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Rigidity of Artin-Schelter Regular Algebras
Ellen Kirkman and James KuzmanovichWake Forest University
James ZhangUniversity of Washington
Rigidity of Artin-Schelter Regular Algebras
Shephard-Todd-Chevalley Theorem
Theorem. The ring of invariants C[x1, · · · , xn]G under a finitegroup G is a polynomial ring if and only if G is generated byreflections.
Question. For an Artin-Schelter regular algebra A, when is AG
isomorphic to A for a finite group of graded automorphisms G ?
Question. For an Artin-Schelter regular algebra A, when is AG
Artin-Schelter regular?
Rigidity of Artin-Schelter Regular Algebras
Results in Another Direction
Theorem. (S. P. Smith, 1989) The Weyl algebra A1(k) is not thefixed subring SG under a finite solvable group of automorphisms ofa domain S.Theorem. (J. Alev and P. Polo, 1995)
1. Let g and g′ be two semisimple Lie algebras. Let G be a finitegroup of algebra automorphisms of U(g) such thatU(g)G ∼= U(g′). Then G is trivial and g ∼= g′.
2. If G is a finite group of algebra automorphisms of An(k), thenthe fixed subring An(k)G is isomorphic to An(k) only when Gis trivial.
Rigidity of Artin-Schelter Regular Algebras
Questions in this Other Direction
Let A be Artin-Schelter regular.
Question. When is it the case that AG is never isomorphic to Afor a finite group G of graded automorphisms?
Call such algebras rigid.
Question. When is it the case that AG is never Artin-Schelterregular for a finite group G of graded automorphisms?
Rigidity of Artin-Schelter Regular Algebras
Example
Let A = C−1[x , y ] and let 〈g〉 where g =
[ξ 00 1
]for a primitive
nth root of unity ξ.
I If n is odd, then A〈g〉 = C−1[xn, y ] which is isomorphic to A.
I If n is even, then A〈g〉 = C[xn, y ], a commutative polynomialring, which is not isomorphic to A.
Rigidity of Artin-Schelter Regular Algebras
Hilbert Series of Regular Algebras
Let B be a graded algebra. The Hilbert series of B is defined by
HB(t) =∞∑k=0
dimBktk .
Proposition. (Stephenson-Zhang, Jing-Zhang, ATV) Let B be anArtin-Schelter regular algebra and let
HB(t) =1
(1− t)np(t)
where p(1) 6= 0. Furhtermore n = GKdim(B) and p(t) is a productof cyclotomic polynomials.
Rigidity of Artin-Schelter Regular Algebras
Traces of Graded Automorphisms
Let g be a graded automorphism of a graded algebra A. Thetrace of g is defined by
Tr(g , t) =∞∑n=0
tr(g |An)tn.
Note HA(t) = Tr(Id , t).
Rigidity of Artin-Schelter Regular Algebras
Molien’s Theorem
Theorem. (Jørgensen-Zhang) Let B be a connected gradedK -algebra and let G be a finite group of graded automorphisms ofB with |G |−1 ∈ K . Then
HBG (t) =1
|G |∑g∈G
TrB(g , t).
Rigidity of Artin-Schelter Regular Algebras
Quasi-Reflections
Let G be an automorphism of an AS-regular algebra A withGKdim(A) = n. We call g a quasi-reflection if
Tr(g , t) =1
(1− t)n−1p(t)
with p(1) 6= 0.
Rigidity of Artin-Schelter Regular Algebras
You Need Quasi-Reflections
Theorem. Let G be a finite group of graded automorphisms of aNoetherian AS-regular algebra A. If AG is AS-regular, then G mustcontain a quasi-reflection.
Lemma. Let f (t) = a0 + a1t + · · ·+ antn be a palindrome
polynomial; that is, an−i = ai for all i . Then f ′(1) = nf (1)
2.
Rigidity of Artin-Schelter Regular Algebras
A Sketch of the Proof
Proof. Assume that G does not contain a quasi-reflection.Let
HA(t) =1
(1− t)np(t).
Suppose that AG is regular. Then
HAG (t) =1
(1− t)nq(t),
where q(t) is a product of cyclotomic polynomials. Since G isnontrivial, ` = deg(q(t)) > deg(p(t)) = k .
Rigidity of Artin-Schelter Regular Algebras
Expand HA(t) and HAG (t) into a Laurent series about t = 1.
HA(t) =1
(1− t)n1
p(1)+
1
(1− t)n−1p′(1)
p(1)2+ · · ·
HAG (t) =1
(1− t)n1
q(1)+
1
(1− t)n−1q′(1)
q(1)2+ · · ·
Rigidity of Artin-Schelter Regular Algebras
If we expand HAG (t) =1
|G |∑g∈G
Tr(g , t) into a Laurent series
around t = 1, the first terms come entirely from the trace of theidentity.
HAG (t) =1
|G |
[1
(1− t)n1
p(1)+
1
(1− t)n−1p′(1)
p(1)2+ · · ·
HAG (t) =1
(1− t)n1
q(1)+
1
(1− t)n−1q′(1)
q(1)2+ · · ·
Equating coefficients q(1) = |G |p(1), andq′(1)
q(1)2=
1
|G |p′(1)
p(1)2.
Rigidity of Artin-Schelter Regular Algebras
Since p(t) and q(t) are products of cyclotomic polynomials, theyare palindrome polynomials, and hence by the Lemma
q′(1) = `q(1)
2and p′(1) = k
p(1)
2.
Substituting inq′(1)
q(1)2=
1
|G |p′(1)
p(1)2we have
`
2q(1)=
1
|G |k
2p(1).
Since q(1) = |G |p(1), it follows that ` = k , which is acontradiction. �
Rigidity of Artin-Schelter Regular Algebras
Jordan Plane
The Jordan Plane CJ [x , y ] is defined by yx − xy = x2.
All graded automorphisms are of the form g = ξId with trace
Tr(g , t) =1
(1− ξt)2.
Hence the Jordan Plane is rigid.
Rigidity of Artin-Schelter Regular Algebras
Graded Down-Up Algebras
Let α, β ∈ C with β 6= 0. Then the down-up algebra A(α, β, 0) isthe algebra generated by two elements u, d subject to the relations
d2u = αdud + βud2
du2 = αudu + βu2d .
Then A is a Noetherian Artin-Schelter domain with gldim(A) = 3.Benkart and Roby have shown that A has a vector space basisconsisting of all monomials of the form ui (du)jdk . Hence
HA(t) =1
(1− t)2(1− t2).
Rigidity of Artin-Schelter Regular Algebras
Proposition. The graded automorphisms of a graded down-upalgebra A = A(α, β, 0) are given by
1.
[w 00 z
]for any A(α, β, 0).
2.
[0 xy 0
]when A = A(0, 1, 0) or A(α,−1, 0) for any α.
3.
[w xy z
], when A = A(0, 1, 0) or A(2,−1, 0).
Furthermore if the eigenvalues of the defining matrix for g are λand µ, then
Tr(g , t) =1
(1− λt)(1− µt)(1− λµt2).
Rigidity of Artin-Schelter Regular Algebras
Graded Down-Up Algebras are Rigid
If g is a graded automorphism, then
Tr(g , t) =1
(1− λt)(1− µt)(1− λµt2).
Hence if g is a quasi-reflection
Tr(g , t) =1
(1− t)(1− µt)(1− µt2).
Thus µ = 1 and we have a pole of order 3, not 2.
Rigidity of Artin-Schelter Regular Algebras
A More General Result
Theorem. Let A be Noetherian regular with gldim(A) = 3. If A isgenerated by two elements of degree 1, then A is rigid.
Nonproof. If g is a quasi-reflection, then
Tr(g , t) =1
(1− t)2(1− ξ1t)(1− ξ2t)
for roots of unity ξ1, ξ2.
Rigidity of Artin-Schelter Regular Algebras
Quantum Polynomial Rings
A quantum polynomial ring of dimension n is a NoetherianAS-regular domain of global dimension n with Hilbert series
HA(t) =1
(1− t)n.
Hence quasi-reflections have traces of the form
TrA(g , t) =1
(1− t)n−1(1− ξt).
Rigidity of Artin-Schelter Regular Algebras
The Quasi-Reflections
Theorem. The quasi-reflections of a quantum polynomial ring Aare of two types. If g is a quasi-reflection, then either
I (Reflections) there is a basis {b1, · · · , bn} of A1, such thatg(b1) = ξb1 and g(bj) = bj for j ≥ 2, or
I (Mystic Reflections) the order of g is 4, and there is a basis{b1, · · · , bn} of A1, such that g(b1) = ib1, g(b2) = −ib2, andg(bj) = bj for j ≥ 3.
I In each case A〈g〉 is regular.
Rigidity of Artin-Schelter Regular Algebras
Example
Let A = C−1[x , y ].
I Let g be given by
[ξ 00 1
]where ξ is an nth root of unity.
Then g is a reflection and
Tr(g , t) =1
(1− t)(1− ξt).
A〈g〉 ∼= C±1[xn, y ].
I Let g be given by
[0 −11 0
].
Rigidity of Artin-Schelter Regular Algebras
Then g is a mystic reflection with
b1 = x − iy , b2 = x + iy .
The trace of g is given by
Tr(g , t) =1
(1− t)(1 + t).
The invariant ring is given by A〈g〉 ∼= C[x2 + y2, xy ].
Rigidity of Artin-Schelter Regular Algebras
Consequences
Theorem. Let A be a quantum polynomial ring.
I If A has a reflection, then A has a normal element of degree 1.I If A has a mystic reflection, then there is a basis{b1, b2, . . . bn} of A1 such that
I b21 = b2
2 is a normal element, andI C〈b1, b2〉 ∼= C−1[x , y ].
Rigidity of Artin-Schelter Regular Algebras
Sklyanin Algebras
I Let A be a non-PI Skylanin algebra of global dimension n ≥ 3.Then A has no element b of degree 1 with b2 normal. HenceA is rigid.
I If A has dimension 4, then the graded automorphisms werefound by Smith and Staniszkis, and their traces, none ofwhich have a pole of order 3, were found by Jing and Zhang.
I Dimension 3 PI calculations?
Rigidity of Artin-Schelter Regular Algebras
Theorem. Let A be a quantum polynomial ring and G a finitegroup of graded automorphisms such that AG is regular with
HAG (t) =1
(1− t)nq(t)
having q(1) 6= 0. Then q(1) = |G | and degree(q(t)) is the numberof quasi-reflections in G .
Rigidity of Artin-Schelter Regular Algebras
Rees Ring of the Weyl Algebra An
Let A the algebra generated by x1, . . . , xn, y1, . . . yn, z withrelations xiyi − yixi = z2 for i = 1, 2, . . . , n.Let all other pairs of generators commute.
The algebra A is AS-regular of dimension 2n + 1.
Proposition. The algebra A is rigid.
Rigidity of Artin-Schelter Regular Algebras
Suppose AG is regular.
I If g is a quasi-reflection, then g is given by matrix of the form[I 0v −1
]where I is a 2n × 2n identity matrix. All have
order 2.
I Any group containing two quasi-reflections is infinite.[I 0v −1
] [I 0u −1
]=
[I 0
v − u 1
].
Rigidity of Artin-Schelter Regular Algebras
I If AG is regular, then G = {e, g} where g is a quasi-reflection.If HAG (t) = 1
(1−t)2n+1q(t), then degree(q) is the number of
quasi-reflections and q(1) = |G |.I In this case AG is isomorphic to the algebra generated by
X1,X2, . . . ,Xn,Y1,Y2, . . . ,Yn, z2 subject to the relations
XiYi − YiXi = z2 with all other pairs of generatorscommuting.
I The algebra AG is regular but cannot be isomorphic to A,since AG can be generated by 2n elements over C whereas Arequires 2n + 1.
Rigidity of Artin-Schelter Regular Algebras
Let g be a finite dimensional Lie algebra over K with bracketoperation [, ]. If b1, b2, . . . , bn is a basis for g over K , then theenveloping algebra U(g) is the associative algebra generated byb1, b2, . . . , bn subject to the relations bibj − bjbi = [bi , bj ].
The homogenization of U(g),H(g) is the associative algebragenerated by , b1, b2, . . . , bn, z subject to the relations biz = zbi
and bibj − bjbi = [bi , bj ]z .Then H(g) is regular of dimension n + 1.
Rigidity of Artin-Schelter Regular Algebras
Proposition. Let g be a finite dimensional Lie algebra with no1-dimensional Lie ideal. Then H(g) is rigid.
The proof consists of showing that there are no quasi-reflections.
Rigidity of Artin-Schelter Regular Algebras