elimination in non--commutative g--algebras and ... · elimination in non–commutative...

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)


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/


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


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


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


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).


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.


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. . .


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.


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


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.


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.


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).


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...

......

...

)


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〉.


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.


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


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


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.


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).


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.


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!


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.


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.


(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.


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


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.


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.


Thank you for your attention!¡Muchas gracias por su atencion!

Please visit the SINGULAR homepagehttp://www.singular.uni-kl.de/


elimination in non--commutative g--algebras and ... · elimination in non–commutative...

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