Elimination in non–commutative G–algebrasand Applications to D–modules
Viktor Levandovskyy
SFB Project F1301 of the Austrian FWFResearch Institute for Symbolic Computation (RISC)
Johannes Kepler UniversityLinz, Austria
Tenth Meeting on Computer Algebra and ApplicationsEACA 2006
8.9.2006, Sevilla, Andalucıa
Viktor Levandovskyy (RISC) Elimination 8.9.2006, Sevilla, Andalucıa 1 / 30
Implementation in PLURAL
What is PLURAL?PLURAL is the kernel extension of SINGULAR
PLURAL is distributed with SINGULAR (from version 3-0-0 on)freely distributable under GNU Public Licenseavailable for most hardware and software platforms
PLURAL as a Grobner engineimplementation of all the Grobner basics availablenew fast algorithm slimgb is available for Pluralinvolutive bases: janet is available for two–sided inputnon–commutative Grobner basics:
I as kernel functions (twostd, opposite etc)I as libraries (NCDECOMP.LIB, NCTOOLS.LIB, NCPREIMAGE.LIB etc)
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Software from RISC Linz
Algorithmic Combinatorics Group, Prof. Peter Paulemost of the software are packages for MATHEMATICA
created by P. Paule, A. Riese, C. Schneider, M. Kauers,K. Wegschaider, S. Gerhold, M. Schorn, F. Caruso, C. Mallinger,B. Zimmermann, C. Koutschan, T. Bayer, C. Weixlbaumer et al.
The Software is freely available for non–commercial usewww.risc.uni-linz.ac.at/research/combinat/software/
Viktor Levandovskyy (RISC) Elimination 8.9.2006, Sevilla, Andalucıa 3 / 30
Symbolic Summation
Hypergeometric SummationFASTZEIL, Gosper’s and Zeilberger’s algorithmsZEILBERGER, Gosper and Zeilberger alg’s for MAXIMA
MULTISUM, proving hypergeometric multi-sum identities
q–Hypergeometric SummationQZEIL, q–analogues of Gosper and Zeilberger alg’sBIBASIC TELESCOPE, generalized Gosper’s algorithm to bibasichypergeometric summationQMULTISUM, proving q–hypergeometric multi-sum identities
Symbolic Summation in Difference FieldsSIGMA, discovering and proving multi-sum identities
Viktor Levandovskyy (RISC) Elimination 8.9.2006, Sevilla, Andalucıa 4 / 30
More Software from RISC LinzSequences and Power Series
ENGEL, q–Engel ExpansionGENERATINGFUNCTIONS, manipulations with univariate holonomicfunctions and sequencesRLANGGFUN, inverse Schutzenberger methodology in MAPLE
Partition Analysis, Permutation GroupsOMEGA, Partition AnalysisPERMGROUP, permutation groups, group actions, Polya theory
Difference/Differential EquationsDIFFTOOLS, solving linear difference eq’s with poly coeffsORESYS, uncoupling systems of linear Ore operator equationsRATDIFF, rat. solutions of lin. difference eq’s after van HoeijSUMCRACKER, identities and inequalities, including summations
Viktor Levandovskyy (RISC) Elimination 8.9.2006, Sevilla, Andalucıa 5 / 30
Algebras in PLURAL: PreliminariesLet K be a field and R be a commutative ring R = K[x1, . . . , xn].
Mon(R) 3 xα = xα11 xα2
2 . . . xαnn 7→ (α1, α2, . . . , αn) = α ∈ Nn.
Definition1 a total ordering ≺ on Nn is called a well–ordering, if
I ∀F ⊆ Nn there exists a minimal element of F ,in particular ∀ a ∈ Nn, 0 ≺ a
2 an ordering ≺ is called a monomial ordering on R, ifI ∀α, β ∈ Nn α ≺ β ⇒ xα ≺ xβ
I ∀α, β, γ ∈ Nn such that xα ≺ xβ we have xα+γ ≺ xβ+γ .3 Any f ∈ R \ {0} can be written uniquely as f = cxα + f ′, with
c ∈ K∗ and xα′ ≺ xα for any non–zero term c′xα
′of f ′. We define
lm(f ) = xα, the leading monomial of flc(f ) = c, the leading coefficient of f
Viktor Levandovskyy (RISC) Elimination 8.9.2006, Sevilla, Andalucıa 6 / 30
Towards G–algebras
Consider the algebra over the field K,
A = K〈x1, . . . , xn | {xjxi = cijxixj + dij} ∀1 ≤ i < j ≤ n〉,
where ∀ 1 ≤ i < j ≤ n, cij ∈ K∗ and dij ∈ R.If there exists a monomial well–ordering ≺ on R such that
∀1 ≤ i < j ≤ n, lm(dij) ≺ xixj ,
then A has a PBW basis {xα11 xα2
2 . . . xαnn } if and only if
∀ 1 ≤ i < j < k ≤ n, NDCijk reduces to 0 w.r.t. the relations, where
NDCijk := cikcjk · dijxk − xkdij + cjk · xjdik − cij · dikxj + djkxi − cijcik · xidjk .
Such algebra A is called a G–algebra (in n variables).
Viktor Levandovskyy (RISC) Elimination 8.9.2006, Sevilla, Andalucıa 7 / 30
Nice Properties of G–algebras
We collect the properties in the following Theorem.
Theorem (Properties of G–algebras)Let A be a G–algebra in n variables. Then
A is left and right Noetherian,A is an integral domain,the Gel’fand–Kirillov dimension GKdim(A) = n + GKdim(K),the global homological dimension gl.dim(A) ≤ n,the Krull dimension Kr.dim(A) ≤ n,A is Auslander-regular and a (GKdim-)Cohen-Macaulay algebra.
We say that a GR–algebra A = A/TA is a factor of a G–algebra in nvariables A by a proper two–sided ideal TA.
Viktor Levandovskyy (RISC) Elimination 8.9.2006, Sevilla, Andalucıa 8 / 30
Examples of GR–algebras
Mora, Apel, Kandri–Rody and Weispfenning, . . .algebras of solvable type, skew polynomial ringsuniv. enveloping algebras of fin. dim. Lie algebrasquasi–commutative algebras, rings of quantum polynomialspositive (resp. negative) parts of quantized enveloping algebrassome iterated Ore extensions, some nonstandard quantumdeformations (Klimyk)many quantum groupsWeyl, Clifford, exterior algebrasWitten’s deformation of U(sl2), Smith algebrasalgebras, associated to (q–)differential, (q–)shift, (q–)differenceand other linear operators. . .
Viktor Levandovskyy (RISC) Elimination 8.9.2006, Sevilla, Andalucıa 9 / 30
Applications of Grobner bases
Grobner Basics are, according to Buchberger, Sturmfels et.al....the most important and fundamental applications of Grobner Bases.
Ideal (resp. module) membership problemIntersection with subrings (elimination of variables)Intersection of ideals (resp. submodules)Quotient and saturation of (two–sided) idealsKernel of a module homomorphismKernel of a ring homomorphismAlgebraic relations between pairwise commuting polynomialsHilbert polynomial of graded ideals and modules
This list is universal for both commutative and non–commutativesituations.
Viktor Levandovskyy (RISC) Elimination 8.9.2006, Sevilla, Andalucıa 10 / 30
Non-commutative Grobner basicsFor the noncommutative PBW world, we need even more basics:
Gel’fand–Kirillov dimension of a module (GKDIM.LIB)Two–sided Grobner basis of a bimodule (e.g. twostd)Annihilator of finite dimensional modulePreimage of one–sided ideal under algebra morphismFinite dimensional representationsGraded Betti numbers (for graded modules over graded algebras)Left and right kernel of the presentation of a moduleCentral Character Decomposition of a module (NCDECOMP.LIB)
Very ImportantExt and Tor modules for centralizing bimodules (NCHOMOLOG.LIB)Hochschild cohomology for modules
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Anomalies With Elimination
Contrast to Commutative CaseIn terminology, we rather use ”intersection with subalgebras” instead of”elimination of variables”, since the latter may have no sense.
Let A = K〈x1, . . . , xn | {xjxi = cijxixj + dij}1≤i<j≤n〉 be a G–algebra.Consider a subalgebra Ar , generated by {xr+1, . . . , xn}.We say that such Ar is an admissible subalgebra, if dij are polynomialsin xr+1, . . . , xn for r + 1 ≤ i < j ≤ n and Ar ( A is a G–algebra.
Definition (Elimination ordering)Let A and Ar be as before and B := K〈x1, . . . , xr | . . . 〉 ⊂ AAn ordering ≺ on A is an elimination ordering for x1, . . . , xrif for any f ∈ A, lm(f ) ∈ B implies f ∈ B.
Viktor Levandovskyy (RISC) Elimination 8.9.2006, Sevilla, Andalucıa 12 / 30
Constructive Elimination Lemma
”Elimination of variables x1, . . . , xr from an ideal I”means the intersection I ∩ Ar with an admissible subalgebra Ar .In contrast to the commutative case:• not every subset of variables determines an admissible subalgebra• there can be no admissible elimination ordering ≺Ar on A
Lemma
Let A be a G–algebra, generated by {x1, . . . , xn} and I ⊂ A be an ideal.Suppose, that the following conditions are satisfied:
{xr+1, . . . , xn} generate an essential subalgebra B,∃ an admissible elimination ordering ≺B for x1, . . . , xr on A.
Then, if S is a left Grobner basis of I with respect to ≺B, we have S ∩Bis a left Grobner basis of I ∩ B.
Viktor Levandovskyy (RISC) Elimination 8.9.2006, Sevilla, Andalucıa 13 / 30
Anomalies With Elimination: Example
Example
Consider the algebra A = K〈a,b | ba = ab + b2〉.It is a G–algebra with respect to any well–ordering, such that b2 ≺ ab,that is b ≺ a. Any elimination ordering for b must satisfy b � a, henceA is not a G–algebra w.r.t. any elimination ordering for b.
The Grobner basis of a two–sided ideal, generated by b2 − ba + ab inK〈a,b〉 w.r.t. an ordering b � a is infinite and equals to
{ban−1b − 1n
(ban − anb) | n ≥ 1}.
Finding an admissible elimination ordering can be done by solving alinear programming problem (ongoing work with J. Lobillo, Granada).
Viktor Levandovskyy (RISC) Elimination 8.9.2006, Sevilla, Andalucıa 14 / 30
Elimination Orderings in G–algebras
1 lexicographical orderings: mainly for (q–) Weyl algebras1 . . . 0...
. . ....
0 . . . 1
2 block orderings MA ⊗MB: a universal tool(
A 00 B
)3 extra weight orderings (a(w1, . . . ,wk ),ord): the champion!(
w1 . . . wk 0 . . . 0...
......
...
)
Viktor Levandovskyy (RISC) Elimination 8.9.2006, Sevilla, Andalucıa 15 / 30
Morphisms of general GR–algebras
SetupLet A = A/TA and B = B/TB be two GR–algebras andΦ : A −→ B be a map (respectively, a map φ : A −→ B).Define fi := NF(Φ(xi),TB) resp. fi := φ(xi).
Let Eo := A⊗K Bopp (a G–algebra), T oE := TA + T opp
B a two–sided idealand Eo := A⊗K Bopp = Eo/〈T o
E 〉 a GR–algebra.
Asymmetric constructionDefine the set So := {xi − φ(xi)
opp | 1 ≤ i ≤ n} ⊂ Eo. We view the(A,B)–bimodule A〈S〉B as the left ideal Io
φ := A⊗KBopp〈So〉.
Viktor Levandovskyy (RISC) Elimination 8.9.2006, Sevilla, Andalucıa 16 / 30
Morphisms of GR–algebras. Asymmetric method
Theorem (Asymmetric construction)Let A,B be GR–algebras (resp. A,B be G–algebras). Then thefollowing assertions hold:
G–algebras:φ ∈ Mor(A,B) if and only if Io
φ ∩ Bopp = 〈0〉,for any φ ∈ Mor(A,B), kerφ = Io
φ ∩ A,GR–algebras:Φ ∈ Mor(A,B) if and only if NF(Io
φ ∩ Bopp | T oppB ) = 〈0〉,
for any Φ ∈ Mor(A,B),
ker Φ = IoΦ ∩ A = NF(TA + (T opp
B + Ioφ) ∩ A | TA).
Limitation: We cannot compute the preimage of a left ideal with theAsymmetric Method.
Viktor Levandovskyy (RISC) Elimination 8.9.2006, Sevilla, Andalucıa 17 / 30
Example (U(sl2) → A1)
Let A1 = K〈x , ∂ | ∂x = x∂ + 1〉 be the first Weyl algebra.
Consider the map U(sl2)φ−→ A1, defined by
e 7→ x , f 7→ −x2∂, h 7→ 2x∂
Performing the elimination Ioφ ∩ Aopp
1 , we obtain zero ideal, hence
φ ∈ Mor(U(sl2),W1).
Computing another elimination Ioφ ∩ U(sl2), we get
kerφ = 〈4ef + h2 − 2h〉.
So, φ induces the embedding
0 → U(sl2)/〈4ef + h2 − 2h〉 −→ A1
Viktor Levandovskyy (RISC) Elimination 8.9.2006, Sevilla, Andalucıa 18 / 30
Symmetric Deformation: Motivation and MethodLet φ : A → B be a map of K–algebras. There are the natural actions
a.b := φ(a)b and b.a := bφ(a).
ObservationThis (A,A)–bimodule structure on B is well–defined iff φ is a morphism.
LemmaConsider the set G = {g − φ(g) | g ∈ A} ⊂ A⊗K B. Then
G = A〈{xi − φ(xi) | 1 ≤ i ≤ n}〉A ⊂ A⊗K B.
For e.g. Weyl/shift algebras, we introduce new relations like
yjxi = xiyj + [yj , fi ] for all 1 ≤ i ≤ n,1 ≤ j ≤ m.
Notation: (A,B, φ) → A⊗φK B
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Symmetric Deformation: Theorem
TheoremLet A,B be GR–algebras and Φ ∈ Mor(A,B).Let IΦ be the (A,A)–bimodule A〈{xi − Φ(xi) | 1 ≤ i ≤ n}〉A ⊂ A⊗K Band fi := Φ(xi). Suppose there exists an elimination ordering for B onA⊗K B, such that
1 ≤ i ≤ n,1 ≤ j ≤ m, lm(lc(fiyj)yj fi − lc(yj fi)fiyj) ≺ xiyj .
Then1) A⊗φK B is a G–algebra (resp. A⊗Φ
K B is a GR–algebra).2) Let J ⊂ B be a left ideal, then
Φ−1(J ) = (IΦ + J ) ∩ A.
Viktor Levandovskyy (RISC) Elimination 8.9.2006, Sevilla, Andalucıa 20 / 30
Elimination: Good Example
Example (U(h2) → A1)
Let A1 = K〈x , ∂ | ∂x = x∂ + 1〉 be the first Weyl algebra.For t ∈ Z let St = K〈a,b | [b,a] = t · a〉.For a fixed t ≥ 2, we consider the map St
ψt−→ A1, a 7→ x t ,b 7→ xd + t .
Computation: ∀t ∈ N, ψt is a morphism and kerψt = 〈0〉.
St ⊗ψtK A1 has new relations {[x ,b] = −x , [d ,a] = tx t−1, [d ,b] = d}.
The ordering condition is satisfied, if x t−1 ≺ ad and x � a,d � a.We come to admissible elimination ordering on St ⊗ψt
K A1 by using theextra weight vector (0,0,1, t), based on any well–ordering on variables(a,b, x ,d).
Viktor Levandovskyy (RISC) Elimination 8.9.2006, Sevilla, Andalucıa 21 / 30
Elimination: Bad Example
Example (U(sl2) → A1)
U(sl2) = K〈e, f ,h | fe = ef − h,he = eh + 2e,hf = fh − 2f 〉.Consider the map U(sl2)
φ−→ A1, defined by
e 7→ x , f 7→ −x2∂, h 7→ 2x∂.
By computing, we show φ ∈ Mor(U(sl2),A1).
Define E ′ = U(sl2)⊗φK A1, by introducing additional relations{[d ,e] = 1, [x , f ] = 2xd , [d , f ] = −d2, [x ,h] = −2x , [d ,h] = 2d}.
The ordering restrictions on E ′ fx � xd and fd � d2 hold iff f � d .But then the elimination condition {x ,d} � {e, f ,h} cannot be satisfiedon E ′ and preimage cannot be computed.
Viktor Levandovskyy (RISC) Elimination 8.9.2006, Sevilla, Andalucıa 22 / 30
A Recent Development:DMOD.LIB
Announced during ICM 2006 in MadridThe SINGULAR 3-0-2 library DMOD.LIB contains algorithms ofalgebraic D–Module Theory.A joint work of V. L. and J. M. Morales (Zaragoza).
Functionality: ANNFS algorithmsOaku–Takayama approach (ANNFSOT command)Briancon–Maisonobe approach (ANNFSBM command)Bernstein polynomial is computed within both approaches
Also there are some convenient auxiliary tools.
Constructively: two bigger rings are constructed and two eliminationsare applied in a sequence. Complexity of such computations is high!
Viktor Levandovskyy (RISC) Elimination 8.9.2006, Sevilla, Andalucıa 23 / 30
Application of Preimage Algorithm to D–modules
The algorithm of Oaku and Takayama (1999)The 1st step of the OT algorithm requires to compute the preimage ofthe left ideal
L = 〈{tj − fj ,p∑
j=1
∂fj∂xi
∂t j + ∂i}〉,1 ≤ j ≤ p,1 ≤ i ≤ n
in the subalgebra K[{tj · ∂t j}]〈{xi , ∂xi | [∂x i , xi ] = 1}〉 of
K〈{tj , ∂t j} | [∂t j , tj ] = 1〉 ⊗K K〈{xi , ∂xi} | [∂xi , xi ] = 1〉
Moreover, in the preimage, tj · ∂t j will be replaced by −sj − 1.
Viktor Levandovskyy (RISC) Elimination 8.9.2006, Sevilla, Andalucıa 24 / 30
Application of Preimage Algorithm to D–modules
Setup with the Symmetric Deformation
A := K〈sj ,Xi ,Di | DiXi = XiDi + 1〉
B := K〈tj ,Dtj , xi ,di | dixi = xidi + 1,Dtj tj = tjDtj + 1〉
Consider the map φ : A → B, where sj 7→ −tjDtj − 1, Xi 7→ xi ,Di 7→ di .
Hence, Iφ = 〈{Xi − xi ,Di − di , tjDtj + sj + 1}〉 ⊂ A⊗φK B =: E .
Due to the structure, we replace E with E ′ = K〈tj ,Dtj , xi ,di , sj〉 subjectto the relations
{[di , xi ] = 1, [Dtj , tj ] = 1, sj tj = tjsj − tj , sjDtj = Dtjsj + Dtj}.
Respectively, Iφ ⊂ E becomes I′φ = 〈{tjDtj + sj + 1}〉 ⊂ E ′.Any ordering ≺ satisfying {tj ,Dtj} � {xi ,di , sj} (which is very easy tofind) satisfies the conditions of the Theorem.
Viktor Levandovskyy (RISC) Elimination 8.9.2006, Sevilla, Andalucıa 25 / 30
(Dancing Flamenco?)
By the Theorem, for any L ⊂ B, φ−1(L) = (Iφ + L) ∩ A. Hence,
Iφ + L = 〈{tj − fj ,p∑
j=1
∂fj∂xi
∂t j + ∂i , tj∂tj + sj + 1}〉 =
= 〈{tj − fj ,p∑
j=1
∂fj∂xi
∂t j + ∂i , fj∂tj + sj}〉
Citing Gago–Vargas, Hartillo and Ucha JSC paper from 2005...
”...As far as we know, the example f = (x2 + y3) · (x3 + y2) isintractable for available computer algebra systems.”→ Demonstration in September 2006.
Viktor Levandovskyy (RISC) Elimination 8.9.2006, Sevilla, Andalucıa 26 / 30
Selected Benchmarks
Computed by M. Brickenstein on Opteron 2,2 Ghz, 16 GB RAM
Example Order SLIMGB STD
Ucha2 block 3s > 200sUcha4 lp 0.015s 0.22sTarasov2 dp 15s > 10000sBernstein5 block 1min � 10000s
V.L.,SINGULAR 3-0-2 on Pentium III 866 MHz, 1GB RAM
Test KAN MACAULAY2 PLURAL 3-0-2y3 − x7 0.77s 1.83s 0.21sy5 − x7 3.05s * 0.74s
Viktor Levandovskyy (RISC) Elimination 8.9.2006, Sevilla, Andalucıa 27 / 30
Sneak Peek: Annihilator of a Holonomic Module
DefinitionLet A be a K–algebra and M be a fin. gen A–module. M is calledholonomic module, if GKdim AnnA(M) = 2 ·GKdim M.
By the work of D. Vogan, Harish–Chandra modules over univ.enveloping algebras of fin. dim. complex Lie algebras are holonomic.
New DevelopmentThere is an algorithm (though a complicated one), computing AnnA(M)for a module M. It terminates, if the GKdim AnnA(M) is given explicitly.
Open ProblemsDetermine algorithmically, whether a given module is holonomic (e.g.via homological algebra).More complicated: compute GKdim AnnA(M) for a given M.
Viktor Levandovskyy (RISC) Elimination 8.9.2006, Sevilla, Andalucıa 28 / 30
Perspectives
Grobner bases for more non–commutative algebras• tensor product of commutative local algebras with certainnon–commutative algebras• different localizations of G–algebras
localization at some ”coordinate” ideal of commutative variables(producing e.g. local Weyl algebras K[x ]〈x〉〈D | Dx = xD + 1〉)
⇒ local orderings and the generalization of standard basisalgorithm, Grobner basics and homological algebralocalization as field of fractions of commutative variables(producing e.g. rational Weyl algebras K(x)〈D | Dx = xD + 1〉),including Ore Algebras (F. Chyzak, B. Salvy)
⇒ global orderings and a generalization Grobner basis algorithm.
Viktor Levandovskyy (RISC) Elimination 8.9.2006, Sevilla, Andalucıa 29 / 30
Thank you for your attention!¡Muchas gracias por su atencion!
Please visit the SINGULAR homepagehttp://www.singular.uni-kl.de/
Viktor Levandovskyy (RISC) Elimination 8.9.2006, Sevilla, Andalucıa 30 / 30