A Factorization Algorithm for G -Algebras and Applications ACA 2016 – Kassel – Germany Albert Heinle Symbolic Computation Group David R. Cheriton School of Computer Science University of Waterloo Canada 2016–08–04 1 / 30
Introduction On Non-Commutative Finite Factorization Domains Non-Commutative Factorized Gr¨ obner Bases Conclusion and Future Work 2 / 30
Introduction 3 / 30
Factorization Properties of Integral Domains For integral domains (in the literature commonly assumed to be commutative rings) many factorization properties have been defined. (c.f. (Anderson et al., 1990; Anderson and Anderson, 1992; Anderson and Mullins, 1996; Anderson, 1997)) Figure : from (Anderson et al., 1990) 4 / 30
Factorization Properties of Integral Domains For integral domains (in the literature commonly assumed to be commutative rings) many factorization properties have been defined. (c.f. (Anderson et al., 1990; Anderson and Anderson, 1992; Anderson and Mullins, 1996; Anderson, 1997)) Figure : Created on https://imgflip.com/ 4 / 30
What has been done for Non-Commutative Rings? ◮ Free associative algebras are unique factorization domains (Cohn, 1963). ◮ Certain Ore domains (like the Weyl algebra) are unique factorization domains (e.g. (Bueso et al., 2003)). 5 / 30
What has been done for Non-Commutative Rings? ◮ Free associative algebras are unique factorization domains (Cohn, 1963). ◮ Certain Ore domains (like the Weyl algebra) are unique factorization domains (e.g. (Bueso et al., 2003)). STOP 5 / 30
What has been done for Non-Commutative Rings? ◮ Free associative algebras are unique factorization domains (Cohn, 1963). ◮ Certain Ore domains (like the Weyl algebra) are unique factorization domains (e.g. (Bueso et al., 2003)). STOP The factors are only unique up to similarity! 5 / 30
What has been done for Non-Commutative Rings? ◮ Free associative algebras are unique factorization domains (Cohn, 1963). ◮ Certain Ore domains (like the Weyl algebra) are unique factorization domains (e.g. (Bueso et al., 2003)). STOP The factors are only unique up to similarity! Definition Let R be a ring. Two elements a , b ∈ R are said to be similar , if R / Ra and R / Rb are isomorphic as left R -modules. However, similarity is a very weak property, as one can e.g. see in (Giesbrecht and Heinle, 2012). 5 / 30
On Non-Commutative Finite Factorization Domains 6 / 30
Definitions Definition (Commutative FFD, cf. (Anderson et al., 1990)) Let R be a commutative integral domain. Then R is a finite factorization domain (FFD) if each nonzero non-unit of R has only a finite number of non-associate divisors and hence, only a finite number of factorizations up to order and associates. Definition (Non-Commutative FFD, cf. (Bell et al., 2014)) Let A be a (not necessarily commutative) domain. We say that A is a finite factorization domain (FFD, for short), if every nonzero, non-unit element of A has at least one factorization into irreducible elements and there are at most finitely many distinct factorizations into irreducible elements up to multiplication of the irreducible factors by central units in A . 7 / 30
Definitions Definition (Commutative FFD, cf. (Anderson et al., 1990)) Let R be a commutative integral domain. Then R is a finite factorization domain (FFD) if each nonzero non-unit of R has only a finite number of non-associate divisors and hence, only a finite number of factorizations up to order and associates. Definition (Non-Commutative FFD, cf. (Bell et al., 2014)) Let A be a (not necessarily commutative) domain. We say that A is a finite factorization domain (FFD, for short), if every nonzero, non-unit element of A has at least one factorization into irreducible elements and there are at most finitely many distinct factorizations into irreducible elements up to multiplication of the irreducible factors by central units in A . 7 / 30
Necessary Conditions for Non-Commutative FFDs Theorem (cf. (Bell et al., 2014)) Let K be an algebraically closed field and let A be a K -algebra. If there exists a finite-dimensional filtration { V n : n ∈ N } on A such that the associated graded algebra B = gr V ( A ) is a (not necessarily commutative) domain over K , then A is a finite factorization domain over K . 8 / 30
Necessary Conditions for Non-Commutative FFDs Theorem (cf. (Bell et al., 2014)) Let K be an algebraically closed field and let A be a K -algebra. If there exists a finite-dimensional filtration { V n : n ∈ N } on A such that the associated graded algebra B = gr V ( A ) is a (not necessarily commutative) domain over K , then A is a finite factorization domain over K . Corollary (cf. (Bell et al., 2014)) Let K be a field and let A be a K -algebra. If there exists a finite-dimensional filtration { V n : n ∈ N } on A such that the associated graded algebra B = gr V ( A ) has the property that B ⊗ K K is a (not necessarily commutative) domain, then A is a finite factorization domain. 8 / 30
Example for a Commutative Non-FFD Example Let K = R and A = R + C [ t ] · t ⊆ C [ t ]. We consider the filtration induced by the degree in t on this algebra. Then the associated graded algebra of A is A itself again, i.e. a domain. But we have infinitely many factorizations of t 2 of the form t 2 = (cos( θ ) + i sin( θ )) t · (cos( θ ) − i sin( θ )) t for any θ ∈ [0 , 2 π ). Notice that the units of A are precisely the nonzero real numbers and hence for θ ∈ [0 , π ) these factorizations are distinct. 9 / 30
Example for a Noncommutative Non-FFD Let K ( x ) � ∂ | ∂ · f ( x ) = f ( x ) ∂ + f ′ ( x ) � . Then there are infinitely many factorizations of ∂ 2 of the form � b � � b � ∂ 2 = ∂ + ∂ − , b , c ∈ K . x + c x + c 10 / 30
G -Algebras Definition For n ∈ N and 1 ≤ i < j ≤ n consider the units c ij ∈ K ∗ and polynomials d ij ∈ K [ x 1 , . . . , x n ]. Suppose, that there exists a monomial total well-ordering ≺ on K [ x 1 , . . . , x n ], such that for any 1 ≤ i < j ≤ n either d ij = 0 or the leading monomial of d ij is smaller than x i x j with respect to ≺ . The K -algebra A := K � x 1 , . . . , x n | { x j x i = c ij x i x j + d ij : 1 ≤ i < j ≤ n }� is called a G - algebra , if { x α 1 · . . . · x α n : α i ∈ N 0 } is a K -basis of A . 1 n Remark ◮ Also known as “algebras of solvable type” and “PBW (Poincar´ e Birkhoff Witt) Algebras” 11 / 30
Examples for G -Algebras ◮ Weyl algebras ( K � x 1 , . . . , x n , ∂ 1 , . . . , ∂ n | ∀ i : ∂ i x i = x i ∂ i + 1 � ) ◮ Shift algebras ( K � x 1 , . . . , x n , s 1 , . . . , s n | ∀ i : s i x i = ( x i + 1) s i � ) ◮ q -Weyl algebras ( K � x 1 , . . . , x n , ∂ 1 , . . . , ∂ n | ∀ i ∃ q i ∈ K ∗ : ∂ i x i = q i x i ∂ i + 1 � ) ◮ q -Shift algebras ( K � x 1 , . . . , x n , s 1 , . . . , s n | ∀ i ∃ q i ∈ K ∗ : s i x i = q i x i s i � ) ◮ Universal enveloping algebras of finite dimensional Lie algebras. ◮ . . . 12 / 30
G -Algebras are FFD Theorem (cf. (Bell et al., 2014)) Let K be a field. Then G-algebras over K and their subalgebras are finite factorization domains. 13 / 30
Consequences ◮ We have now more than just the similarity property to characterize factorizations in G -algebras. ◮ New algorithmic problem: Calculate all factorizations of an element in a given G -algebra. ◮ With this knowledge, study how algorithms from commutative algebra can be generalized to certain non-commutative algebras. 14 / 30
Non-Commutative Factorized Gr¨ obner Bases 15 / 30
Factorized Gr¨ obner bases – Commutative ◮ The factorized Gr¨ obner approach has been studied extensively for the commutative case (Czapor, 1989b,a; Davenport, 1987; Gr¨ abe, 1995a,b). ◮ Application: Obtaining triangular sets. ◮ Possible extension: Allowing constraints on the solutions. ◮ Implementations: e.g. in Singular and Reduce . ◮ Idea: For each factor ˜ g of a reducible element g during a Gr¨ obner computation, recursively call algorithm on the same generator set, with g being replaced by ˜ g . 16 / 30
Generalization to Non-Commutative Rings ◮ Ideals in commutative ring ↔ Varieties ◮ Ideals in Non-Commutative ring ↔ Solutions ◮ Formal notion of solutions: Let F be a left A -module for a K -algebra A (space of solutions). Let a left A -module M be finitely presented by an n × m matrix P . Then Sol A ( P , F ) = { f ∈ F m : Pf = 0 } ◮ Divisors for commutative rings ↔ Right divisors for non-commutative rings. 17 / 30
Picking the Right Right Divisors There are different strategies: ◮ Split Gr¨ obner computation with respect to different irreducible right divisors. 18 / 30
Picking the Right Right Divisors There are different strategies: ◮ Split Gr¨ obner computation with respect to different irreducible right divisors. ⇒ This approach may cause lost of possible solutions to the whole system. 18 / 30
Picking the Right Right Divisors There are different strategies: ◮ Split Gr¨ obner computation with respect to different irreducible right divisors. ⇒ This approach may cause lost of possible solutions to the whole system. ◮ Split Gr¨ obner computation with respect to all possible maximal right divisors. 18 / 30
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