Abstract
Let W+ be the positive Witt algebra, which has a \(\mathcal {C}\)-basis \(\{e_{n}: n \in \mathcal {Z}_{\geq 1}\}\), with Lie bracket [ei,ej] = (j − i)ei+j. We study the two-sided ideal structure of the universal enveloping algebra U(W+) of W+. We show that if I is a (two-sided) ideal of U(W+) generated by quadratic expressions in the ei, then U(W+)/I has finite Gelfand-Kirillov dimension, and that such ideals satisfy the ascending chain condition. We conjecture that analogous facts hold for arbitrary ideals of U(W+), and verify a version of these conjectures for radical Poisson ideals of the symmetric algebra S(W+).
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Acknowledgments
The first author was supported by Leverhulme Trust Grant RPG-2013-293 and RFBR grant 16-01-00818. The second author was supported by EPSRC grant EP/M008460/1.
We would like to thank Jacques Alev, Tom Lenagan, Omar Leon Sanchez, Paul Smith and Toby Stafford for helpful discussions. We would particularly like to thank Ioan Stanciu, whose computer experiments, done as part of his MMath dissertation at the University of Edinburgh, gave us experimental evidence for Conjecture 1.2.
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Appendix: Macaulay2 computations
Appendix: Macaulay2 computations
We present the routines needed for the proof of Lemma 4.3.
Routine A.1
The following Macaulay2 code is used in the proof of the general case of Lemma 4.3.
We first define the vectors v0,…v6 from the proof of Lemma 4.3.
Macaulay2, version 1.7
with packages: ConwayPolynomials, Elimination, IntegralClosure,
LLLBases, PrimaryDecomposition, ReesAlgebra,
TangentCone
i1 : R = QQ[I,J];
i2 : v0={(J-6),
(J + 4)⋆(J-5),
(J + 2)⋆(J-4),
(J + 4)⋆(J + 3)⋆(J-4),
J⋆(J-3),
(J + 4)⋆(J + 1)⋆(J-3),
(J + 4)⋆(J + 3)⋆(J + 2)⋆(J-3),
(J + 2)⋆J⋆(J-2),
(J + 4)⋆(J + 3)⋆J⋆(J-2),
(J + 4)⋆(J + 3)⋆(J + 2)⋆(J + 1)⋆(J-2),
(J + 4)⋆(J + 3)⋆(J + 2)⋆(J + 1)⋆J⋆(J-1)};
i3 : v1={0, (I-1)⋆(J-5),
0, 2⋆(I-1)⋆(J + 3)⋆(J-4),
0, (I-1)⋆(J + 1)⋆(J-3),
3⋆(I-1)⋆(J + 3)⋆(J + 2)⋆(J-3),
0, 2⋆(I-1)⋆(J + 3)⋆J⋆(J-2),
4⋆(I-1)⋆(J + 3)⋆(J + 2)⋆(J + 1)⋆(J-2),
6⋆(I-1)⋆(J + 3)⋆(J + 2)⋆(J + 1)⋆J⋆(J-1)};
i4 : v2={0, 0, (I-2)⋆(J-4),
I⋆(I-1)⋆(J-4),
0, (I-2)⋆(J + 2)⋆(J-3),
3⋆I⋆(I-1)⋆(J + 2)⋆(J-3),
3⋆(I-2)⋆J⋆(J-2),
I⋆(I-1)⋆J⋆(J-2)+ 2⋆(I-2)⋆(J + 2)⋆(J + 1)⋆(J-2),
6⋆I⋆(I-1)⋆(J + 2)⋆(J + 1)⋆(J-2)
+(I-2)⋆(J + 2)⋆(J + 1)⋆J⋆(J-1),
15⋆I⋆(I-1)⋆(J + 2)⋆(J + 1)⋆J⋆(J-1)};
i5 : v3={0, 0, 0, 0, 2⋆(I-3)⋆(J-3),
(I + 1)⋆(I-2)⋆(J-3)+(I-3)⋆(J + 1)⋆(J-2),
(I + 1)⋆I⋆(I-1)⋆(J-3)+(I-3)⋆(J + 1)⋆J⋆(J-1),
0, 4⋆(I + 1)⋆(I-2)⋆(J + 1)⋆(J-2),
4⋆(I + 1)⋆I⋆(I-1)⋆(J + 1)⋆(J-2)
+ 4⋆(I + 1)⋆(I-2)⋆(J + 1)⋆J⋆(J-1),
20⋆(I + 1)⋆I⋆(I-1)⋆(J + 1)⋆J⋆(J-1)};
i6 : v4={0, 0, (I-4)⋆(J-2),
(I-4)⋆J⋆(J-1),
0, (I + 2)⋆(I-3)⋆(J-2),
3⋆(I + 2)⋆(I-3)⋆J⋆(J-1),
3⋆I⋆(I-2)⋆(J-2),
2⋆(I + 2)⋆(I + 1)⋆(I-2)⋆(J-2)+I⋆(I-2)⋆J⋆(J-1),
(I + 2)⋆(I + 1)⋆I⋆(I-1)⋆(J-2)
+ 6⋆(I + 2)⋆(I + 1)⋆(I-2)⋆J⋆(J-1),
15⋆(I + 2)⋆(I + 1)⋆I⋆(I-1)⋆J⋆(J-1)};
i7 : v5={0, (I-5)⋆(J-1),
0, 2⋆(I + 3)⋆(I-4)⋆(J-1),
0, (I + 1)⋆(I-3)⋆(J-1),
3⋆(I + 3)⋆(I + 2)⋆(I-3)⋆(J-1),
0, 2⋆(I + 3)⋆I⋆(I-2)⋆(J-1),
4⋆(I + 3)⋆(I + 2)⋆(I + 1)⋆(I-2)⋆(J-1),
6⋆(I + 3)⋆(I + 2)⋆(I + 1)⋆I⋆(I-1)⋆(J-1)};
i8 : v6={(I-6),
(I + 4)⋆(I-5),
(I + 2)⋆(I-4),
(I + 4)⋆(I + 3)⋆(I-4),
I⋆(I-3),
(I + 4)⋆(I + 1)⋆(I-3),
(I + 4)⋆(I + 3)⋆(I + 2)⋆(I-3),
(I + 2)⋆I⋆(I-2),
(I + 4)⋆(I + 3)⋆I⋆(I-2),
(I + 4)⋆(I + 3)⋆(I + 2)⋆(I + 1)⋆(I-2),
(I + 4)⋆(I + 3)⋆(I + 2)⋆(I + 1)⋆I⋆(I-1)};
We define an automorphism f of Z[i,j] which sends i↦i + 1,j↦j − 1.
i9 : f=map(R,R,{I + 1,J-1});
i10 : dof = L -> toList apply(0..10, i->f(L#i));
We compute the locus on which the vectors v0,v1,v2,v3,f(v0),f(v1),f(v2), f2(v0),f2(v1),f3(v0) are linearly independent, and find the top-dimensional components of this locus.
i11 : M=matrix{v0,v1,v2,v3,dof(v0),dof(v1),
dof(v2), dof(dof(v0)),dof(dof(v1)),
dof(dof(dof(v0)))};
i12 : K=minors(10,M);
i13 : KK=topComponents K;
i14 : associatedPrimes KK
o14 = {ideal I, ideal(J - 1), ideal(J + 1),
ideal(I + 1), ideal(I - 1), ideal(I - J + 3)
Routine A.2
This routine is used for the case j = i + 6 of Lemma 4.3. i15 : S=QQ[I];
i16 : g6=map(S,R,{I,I + 6});
i17 : dg6 = L -> toList apply(0..10, i->g6(L#i));
i18 : N6=matrix{dg6(v0), dg6(v1), dg6(v2),dg6(v3),
dg6(dof(v0)), dg6(dof(v1)), dg6(dof(v2)),
dg6(dof(dof(v0))), dg6(dof(dof(v1))),
dg6(dof(dof(dof(v0+v6))))};
i19 : J6=minors(10,N6);
i20 : associatedPrimes J6
o20 = {ideal I, ideal(I - 1), ideal(I + 5),
ideal(I + 1), ideal(I + 7)}
Routine A.3
This routine is used for the case j = i + 5 of Lemma 4.3.
i21 : g5=map(S,R,{I,I + 5});
i22 : dg5 = L -> toList apply(0..10, i->g5(L#i));
i23 : N5=matrix{dg5(v0), dg5(v1), dg5(v2), dg5(v3),
dg5(dof(v0)), dg5(dof(v1)), dg5(dof(v2)),
dg5(dof(dof(v0))),dg5(dof(dof(v1+v6)))};
i24 : J5=minors(9,N5);
i25 : associatedPrimes J5 o25 = {ideal I, ideal(I - 1)}
Routine A.4
This routine is used for the case j = i + 4 of Lemma 4.3.
i26 : g4=map(S,R,{I,I + 4});
i27 : dg4 = L -> toList apply(0..10, i->g4(L#i));
i28 : N4=matrix{dg4(v0), dg4(v1), dg4(v2), dg4(v3),
dg4(dof(v0)), dg4(dof(v1)), dg4(dof(v2+v6)),
dg4(dof(dof(v0+v6))),dg4(dof(dof(v1+v5)))};
i29 : J4=minors(9,N4);
i30 : associatedPrimes J4
o30 = {ideal I, ideal(I - 1), ideal(I + 1)}
Routine A.5
This routine is used for the case j = i + 3 of Lemma 4.3.
i31 : g3=map(S,R,{I,I + 3});
i32 : dg3 = L -> toList apply(0..10, i->g3(L#i));
i33 : N3=matrix{dg3(v0), dg3(v1), dg3(v2), dg3(v3+v6),
dg3(dof(v0)), dg3(dof(v1+v6)), dg3(dof(v2+v5))};
i34 : J3=minors(7,N3);
i35 : associatedPrimes J3
o35 = {ideal(I - 1), ideal(2I + 3)}
Routine A.6
This routine is used for the case j = i + 2 of Lemma 4.3.
i36 : g2=map(S,R,{I,I + 2});
i37 : dg2 = L -> toList apply(0..10, i->g2(L#i));
i38 : N2=matrix{dg2(v0), dg2(v1), dg2(v2+v6), dg2(v3+v5),
dg2(dof(v0+v6)),dg2(dof(v1+v5)),
dg2(dof(v2+v4))};
i39 : J2=minors(7,N2);
i40 : associatedPrimes J2
o40 = {ideal I, ideal(I - 1), ideal(I + 1)}
Routine A.7
This routine is used for the case j = i + 1 of Lemma 4.3.
i41 : g1=map(S,R,{I,I + 1});
i42 : dg1 = L -> toList apply(0..10, i->g1(L#i));
i43 : N1=matrix{dg1(v0), dg1(v1+v6), dg1(v2+v5),
dg1(v3+v4)};
i44 : J1=minors(4,N1);
i45 : associatedPrimes J1 o45 = {}
Routine A.8
This routine is used for the case j = i of Lemma 4.3.
i46 : g0=map(S,R,{I,I});
i47 : dg0 = L -> toList apply(0..10, i->g0(L#i));
i48 : N0=matrix{dg0(v0+v6),dg0(v1+v5),
dg0(v2+v4), dg0(v3)};
i49 : J0=minors(4,N0);
i50 : associatedPrimes J0
o50 = {ideal(I - 1)}
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Petukhov, A.V., Sierra, S.J. Ideals in the Enveloping Algebra of the Positive Witt Algebra. Algebr Represent Theor 23, 1569–1599 (2020). https://doi.org/10.1007/s10468-019-09896-2
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DOI: https://doi.org/10.1007/s10468-019-09896-2
Keywords
- Witt algebra
- Positive Witt algebra
- Poisson algebra
- Poisson Gelfand-Kirillov dimension
- Ascending chain condition