A Factorization Algorithm for G -Algebras and Applications ACA 2016 - - PowerPoint PPT Presentation
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 20160804 1 / 30
ACA 2016 – Kassel – Germany Albert Heinle
Symbolic Computation Group David R. Cheriton School of Computer Science University of Waterloo Canada
2016–08–04
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Introduction On Non-Commutative Finite Factorization Domains Non-Commutative Factorized Gr¨
Conclusion and Future Work
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For integral domains (in the literature commonly assumed to be commutative rings) many factorization properties have been
1992; Anderson and Mullins, 1996; Anderson, 1997))
Figure : from (Anderson et al., 1990)
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For integral domains (in the literature commonly assumed to be commutative rings) many factorization properties have been
1992; Anderson and Mullins, 1996; Anderson, 1997))
Figure : Created on https://imgflip.com/
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◮ 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)).
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◮ 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)).
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◮ 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)).
The factors are only unique up to similarity!
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◮ 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)).
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).
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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.
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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.
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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 {Vn : n ∈ N} on A such that the associated graded algebra B = grV (A) is a (not necessarily commutative) domain over K, then A is a finite factorization domain over K.
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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 {Vn : n ∈ N} on A such that the associated graded algebra B = grV (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 {Vn : n ∈ N} on A such that the associated graded algebra B = grV (A) has the property that B ⊗K K is a (not necessarily commutative) domain, then A is a finite factorization domain.
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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 t2 of the form t2 = (cos(θ) + i sin(θ))t · (cos(θ) − i sin(θ))t for any θ ∈ [0, 2π). Notice that the units of A are precisely the nonzero real numbers and hence for θ ∈ [0, π) these factorizations are distinct.
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Let K(x)∂ | ∂ · f (x) = f (x)∂ + f ′(x). Then there are infinitely many factorizations of ∂2 of the form ∂2 =
b x + c ∂ − b x + c
b, c ∈ K.
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Definition
For n ∈ N and 1 ≤ i < j ≤ n consider the units cij ∈ K∗ and polynomials dij ∈ K[x1, . . . , xn]. Suppose, that there exists a monomial total well-ordering ≺ on K[x1, . . . , xn], such that for any 1 ≤ i < j ≤ n either dij = 0 or the leading monomial of dij is smaller than xixj with respect to ≺. The K-algebra A := Kx1, . . . , xn | {xjxi = cijxixj + dij : 1 ≤ i < j ≤ n} is called a G-algebra, if {xα1
1
· . . . · xαn
n
: αi ∈ N0} is a K-basis of A.
Remark
◮ Also known as “algebras of solvable type” and “PBW
(Poincar´ e Birkhoff Witt) Algebras”
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◮ Weyl algebras (Kx1, . . . , xn, ∂1, . . . , ∂n | ∀i : ∂ixi = xi∂i + 1) ◮ Shift algebras (Kx1, . . . , xn, s1, . . . , sn | ∀i : sixi = (xi + 1)si) ◮ q-Weyl algebras
(Kx1, . . . , xn, ∂1, . . . , ∂n | ∀i∃qi ∈ K∗ : ∂ixi = qixi∂i + 1)
◮ q-Shift algebras
(Kx1, . . . , xn, s1, . . . , sn | ∀i∃qi ∈ K∗ : sixi = qixisi)
◮ Universal enveloping algebras of finite dimensional Lie
algebras.
◮ . . .
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Theorem (cf. (Bell et al., 2014))
Let K be a field. Then G-algebras over K and their subalgebras are finite factorization domains.
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◮ 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.
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◮ The factorized Gr¨
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¨
generator set, with g being replaced by ˜ g.
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◮ 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 SolA(P, F) = {f ∈ Fm : Pf = 0}
◮ Divisors for commutative rings ↔ Right divisors for
non-commutative rings.
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There are different strategies:
◮ Split Gr¨
right divisors.
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There are different strategies:
◮ Split Gr¨
right divisors. ⇒ This approach may cause lost of possible solutions to the whole system.
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There are different strategies:
◮ Split Gr¨
right divisors. ⇒ This approach may cause lost of possible solutions to the whole system.
◮ Split Gr¨
maximal right divisors.
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There are different strategies:
◮ Split Gr¨
right divisors. ⇒ This approach may cause lost of possible solutions to the whole system.
◮ Split Gr¨
maximal right divisors. ⇒ Less possible solutions may be lost.
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There are different strategies:
◮ Split Gr¨
right divisors. ⇒ This approach may cause lost of possible solutions to the whole system.
◮ Split Gr¨
maximal right divisors. ⇒ Less possible solutions may be lost.
◮ Split Gr¨
non-unique maximal right divisors.
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There are different strategies:
◮ Split Gr¨
right divisors. ⇒ This approach may cause lost of possible solutions to the whole system.
◮ Split Gr¨
maximal right divisors. ⇒ Less possible solutions may be lost.
◮ Split Gr¨
non-unique maximal right divisors. ⇒ Our choice!
Remark
This methodology also appears in the context of semifirs, where the concept of so called block factorizations or cleavages has been introduced to study the reducibility of a principal ideal (Cohn, 2006, Chapter 3.5).
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In the commutative case, for an ideal I and the output B1, . . . , Bm
√ I =
m
We would like to have something similar for the non-commutative case. However, as the next example depicts, we do not have it in our setting.
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Let p =(x6 + 2x4 − 3x2)∂2 − (4x5 − 4x4 − 12x2 − 12x)∂ + (6x4 − 12x3 − 6x2 − 24x − 12) in the polynomial first Weyl algebra. This polynomial appears in (Tsai, 2000, Example 5.7) and has two different factorizations, namely p =(x4∂ − x3∂ − 3x3 + 3x2∂ + 6x2 − 3x∂ − 3x + 12)· (x2∂ + x∂ − 3x − 1) =(x4∂ + x3∂ − 4x3 + 3x2∂ − 3x2 + 3x∂ − 6x − 3)· (x2∂ − x∂ − 2x + 4).
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A reduced Gr¨
x2∂ + x∂ − 3x − 1 ∩ x2∂ − x∂ − 2x + 4, computed with Singular, is given by {3x5∂2 + 2x4∂3 − x4∂2 − 12x4∂ + x3∂2 − 2x2∂3 + 16x3∂ + 9x2∂2 + 18x3 + 4x2∂ + 4x∂2 − 42x2 − 4x∂ − 12x − 12, 2x4∂4 − 2x4∂3 + 11x4∂2 + 12x3∂3 − 2x2∂4 − 2x3∂2 + 10x2∂3 − 44x3∂ − 17x2∂2 + 64x2∂ + 12x∂2 + 66x2 + 52x∂ + 4∂2 − 168x − 16∂ − 60}.
Remark
The space of holomorphic solutions of the differential equation associated to p in fact coincides with the union of the solution spaces of the two generators of the intersection.
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Definition
Let B, C be finite subsets in G. We call the tuple (B, C) a constrained Gr¨
NF(g, B) = 0 for every g ∈ C. We call a constrained Gr¨
tuple factorized, if every f ∈ B is either irreducible or has a unique irreducible left divisor.
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◮
Input: B := {f1, . . . , fk } ⊂ G, C := {g1, . . . , gl } ⊂ G.
◮
Output: R := {(˜ B, ˜ C) | (˜ B, ˜ C) is factorized constrained Gr¨
(˜ B,˜ C)∈R ˜
B
◮
Assumption: We can find all factorizations of an element in G.
◮
Algorithm: ◮ If one of the fi is reducible and has more than one distinct factorization, set M := {(f (1)
i
, f (2)
i
| f (1)
i
, f (2)
i
∈ G\K, lc(f (1)
i
) = lc(f (2)
i
) = 1, f (1)
i
·f (2)
i
= fi , f (1)
i
is irreducible} and return
FGBG (B \ {fi }) ∪ {b}, C ∪
a,˜ b)∈M b=˜ b
{˜ b} ◮ P := {(fi , fj ) | i, j ∈ {1, . . . , k}, i < j} ◮ While P = ∅: ◮ Pick (f , g) ∈ P and remove it from P, compute the S-polynomial of f and g and its normal form h with respect to B. ◮ If h = 0 and h is reducible, return FGBG(B ∪ {h}, C). ◮ If h = 0 and h is irreducible, P := P ∪ {(h, f ) | f ∈ B} and B := B ∪ {h} ◮ If there exists i ∈ {1, . . . , l} with NF(gi , B) = 0, return ∅. ◮ Return (B, C) 23 / 30
We consider the first Weyl algebra. Let B := {∂4 + x∂2 − 2∂3 − 2x∂ + ∂2 + x + 2∂ − 2, x∂3 + x2∂ − x∂2 + ∂3 − x2 + x∂ − 2∂2 − x + 1} and C := {∂ − 1}. Each element factors separately as f1 :=∂4 + x∂2 − 2∂3 − 2x∂ + ∂2 + x + 2∂ − 2 =(∂3 + x∂ − ∂2 − x + 2) · (∂ − 1) =(∂ − 1) · (∂3 + x∂ − ∂2 − x + 1), respectively f2 :=x∂3 + x2∂ − x∂2 + ∂3 − x2 + x∂ − 2∂2 − x + 1 =(x∂2 + x2 + ∂2 + x − ∂ − 1) · (∂ − 1) =(x∂ − x + ∂ − 2) · (∂2 + x).
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Hence, FGBG will return two recursive calls of itself, namely
◮ FGBG({∂ − 1, f2}, {∂ − 1, ∂3 + x∂ − ∂2 − x + 1}) ◮ FGBG({∂3 + x∂ − ∂2 − x + 1, f2}, C)
∂3 + x∂ − ∂2 − x + 1 has only one possible factorization. Considering factorizations of f2, we get two further recursive calls:
◮ FGBG({b1, ∂ − 1}, {∂ − 1, ∂2 + x}) ◮ FGBG({∂3 + x∂ − ∂2 − x + 1, ∂2 + x}, C)
Since ∂2 + x divides ∂3 + x∂ − ∂2 − x + 1 from the right, our algorithm returns {({∂2 + x}, C)} as final output.
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◮ Let p1, p2 ∈ Q be non-square numbers, which are negative
and have either 1,2 or 4 in the denominator.
◮ Define
A := Qx, y, z, u |xy + yx = xz + zx = yz + zy = 0, ux + xu = 0, uy + yu = y2, uz + zu = z2, x2 = p1y2 + p2z2.
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◮ Let p1, p2 ∈ Q be non-square numbers, which are negative
and have either 1,2 or 4 in the denominator.
◮ Define
A := Qx, y, z, u |xy + yx = xz + zx = yz + zy = 0, ux + xu = 0, uy + yu = y2, uz + zu = z2, x2 = p1y2 + p2z2. Proof that A is a finite factorization domain.
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◮ FFDs are generalized... What about BFDs, HFDs, etc.? ◮ More non-commutative FFDs are to be identified. ◮ More efficient algorithms to factor (certain) G-algebras. ◮ Study the output of non-commutative factorized Gr¨
basis algorithm. What does it say about the ideal structure? What is the connection to the solution space?
◮ Implementation of all the algorithms (partly done). Latest
ncfactor.lib can be found in the Singular GitHub repository1.
1https://github.com/Singular/Sources/blob/spielwiese/Singular/
LIB/ncfactor.lib
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Anderson, D. (1997). Factorization in integral domains, volume 189. CRC Press. Anderson, D. and Anderson, D. (1992). Elasticity of factorizations in integral domains. Journal of pure and applied algebra, 80(3):217–235. Anderson, D., Anderson, D., and Zafrullah, M. (1990). Factorization in integral domains. Journal of pure and applied algebra, 69(1):1–19. Anderson, D. and Mullins, B. (1996). Finite factorization domains. Proceedings of the American Mathematical Society, 124(2):389–396. Bell, J. P., Heinle, A., and Levandovskyy, V. (2014). On noncommutative finite factorization domains. To Appear in the Transactions of the American Mathematical Society; arXiv preprint arXiv:1410.6178. Bueso, J., G´
Applications to quantum groups. Dordrecht: Kluwer Academic Publishers. Cohn, P. (1963). Noncommutative unique factorization domains. Transactions of the American Mathematical Society, 109(2):313–331. Cohn, P. M. (2006). Free ideal rings and localization in general rings, volume 3. Cambridge University Press. Czapor, S. R. (1989a). Solving algebraic equations: combining Buchberger’s algorithm with multivariate
Czapor, S. R. (1989b). Solving algebraic equations via Buchberger’s algorithm. In Eurocal’87, pages 260–269. Springer. Davenport, J. H. (1987). Looking at a set of equations. Technical report, School of Mathematical Sciences, The University of Bath. Giesbrecht, M. and Heinle, A. (2012). A Polynomial-Time Algorithm for the Jacobson Form of a Matrix of Ore
Giesbrecht, M., Heinle, A., and Levandovskyy, V. (2015). Factoring linear partial differential operators in n
Gr¨ abe, H.-G. (1995a). On factorized Gr¨
World Scientific. Citeseer. 29 / 30
Gr¨ abe, H.-G. (1995b). Triangular systems and factorized Gr¨
Lazard, D. (1991). A new method for solving algebraic systems of positive dimension. Discrete Applied Mathematics, 33(1-3):147–160. Lazard, D. (1992). Solving zero-dimensional algebraic systems. Journal of symbolic computation, 13(2):117–131. M¨
Algebra in Engineering, Communication and Computing, 4(4):217–230. Tsai, H. (2000). Weyl closure of a linear differential operator. Journal of Symbolic Computation, 29:747–775. 30 / 30