystic Reflections) the order of g is 4, and there is a basis {b,, b n } of A, such that g(b ) = ib, g(b 2 ) = ib 2, and g(b j ) = b j for j 3. In each case A g is regular.
Example Let A = C [x, y]. [ ] ξ 0 Let g be given by where ξ is an nth root of unity. 0 Then g is a reflection and Tr(g, t) = ( t)( ξt). A g = C ± [x n, y]. [ 0 Let g be given by 0 ].
Then g is a mystic reflection with b = x iy, b 2 = x + iy. The trace of g is given by Tr(g, t) = ( t)( + t). The invariant ring is given by A g = C[x 2 + y 2, xy].
Consequences Theorem. Let A be a quantum polynomial ring. If A has a reflection, then A has a normal element of degree. If A has a mystic reflection, then there is a basis {b, b 2,... b n } of A such that b 2 = b2 2 is a normal element, and C b, b 2 = C [x, y].
Sklyanin Algebras Let A be a non-pi Skylanin algebra of global dimension n 3. Then A has no element b of degree with b 2 normal. Hence A is rigid. If A has dimension 4, then the graded automorphisms were found by Smith and Staniszkis, and their traces, none of which have a pole of order 3, were found by Jing and Zhang. Dimension 3 PI calculations?
Theorem. Let A be a quantum polynomial ring and G a finite group of graded automorphisms such that A G is regular with H A G (t) = ( t) n q(t) having q() 0. Then q() = G and degree(q(t)) is the number of quasi-reflections in G.
Rees Ring of the Weyl Algebra A n Let A the algebra generated by x,..., x n, y,... y n, z with relations x i y i y i x i = z 2 for i =, 2,..., n. Let all other pairs of generators commute. The algebra A is AS-regular of dimension 2n +. Proposition. The algebra A is rigid.
Suppose A G is regular. If [ g is a quasi-reflection, ] then g is given by matrix of the form I 0 where I is a 2n 2n identity matrix. All have v order 2. Any group containing two quasi-reflections is infinite. [ ] [ ] [ ] I 0 I 0 I 0 =. v ū v ū
If A G is regular, then G = {e, g} where g is a quasi-reflection. If H A G (t) =, then degree(q) is the number of ( t) 2n+ q(t) quasi-reflections and q() = G. In this case A G is isomorphic to the algebra generated by X, X 2,..., X n, Y, Y 2,..., Y n, z 2 subject to the relations X i Y i Y i X i = z 2 with all other pairs of generators commuting. The algebra A G is regular but cannot be isomorphic to A, since A G can be generated by 2n elements over C whereas A requires 2n +.
Let g be a finite dimensional Lie algebra over K with bracket operation [, ]. If b, b 2,..., b n is a basis for g over K, then the enveloping algebra U(g) is the associative algebra generated by b, b 2,..., b n subject to the relations b i b j b j b i = [b i, b j ]. The homogenization of U(g), H(g) is the associative algebra generated by, b, b 2,..., b n, z subject to the relations b i z = zb i and b i b j b j b i = [b i, b j ]z. Then H(g) is regular of dimension n +.
Proposition. Let g be a finite dimensional Lie algebra with no -dimensional Lie ideal. Then H(g) is rigid. The proof consists of showing that there are no quasi-reflections.
Reference E. Kirkman, J. Kuzmanovich and J. J. Zhang, Rigidity of Graded Regular Algebras, Trans. Amer. Math. Soc. 360 (2008), No. 2, pages 633-6369.