P LURAL , a Non–commutative Extension of S INGULAR : Past, Present and Future Viktor Levandovskyy SFB Project F1301 of the Austrian FWF Research Institute for Symbolic Computation (RISC) Johannes Kepler University Linz, Austria International Congress on Mathematical Software 2006 3.09.2006, Castro Urdiales Viktor Levandovskyy (RISC) P LURAL 3.09.2006, Castro Urdiales 1 / 29
What is P LURAL ? What is P LURAL ? P LURAL is the kernel extension of S INGULAR , providing a wide range of symbolic algoritms with non–commutative polynomial algebras ( GR –algebras). Gr¨ obner bases, Gr¨ obner basics, non–commutative Gr¨ obner basics more advanced algorithms for non–commutative algebras, P LURAL is distributed with S INGULAR (from version 3-0-0 on) freely distributable under GNU Public License available for most hardware and software platforms Viktor Levandovskyy (RISC) P LURAL 3.09.2006, Castro Urdiales 2 / 29
Preliminaries Let K be a field and R be a commutative ring R = K [ x 1 , . . . , x n ] . Mon ( R ) ∋ x α = x α 1 �→ ( α 1 , α 2 , . . . , α n ) = α ∈ N n . 1 x α 2 2 . . . x α n n Definition a total ordering ≺ on N n is called a well–ordering , if 1 ◮ ∀ F ⊆ N n there exists a minimal element of F , in particular ∀ a ∈ N n , 0 ≺ a an ordering ≺ is called a monomial ordering on R , if 2 ◮ ∀ α, β ∈ N n α ≺ β ⇒ x α ≺ x β ◮ ∀ α, β, γ ∈ N n such that x α ≺ x β we have x α + γ ≺ x β + γ . Any f ∈ R \ { 0 } can be written uniquely as f = cx α + f ′ , with 3 c ∈ K ∗ and x α ′ ≺ x α for any non–zero term c ′ x α ′ of f ′ . We define lm ( f ) = x α , the leading monomial of f lc ( f ) = c , the leading coefficient of f Viktor Levandovskyy (RISC) P LURAL 3.09.2006, Castro Urdiales 3 / 29
Towards GR –algebras Suppose we are given the following data a field K and a commutative ring R = K [ x 1 , . . . , x n ] , 1 a set C = { c ij } ⊂ K ∗ , 1 ≤ i < j ≤ n 2 a set D = { d ij } ⊂ R , 1 ≤ i < j ≤ n 3 Assume, that there exists a monomial well–ordering ≺ on R such that ∀ 1 ≤ i < j ≤ n , lm ( d ij ) ≺ x i x j . The Construction To the data ( R , C , D , ≺ ) we associate an algebra A = K � x 1 , . . . , x n | { x j x i = c ij x i x j + d ij } ∀ 1 ≤ i < j ≤ n � Viktor Levandovskyy (RISC) P LURAL 3.09.2006, Castro Urdiales 4 / 29
PBW Bases and G –algebras Define the ( i , j , k ) – nondegeneracy condition to be the polynomial NDC ijk := c ik c jk · d ij x k − x k d ij + c jk · x j d ik − c ij · d ik x j + d jk x i − c ij c ik · x i d jk . Theorem A = A ( R , C , D , ≺ ) has a PBW basis { x α 1 1 x α 2 2 . . . x α n n } if and only if ∀ 1 ≤ i < j < k ≤ n , NDC ijk reduces to 0 w.r.t. relations Definition An algebra A = A ( R , C , D , ≺ ) , where nondegeneracy conditions vanish, is called a G –algebra (in n variables). Viktor Levandovskyy (RISC) P LURAL 3.09.2006, Castro Urdiales 5 / 29
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 Cohen-Macaulay algebra. We say that a GR –algebra A = A / T A is a factor of a G –algebra in n variables A by a proper two–sided ideal T A . Viktor Levandovskyy (RISC) P LURAL 3.09.2006, Castro Urdiales 6 / 29
Examples of GR –algebras Mora, Apel, Kandri–Rody and Weispfenning, . . . algebras of solvable type, skew polynomial rings univ. enveloping algebras of fin. dim. Lie algebras quasi–commutative algebras, rings of quantum polynomials positive (resp. negative) parts of quantized enveloping algebras some iterated Ore extensions, some nonstandard quantum deformations many quantum groups Weyl, Clifford, exterior algebras Witten’s deformation of U ( sl 2 ) , Smith algebras algebras, associated to ( q –)differential, ( q –)shift, ( q –)difference and other linear operators Viktor Levandovskyy (RISC) P LURAL 3.09.2006, Castro Urdiales 7 / 29
Criteria for detecting useless critical pairs Generalized Product Criterion Let A be a G –algebra of Lie type (that is, all c ij = 1). Let f , g ∈ A . Suppose that lm ( f ) and lm ( g ) have no common factors, then spoly ( f , g ) → { f , g } [ g , f ] , where [ g , f ] := gf − fg is the Lie bracket. Chain Criterion If ( f i , f j ) , ( f i , f k ) and ( f j , f k ) are in the set of pairs P and x α j | lcm ( x α i , x α k ) , then we can delete ( f i , f k ) from P . The Chain Criterion can be proved with the Schreyer’s construction of the first syzygy module of a given module, which generalizes to the case of G –algebras. Viktor Levandovskyy (RISC) P LURAL 3.09.2006, Castro Urdiales 8 / 29
Left, right and twosided structures It suffices to have implemented left Gr¨ obner bases functionality for opposite algebras A op functionality for enveloping algebras A env = A ⊗ K A op mapping A → A op → A Then � op � LGB A op ( F op ) for a finite set F ⊂ A , RGB A ( F ) = 1 the two–sided Gr¨ obner can be computed, for instance, with the 2 algorithm by Manuel and Maria Garcia Roman in A env . Viktor Levandovskyy (RISC) P LURAL 3.09.2006, Castro Urdiales 9 / 29
Gr¨ obner Trinity With essentially the same algorithm, we can compute GB left Gr¨ obner basis G of a module M 1 SYZ left Gr¨ obner basis of the 1st syzygy module of M 2 LIFT the transformation matrix between two bases G and M 3 The algorithm for Gr¨ obner Trinity must be able to compute ... with submodules of free modules ◮ accept monomial module orderings as input ◮ distinguish preferred module components within factor algebras with extra weights for the ordering / module generators and to use the information on Hilbert polynomial Viktor Levandovskyy (RISC) P LURAL 3.09.2006, Castro Urdiales 10 / 29
Gr¨ obner basis engine ...is an (implementation of an) algorithm, designed to compute the Gr¨ obner Trinity and having the prescribed functionality. Gr¨ obner basis engine(s) behind S INGULAR ’s std command Gr¨ obner bases (non–negatively graded orderings) standard bases (local and mixed orderings) P LURAL (left Gr¨ obner bases for non–negatively graded orderings over GR –algebras) Viktor Levandovskyy (RISC) P LURAL 3.09.2006, Castro Urdiales 11 / 29
Potential Gr¨ obner basis engines slimgb — Slim Gr¨ obner basis implemented by M. Brickenstein uses t –representation and generalized t –Chain Criterion ”exchanging” normal form selection strategy prefers ”shorter” polynomials performs simultaneous reductions of a group of polys by a poly controls the size of coefficients janet — Janet involutive basis implemented by D. Yanovich, following the ideas of V. P . Gerdt an enhanced implementation is planned Viktor Levandovskyy (RISC) P LURAL 3.09.2006, Castro Urdiales 12 / 29
Gr¨ obner basics Buchberger, Sturmfels, ... GBasics are the most important and fundamental applications of Gr¨ obner Bases. Universal Gr¨ obner Basics Ideal (resp. module) membership problem (NF, REDUCE ) Intersection with subrings (elimination of variables) ( ELIMINATE ) Intersection of ideals (resp. submodules) ( INTERSECT ) Quotient and saturation of ideals ( QUOT ) Kernel of a module homomorphism ( MODULO ) Kernel of a ring homomorphism ( NCPREIMAGE . LIB ) Algebraic relations between pairwise commuting polynomials Hilbert polynomial of graded ideals and modules Viktor Levandovskyy (RISC) P LURAL 3.09.2006, Castro Urdiales 13 / 29
Anomalies With Elimination Admissible Subalgebras Let A = K � x 1 , . . . , x n | { x j x i = c ij x i x j + d ij } 1 ≤ i < j ≤ n � be a G –algebra. Consider a subalgebra A r , generated by { x r + 1 , . . . , x n } . We say that such A r is an admissible subalgebra , if d ij are polynomials in x r + 1 , . . . , x n for r + 1 ≤ i < j ≤ n and A r � A is a G –algebra. Definition (Elimination ordering) Let A and A r be as before and B := K � x 1 , . . . , x r | . . . � ⊂ A An ordering ≺ on A is an elimination ordering for x 1 , . . . , x r if for any f ∈ A , lm ( f ) ∈ B implies f ∈ B . Viktor Levandovskyy (RISC) P LURAL 3.09.2006, Castro Urdiales 14 / 29
Constructive Elimination Lemma ”Elimination of variables x 1 , . . . , x r from an ideal I ” means the intersection I ∩ A r with an admissible subalgebra A r . In contrast to the commutative case: • not every subset of variables determines an admissible subalgebra • there can be no admissible elimination ordering ≺ A r on A Lemma Let A be a G–algebra, generated by { x 1 , . . . , x n } and I ⊂ A be an ideal. Suppose, that the following conditions are satisfied: { x r + 1 , . . . , x n } generate an essential subalgebra B, ∃ an admissible elimination ordering ≺ B for x 1 , . . . , x r on A. Then, if S is a left Gr¨ obner basis of I with respect to ≺ B , we have S ∩ B is a left Gr¨ obner basis of I ∩ B. Viktor Levandovskyy (RISC) P LURAL 3.09.2006, Castro Urdiales 15 / 29
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