Letterplace ideals and non-commutative Gröbner bases

Transcription

1 Letterplace ideals and non-commutative Gröbner bases Viktor Levandovskyy and Roberto La Scala (Bari) RWTH Aachen , NOCAS, Passau, Niederbayern La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

2 Notations X = {x 0, x 1,...} a finite or countable set, K a field K X the free associative algebra generated by X I a two-sided ideal of K X All associative algebras, generated by a finite or countable number of elements, can be presented as K X /I. Examples algebras of finite or countable dimension (quantized) universal enveloping algebras of Lie algebras of finite or countable dimension relatively free algebras defined by the polynomial identities satified by an associative algebra etc. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

3 Notations X = {x 0, x 1,...} a finite or countable set, K a field K X the free associative algebra generated by X I a two-sided ideal of K X All associative algebras, generated by a finite or countable number of elements, can be presented as K X /I. Examples algebras of finite or countable dimension (quantized) universal enveloping algebras of Lie algebras of finite or countable dimension relatively free algebras defined by the polynomial identities satified by an associative algebra etc. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

4 Trivial surjective homomorphism K X F[X] 1-to-1 correspondence between all ideals J F[X] and a class of two-sided ideals I K X any ideal I contains all the commutators [x i, x j ] = x i x j x j x i Problem Is there a 1-to-1 correspondence between all two-sided ideals I K X and a class of ideals J F[Y ] for some Y? Is there a good correspondence given between generating sets? in particular, between their Gröbner bases? We propose a solution for the case I is an homogeneous ideal. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

5 Trivial surjective homomorphism K X F[X] 1-to-1 correspondence between all ideals J F[X] and a class of two-sided ideals I K X any ideal I contains all the commutators [x i, x j ] = x i x j x j x i Problem Is there a 1-to-1 correspondence between all two-sided ideals I K X and a class of ideals J F[Y ] for some Y? Is there a good correspondence given between generating sets? in particular, between their Gröbner bases? We propose a solution for the case I is an homogeneous ideal. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

6 Notations X the set of letters, P = N = {0, 1,...} the set of places (x i j) = (x i, j) element of the product set X P K [X P] the polynomial ring in the (commutative) variables (x i j) X the set of words, [X P] the set of letterplace monomials Multi-gradings K X µ := vector space, generated by the words of multidegree µ K [X P] µ,ν := vector space, generated by the monomials of multidegree µ for the letters and ν for the places. Example If m = (x 2 0)(x 0 0)(x 4 2)(x 2 4) K [X P], then µ(m) = (1, 0, 2, 0, 1) and ν(m) = (2, 0, 1, 0, 1). La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

7 Notations X the set of letters, P = N = {0, 1,...} the set of places (x i j) = (x i, j) element of the product set X P K [X P] the polynomial ring in the (commutative) variables (x i j) X the set of words, [X P] the set of letterplace monomials Multi-gradings K X µ := vector space, generated by the words of multidegree µ K [X P] µ,ν := vector space, generated by the monomials of multidegree µ for the letters and ν for the places. Example If m = (x 2 0)(x 0 0)(x 4 2)(x 2 4) K [X P], then µ(m) = (1, 0, 2, 0, 1) and ν(m) = (2, 0, 1, 0, 1). La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

8 Known ideas Equivalent representations Put V = n N K [X P],1 n ( = any, 1n = (1, 1,..., 1)). If m = #X, the groups GL m and S n act resp. from left and right over the spaces K X n and V n. One has the bijection (mentioned by Feynmann, Rota) ι : K X V w = x i1 x in (x i1 0) (x in n 1). Restricted map ι n : K X n V n is a module isomorphism. Clearly ι : K X V K [X P] is not a ring homomorphism. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

9 New Ideas Act by shift The monoid N has a faithful action over the graded algebra K [X P] (total degree). For each variable (x i j) and s N we put s (x i j) := (x i s + j) In other words, one has a monomorphism N End(K [X P]). Decompose by shift If m = (x i1 j 1 ) (x in j n ) [X P] we define the shift of m to be the integer sh(m) = min{j 1,..., j n }. K [X P] (s) space, gen. by the monomials with shift s N. One has K [X P] = s N K [X P] (s) s K [X P] (t) = K [X P] (s+t) La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

10 New Ideas Act by shift The monoid N has a faithful action over the graded algebra K [X P] (total degree). For each variable (x i j) and s N we put s (x i j) := (x i s + j) In other words, one has a monomorphism N End(K [X P]). Decompose by shift If m = (x i1 j 1 ) (x in j n ) [X P] we define the shift of m to be the integer sh(m) = min{j 1,..., j n }. K [X P] (s) space, gen. by the monomials with shift s N. One has K [X P] = s N K [X P] (s) s K [X P] (t) = K [X P] (s+t) La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

11 Example If m = (x 2 2)(x 1 2)(x 2 4) then sh(m) = 2. Moreover 3 m = (x 2 5)(x 1 5)(x 2 7) and sh(3 m) = 3 + sh(m) = 5. Definition An ideal J K [X P] is called place-multigraded, if J = ν J,ν with J,ν = J K [X P],ν shift-decomposable, if J = s J(s) with J (s) = J K [X P] (s). A place-multigraded ideal is also graded and shift-decomposable. Definition A shift-decomposable ideal J K [X P] is called shift-invariant if s J (t) = J (s+t) for all s, t N. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

12 Example If m = (x 2 2)(x 1 2)(x 2 4) then sh(m) = 2. Moreover 3 m = (x 2 5)(x 1 5)(x 2 7) and sh(3 m) = 3 + sh(m) = 5. Definition An ideal J K [X P] is called place-multigraded, if J = ν J,ν with J,ν = J K [X P],ν shift-decomposable, if J = s J(s) with J (s) = J K [X P] (s). A place-multigraded ideal is also graded and shift-decomposable. Definition A shift-decomposable ideal J K [X P] is called shift-invariant if s J (t) = J (s+t) for all s, t N. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

13 Proposition Let J be an ideal of K [X P]. We put I = ι 1 (J V ) K X. Proof. If J is shift-invariant, then I is a left ideal. If J is place-multigraded, then I is graded right ideal. Assume J is shift-invariant and let f I, w X. Denote g = ι(f ) J V and m = ι(w). If deg(w) = s we have clearly ι(wf ) = m(s g) J V and hence wf I. Suppose now that J is place-multigraded and hence graded. Since V is a graded subspace, it follows that J V = d (J d V ) and hence, setting I d = ι 1 (J d V ) we obtain I = d I d. Let f I d that is ι(f ) = g J d V. For all w X we have that ι(fw) = g(d m) J V that is fw I. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

14 Proposition Let J be an ideal of K [X P]. We put I = ι 1 (J V ) K X. Proof. If J is shift-invariant, then I is a left ideal. If J is place-multigraded, then I is graded right ideal. Assume J is shift-invariant and let f I, w X. Denote g = ι(f ) J V and m = ι(w). If deg(w) = s we have clearly ι(wf ) = m(s g) J V and hence wf I. Suppose now that J is place-multigraded and hence graded. Since V is a graded subspace, it follows that J V = d (J d V ) and hence, setting I d = ι 1 (J d V ) we obtain I = d I d. Let f I d that is ι(f ) = g J d V. For all w X we have that ι(fw) = g(d m) J V that is fw I. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

15 Proposition Let I be a left ideal of K X and put I = ι(i). Define J to be the ideal of K [X P] generated by s N s I. Then J is a shift-invariant ideal. Moreover, if I is graded then J is place-multigraded. Example If f = 2x 2 x 3 x 1 3x 3 x 1 x 3 I, all the following polynomials belong to J: ι(f ) = 2(x 2 0)(x 3 1)(x 1 2) 3(x 3 0)(x 1 1)(x 3 2) 1 ι(f ) = 2(x 2 1)(x 3 2)(x 1 3) 3(x 3 1)(x 1 2)(x 3 3) 2 ι(f ) = 2(x 2 2)(x 3 3)(x 1 4) 3(x 3 2)(x 1 3)(x 3 4) etc La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

16 Proposition Let I be a left ideal of K X and put I = ι(i). Define J to be the ideal of K [X P] generated by s N s I. Then J is a shift-invariant ideal. Moreover, if I is graded then J is place-multigraded. Example If f = 2x 2 x 3 x 1 3x 3 x 1 x 3 I, all the following polynomials belong to J: ι(f ) = 2(x 2 0)(x 3 1)(x 1 2) 3(x 3 0)(x 1 1)(x 3 2) 1 ι(f ) = 2(x 2 1)(x 3 2)(x 1 3) 3(x 3 1)(x 1 2)(x 3 3) 2 ι(f ) = 2(x 2 2)(x 3 3)(x 1 4) 3(x 3 2)(x 1 3)(x 3 4) etc La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

17 Definition Let I K X be a graded two-sided ideal. Denote ι(i) the shift-invariant ideal J K [X P] generated by s N s ι(i) and call J the letterplace analogue of the ideal I. Let J K [X P] be a shift-invariant place-multigraded ideal. Denote ι 1 (J) the graded two-sided ideal I = ι 1 (J V ) K X. Theorem The following inclusions hold: ι 1 ( ι(i)) = I, ι( ι 1 (J)) J, ι( ι 1 (J)) = J if and only if J is generated by s N s (J V ). La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

18 Definition Let I K X be a graded two-sided ideal. Denote ι(i) the shift-invariant ideal J K [X P] generated by s N s ι(i) and call J the letterplace analogue of the ideal I. Let J K [X P] be a shift-invariant place-multigraded ideal. Denote ι 1 (J) the graded two-sided ideal I = ι 1 (J V ) K X. Theorem The following inclusions hold: ι 1 ( ι(i)) = I, ι( ι 1 (J)) J, ι( ι 1 (J)) = J if and only if J is generated by s N s (J V ). La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

19 Definition A graded ideal J of K [X P] is called a letterplace ideal, if J is generated by s N s (J V ). In this case J is shift-invariant and place-multigraded. We obtain finally Corollary The map ι : K X V induces a 1-to-1 correspondence ι between graded two-sided ideals I of the free associative algebra K X and letterplace ideals J of the polynomial ring K [X P]. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

20 Definition A graded ideal J of K [X P] is called a letterplace ideal, if J is generated by s N s (J V ). In this case J is shift-invariant and place-multigraded. We obtain finally Corollary The map ι : K X V induces a 1-to-1 correspondence ι between graded two-sided ideals I of the free associative algebra K X and letterplace ideals J of the polynomial ring K [X P]. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

21 How generating sets behave under the ideal correspondence ι? Definition Let J be a letterplace ideal of K [X P] and H K [X P]. We say that H is a letterplace basis of J if H J V, H homogeneous and s H is a generating set of the ideal J. s N Proposition Let I be a graded two-sided ideal of K X and put J = ι(i). Moreover, let G I, G homogeneous and define H = ι(g) J V. Then G is a generating set of I as two-sided ideal if and only if H is a letterplace basis of J. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

22 How generating sets behave under the ideal correspondence ι? Definition Let J be a letterplace ideal of K [X P] and H K [X P]. We say that H is a letterplace basis of J if H J V, H homogeneous and s H is a generating set of the ideal J. s N Proposition Let I be a graded two-sided ideal of K X and put J = ι(i). Moreover, let G I, G homogeneous and define H = ι(g) J V. Then G is a generating set of I as two-sided ideal if and only if H is a letterplace basis of J. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

23 Higmans lemma Definitions A quasi-ordering is a binary relation, which is reflexive (a a) and transitive (a b, b c a c). An ordering is well-founded, if every nonempty set has a minimal element. A well-quasi-ordering is a well-founded quasi-ordering, such that there is no infinite sequence {x i } with x i x j for all i < j Higmans lemma (1952) The set of finite sequences over a well-quasi-ordered set of labels is itself well-quasi-ordered. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

24 Now, we enter the realm of Gröbner bases. A = K X or F[Y ] (Y = X P). M is the monoid of all monomials of A. A term-ordering on A is a total ordering on M which is a well-ordering, compatible with multiplication. Precisely one has: (i) either u v or v u, for any u, v M, u v; (ii) if u v then wu wv and uw vw, for all u, v, w M; (iii) every non-empty subset of M has a minimal element. Remark Even if the number of variables of the polynomial algebra A is infinite, there exist term-orderings. By Higman s lemma, any total ordering on M, which is compatible with multiplication and such that 1 x 0 x 1... holds, is a term-ordering. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

25 Now, we enter the realm of Gröbner bases. A = K X or F[Y ] (Y = X P). M is the monoid of all monomials of A. A term-ordering on A is a total ordering on M which is a well-ordering, compatible with multiplication. Precisely one has: (i) either u v or v u, for any u, v M, u v; (ii) if u v then wu wv and uw vw, for all u, v, w M; (iii) every non-empty subset of M has a minimal element. Remark Even if the number of variables of the polynomial algebra A is infinite, there exist term-orderings. By Higman s lemma, any total ordering on M, which is compatible with multiplication and such that 1 x 0 x 1... holds, is a term-ordering. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

26 Notations lm(f ) the leading (greatest) monomial of f K X, f 0 lm(g) = {lm(g) g G, g 0} with G K X LM(G) the two-sided ideal generated by lm(g) Definition Let I be a two-sided ideal of K X and G I. If lm(g) is a basis of LM(I) then G is called a Gröbner basis of I. In other words, for all f I \ {0} there are w 1, w 2 X, g G \ {0} such that lm(f ) = w 1 lm(g)w 2. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

27 Notations lm(f ) the leading (greatest) monomial of f K X, f 0 lm(g) = {lm(g) g G, g 0} with G K X LM(G) the two-sided ideal generated by lm(g) Definition Let I be a two-sided ideal of K X and G I. If lm(g) is a basis of LM(I) then G is called a Gröbner basis of I. In other words, for all f I \ {0} there are w 1, w 2 X, g G \ {0} such that lm(f ) = w 1 lm(g)w 2. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

28 Gröbner basis computation in K X : Example Let X = {x, y}. Consider f 1 = x 3 y 3 = xxx yyy, f 2 = xyx yxy and I = f 1, f 2 K X with respect to the graded left lexicographical ordering. We compute truncated Gröbner basis up to degree d = 5. Overlap Two words w, v X have an an overlap at word o, if w = ow and v = v o. We denote the overlapping below by v o w. Let G = {f 1, f 2 }. (f 1, f 1 ) : lm(f 1 ) = xxx, so there are two self-overlaps o 1 := o 1,1 = f 1 x xf 1 = xy 3 y 3 x, o 1,2 = f 1 x 2 x 2 f 1 = x 2 y 3 y 3 x 2. Moreover, o 1,2 xo 1,1 = xy 3 x y 3 x 2 = o 1,1 x, so o 1,2 reduces to 0. Hence G = G {o 1 }. (f 2, f 2 ) : lm(f 2 ) = xyx, there are two self-overlaps. Symmetry implies that both of them originate from the overlap xy x yx of lm(f 2 ). Then o 2 = f 2 yx xyf 2 = xyyxy yxyyx. So G = G {o 2 }. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

29 Gröbner basis in K X : Example continued (f 1, f 2 ) : lm(f 1 ) and lm(f 2 ) have two overlaps xx x yx and xy x xx, hence o 3,1 = f 1 yx xxf 2 = xxyxy y 4 x and o 3,2 = f 2 xx xyf 1 = xy 4 yxyxx. Performing reductions, we see that o 3,1 xf 2 y f 2 yy yo 1 = 0 and o 32 o 1 y + yf 2 x + yyf 2 = yyyxy yyyxy = 0. (f 1, o 1 ) has overlap xx x yyy, (f 2, o 1 ) has overlap xy x yyy, (f 1, o 2 ) has overlap xx x yyxy, (o 1, o 2 ) has overlap xyy xy yy, o 2 has a self-overlap xyy xy yxy and (f 2, o 2 ) has two overlaps xy x yyxy and xyy xy x. Since all these elements are of degree 6 and we are in the graded case, we conclude that G = {f 1, f 2, o 1, o 2 } is truncated Gröbner basis up to degree 5. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

30 In a similar way, the notion of Gröbner basis is defined for an ideal of the commutative polynomial ring F[Y ]. Definition Let G F[Y ], f F[Y ]. By definition f has a Gröbner representation with respect to G if f = 0 or there are f i F[Y ], g i G such that f = n i=1 f ig i, with f i g i = 0 or lm(f ) lm(f i )lm(g i ) otherwise. Proposition (Buchberger s criterion) Let G be a basis of an ideal J F[Y ]. Then G is a Gröbner basis of J if and only if for all f, g G \ {0}, f g the S-polynomial S(f, g) has a Gröbner representation with respect to G. This criterion implies a critical pair & completion algorithm, transforming a generating set G 0 into a Gröbner basis G. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

31 In a similar way, the notion of Gröbner basis is defined for an ideal of the commutative polynomial ring F[Y ]. Definition Let G F[Y ], f F[Y ]. By definition f has a Gröbner representation with respect to G if f = 0 or there are f i F[Y ], g i G such that f = n i=1 f ig i, with f i g i = 0 or lm(f ) lm(f i )lm(g i ) otherwise. Proposition (Buchberger s criterion) Let G be a basis of an ideal J F[Y ]. Then G is a Gröbner basis of J if and only if for all f, g G \ {0}, f g the S-polynomial S(f, g) has a Gröbner representation with respect to G. This criterion implies a critical pair & completion algorithm, transforming a generating set G 0 into a Gröbner basis G. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

32 If Y is infinite, then the ring F [Y ] is not noetherian. Hence, it is not guaranteed that G 0, G are finite sets, that is the procedure to terminate in a finite number of steps. If G 0 is a finite basis of the ideal J, then its Gröbner basis G is contained in F[Y ], where Y is the set of variables occuring in G 0. Therefore G is also finite by noetherianity of F[Y ]. Assume J is graded and has a finite number of generators of degree d. Then, the number of elements in the Gröbner basis of J of degree d is finite and the truncated algorithm terminates up to degree d. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

33 When a monoid S acts (by algebra endomorphisms) on a polynomial ring, one has the notion of Gröbner S-basis. In a paper by Drensky and La Scala (JSC, 2006), such notion has been introduced and applied to the ideal I K X defining the universal enveloping algebra of the free 2-nilpotent Lie algebra. The ideal I is generated by all commutators [x i, x j, x k ] and hence it is stable under the action of all endomorphisms x i x j. We introduced here this notion for the specific action of N over K [X P]. Definition Let J be an ideal of K [X P] and H J. Then H is said a shift-basis (resp. a Gröbner shift-basis) of J if s N s H is a basis (resp. a Gröbner basis) of J (then N J = J). If J is a letterplace ideal, then any letterplace basis of J is a shift-basis but not generally a Gröbner shift-basis of J. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

34 When a monoid S acts (by algebra endomorphisms) on a polynomial ring, one has the notion of Gröbner S-basis. In a paper by Drensky and La Scala (JSC, 2006), such notion has been introduced and applied to the ideal I K X defining the universal enveloping algebra of the free 2-nilpotent Lie algebra. The ideal I is generated by all commutators [x i, x j, x k ] and hence it is stable under the action of all endomorphisms x i x j. We introduced here this notion for the specific action of N over K [X P]. Definition Let J be an ideal of K [X P] and H J. Then H is said a shift-basis (resp. a Gröbner shift-basis) of J if s N s H is a basis (resp. a Gröbner basis) of J (then N J = J). If J is a letterplace ideal, then any letterplace basis of J is a shift-basis but not generally a Gröbner shift-basis of J. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

35 Problem Question: is it possible to reduce by symmetry the Buchberger algorithm with respect to the action by shifting? Answer: YES, if the term-ordering is compatible with such action. Definition A term-ordering on K [X P] is called shift-invariant, when u v if and only if s u s v for any u, v [X P] and s N. In this case, one has that lm(s f ) = s lm(f ) for all f K [X P] \ {0} and s N. It is clear that many of the usual term-orderings are shift-invariant. From now on we assume K [X P] endowed with a shift-invariant term-ordering. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

36 Problem Question: is it possible to reduce by symmetry the Buchberger algorithm with respect to the action by shifting? Answer: YES, if the term-ordering is compatible with such action. Definition A term-ordering on K [X P] is called shift-invariant, when u v if and only if s u s v for any u, v [X P] and s N. In this case, one has that lm(s f ) = s lm(f ) for all f K [X P] \ {0} and s N. It is clear that many of the usual term-orderings are shift-invariant. From now on we assume K [X P] endowed with a shift-invariant term-ordering. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

37 One proves immediately: Proposition Let J K [X P] be an ideal and H J. We have that H is a Gröbner shift-basis of J if and only if lm(h) is a shift-basis of LM(J). Lemma Let f 1, f 2 K [X P] \ {0}, f 1 f 2. Then S(s f 1, s f 2 ) = s S(f 1, f 2 ). It follows that we can reduce by symmetry the Buchberger s criterion with respect to the shift action. Proposition Let H be a shift-basis of an ideal J K [X P]. Then H is a Gröbner shift-basis of J if and only if for all f, g H \ {0}, s N, f s g the S-polynomial S(f, s g) has a Gröbner representation with respect to t N t H. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

38 One proves immediately: Proposition Let J K [X P] be an ideal and H J. We have that H is a Gröbner shift-basis of J if and only if lm(h) is a shift-basis of LM(J). Lemma Let f 1, f 2 K [X P] \ {0}, f 1 f 2. Then S(s f 1, s f 2 ) = s S(f 1, f 2 ). It follows that we can reduce by symmetry the Buchberger s criterion with respect to the shift action. Proposition Let H be a shift-basis of an ideal J K [X P]. Then H is a Gröbner shift-basis of J if and only if for all f, g H \ {0}, s N, f s g the S-polynomial S(f, s g) has a Gröbner representation with respect to t N t H. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

39 Proof. We have to prove now that G = s s H is a Gröbner basis of J, that is for any f, g H \ {0}, s, t N, s f t g the S-polynomial S(s f, t g) has a Gröbner representation with respect to G. Assume s t and put u = t s. By the previous lemma we have S(s f, t g) = S(s f, s (u g)) = s S(f, u g). By hypothesis, the S-polynomial S = S(f, u g) is zero or S = i f ig i, where f i K [X P], g i G and lm(s) lm(f i )lm(g i ) for all i such that f i g i 0. By acting with the shift s (algebra endomorphism), it is clear that s S has also a Gröbner representation with respect to G. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

40 By the above proposition, we obtain the correctness of the following Buchberger algorithm reduced by symmetry. Algorithm SGBASIS Input: H 0, a shift-basis of an ideal J K [X P]. Output: H, a Gröbner shift-basis of J. H := H 0 \ {0}; P := {(f, s g) f, g H, s N, f s g, gcd(lm(f ), lm(s g)) 1}; while P do choose (f, s g) P; P := P \ {(f, s g)}; h := REDUCE(S(f, s g), t t H); if h 0 then P := P {(h, s g) g H, s N, gcd(lm(h), lm(s g)) 1}; P := P {(s h, g) g H, s N, gcd(lm(s h), lm(g)) 1}; H := H {h}; return H. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

41 Let us see what happens when we apply this algorithm to letterplace ideals. Notations If ν = (ν k ) is a multidegree, denote ν = (η k ) the multidegree defined as η k = 1 if ν k > 0 and η k = 0 otherwise. Define then V = ν=1 n, n N K [X P],ν Example (x 2 0)(x 0 1)(x 4 1)(x 2 2) V, but (x 2 0)(x 0 1)(x 4 3)(x 2 4) / V. Proposition Let J K [X P] be a letterplace ideal. There exists a Gröbner shift-basis of J contained in ν J,ν V. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

42 Let us see what happens when we apply this algorithm to letterplace ideals. Notations If ν = (ν k ) is a multidegree, denote ν = (η k ) the multidegree defined as η k = 1 if ν k > 0 and η k = 0 otherwise. Define then V = ν=1 n, n N K [X P],ν Example (x 2 0)(x 0 1)(x 4 1)(x 2 2) V, but (x 2 0)(x 0 1)(x 4 3)(x 2 4) / V. Proposition Let J K [X P] be a letterplace ideal. There exists a Gröbner shift-basis of J contained in ν J,ν V. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

43 Definition Let J be a letterplace ideal of K [X P] and H J. We say that H is a Gröbner letterplace basis of J if H ν J,ν V and H is a Gröbner shift-basis of J. From such a basis we want to obtain a Gröbner basis of the graded two-sided ideal I = ι 1 (J). Definition Fix the term-orderings < on K X and on K [X P]. They are called compatible with ι, when v < w holds if and only if ι(v) ι(w) for any v, w X. In this case, it follows that lm(ι(f )) = ι(lm(f )) for all f K X \ {0}. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

44 Definition Let J be a letterplace ideal of K [X P] and H J. We say that H is a Gröbner letterplace basis of J if H ν J,ν V and H is a Gröbner shift-basis of J. From such a basis we want to obtain a Gröbner basis of the graded two-sided ideal I = ι 1 (J). Definition Fix the term-orderings < on K X and on K [X P]. They are called compatible with ι, when v < w holds if and only if ι(v) ι(w) for any v, w X. In this case, it follows that lm(ι(f )) = ι(lm(f )) for all f K X \ {0}. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

45 Assume: K X, K [X P] are endowed with term-orderings compatible with ι and the one of K [X P] is shift-invariant. Proposition Let I K X be a graded two-sided ideal and put J = ι(i). Moreover, let H be a Gröbner letterplace basis of J and put G = ι 1 (H V ). Then G is a Gröbner basis of I as two-sided ideal. Remark Let G 0 be a homogeneous basis of I. Then, the computation of a homogeneous Gröbner basis G of I can be done by applying the algorithm SGBASIS to H 0 = ι(g 0 ). If H = SGBASIS(H 0 ) then G = ι 1 (H V ), and hence one is interested to compute only the elements of H V. We prove: all such elements are obtained from S-polynomials S(f, s g) where f, g are already elements of V. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

46 Assume: K X, K [X P] are endowed with term-orderings compatible with ι and the one of K [X P] is shift-invariant. Proposition Let I K X be a graded two-sided ideal and put J = ι(i). Moreover, let H be a Gröbner letterplace basis of J and put G = ι 1 (H V ). Then G is a Gröbner basis of I as two-sided ideal. Remark Let G 0 be a homogeneous basis of I. Then, the computation of a homogeneous Gröbner basis G of I can be done by applying the algorithm SGBASIS to H 0 = ι(g 0 ). If H = SGBASIS(H 0 ) then G = ι 1 (H V ), and hence one is interested to compute only the elements of H V. We prove: all such elements are obtained from S-polynomials S(f, s g) where f, g are already elements of V. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

47 NCGBASIS Input: G 0, a homogeneous basis of a graded two-sided ideal I K X. Output: G, a homogeneous Gröbner basis of I. H := ι(g 0 \ {0}); P := {(f, s g) f, g H, s N, f s g, gcd(lm(f ), lm(s g)) 1, lcm(lm(f ), lm(s g)) V }; while P do choose (f, s g) P; P := P \ {(f, s g)}; h := REDUCE(S(f, s g), t t H); if h 0 then P := P {(h, s g) g H, s N, gcd(lm(h), lm(s g)) 1, lcm(lm(h), lm(s g)) V }; P := P {(s h, g) g H, s N, gcd(lm(s h), lm(g)) 1, lcm(lm(s h), lm(g)) V }; H := H {h}; G := ι 1 (H); return G La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

48 Remark The termination of a procedure that computes non-commutative (homogeneous) Gröbner bases is not provided in general, even if the set of variables X and a basis G 0 of I are both finite (K X is not noetherian). From the viewpoint of our method, this corresponds to the fact that the set of commutative variables X P is infinite, and the letterplace ideal J = ι(i) is generated by s N s ι(g 0) which is also an infinite set. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

49 Proposition Let I K X be a graded two-sided ideal and d > 0 an integer. If I has a finite number of homogeneous generators of degree d then the algorithm NCGBASIS computes in a finite number of steps all elements of degree d of a homogeneous Gröbner basis of I. Proof. Consider the elements f, g H V at the current step. If both these polynomials have degree d then the condition gcd(lm(h), lm(s g)) 1 implies that s d 1. It follows that the computation actually runs over the variables set X {0,..., d 1}, where X is the finite set of variables occurring in the generators of I of degree d. By noetherianity of the ring F[X {0,..., d 1}] we conclude that the truncated procedure, up to degree d, stops after a finite number of steps. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

50 Proposition Let I K X be a graded two-sided ideal and d > 0 an integer. If I has a finite number of homogeneous generators of degree d then the algorithm NCGBASIS computes in a finite number of steps all elements of degree d of a homogeneous Gröbner basis of I. Proof. Consider the elements f, g H V at the current step. If both these polynomials have degree d then the condition gcd(lm(h), lm(s g)) 1 implies that s d 1. It follows that the computation actually runs over the variables set X {0,..., d 1}, where X is the finite set of variables occurring in the generators of I of degree d. By noetherianity of the ring F[X {0,..., d 1}] we conclude that the truncated procedure, up to degree d, stops after a finite number of steps. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

51 Gröbner basis in K [X P]: Example Example (f 1 = x 3 y 3 = xxx yyy, f 2 = xyx yxy) In the case d = 5, one considers then the polynomial ring K [X P 5 ] = K [x(1), y(1),..., x(5), y(5)] and the polynomials f 1 = x(1)x(2)x(3) y(1)y(2)y(3), f 2 = 1 f 1 = x(2)x(3)x(4) y(2)y(3)y(4), f 3 = 2 f 1 = x(3)x(4)x(5) y(3)y(4)y(5), f 4 = x(1)y(2)x(3) y(1)x(2)y(3), f 5 = 1 f 4 = x(2)y(3)x(4) y(2)x(3)y(4), f 6 = 2 f 4 = x(3)y(4)x(5) y(3)x(4)y(5). We write (i, j) for the pair (i, j). There are 15 pairs to consider. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

52 Gröbner basis in K [X P]: Example continued Criteria the pairs (1, 4), (1, 5), (2, 4), (4, 5) are discarded by the V -criterion, since lcm s of their lm s are non-multilinear in places. the pairs (2, 3), (2, 5), (2, 6), (3, 5), (3, 6), (5, 6) are discarded by the V -criterion, since lcm s of their lm s have nonzero shift. (4, 5) and (5, 6) can also be discarded by the product criterion. Thus it remains to consider P = {(1, 2), (1, 3), (1, 6), (3, 4), (4, 6)}. Following the Algorithm NCGBasis, H := {f 1, f 4 }. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

53 Gröbner basis in K [X P]: Example continued Example (1, 2) : spoly(1, 2) = f 1 x(4) x(1)f 2 = x(1)y(2)y(3)y(4) y(1)y(2)y(3)x(4) =: g 1, hence H := H {g 1 } and g 2 := 1 spoly(1, 2) = x(2)y(3)y(4)y(5) y(2)y(3)y(4)x(5) is the only admissible shift of g 1. (1, 3) : spoly(1, 3) = f 1 x(4)x(5) x(1)x(2)f 3 = x(1)x(2)y(3)y(4)y(5) y(1)y(2)y(3)x(4)x(5) = x(1)g 2 + g 1 x(5) 0. Note, that since lm(f 2 ) lcm(lm(f 1 ), lm(f 3 )), by the Chain criterion we can skip the pair (1, 3) from the pairset (1, 2), (1, 3), (2, 3). The pair (2, 3) is skipped by the V -criterion above. (1, 6) : spoly(1, 6) = f 1 y(4)x(5) x(1)x(2)f 6 = x(1)x(2)y(3)x(4)x(5) x(1)x(2)x(3)y(4)x(5). Indeed, spoly(1, 6) = x(1)f 5 y(5) + f 4 y(4)y(5) + y(1)g 2 0. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

54 Gröbner basis in K [X P]: Example continued Example (3, 4) : spoly(3, 4) = f 4 x(4)x(5) x(1)y(2)f 3 = x(1)y(2)y(3)y(4)y(5) y(1)x(2)y(3)x(4)x(5) and spoly(3, 4) = g 1 y(5) y(1)f 5 x(5) y(1)y(2)f 6 0. (4, 6) : spoly(4, 6) = f 4 y(4)x(5) x(1)y(2)f 6 = x(1)y(2)y(3)x(4)y(5) y(1)x(2)y(3)y(4)x(5) cannot be reduced, hence g 3 := spoly(4, 6) and H becomes {f 1, f 4, g 1, g 3 }. Note, that there are no admissible shifts for g 3. The pairs (g 1, f 1 ),..., (g 1, f 6 ), (f 1, g 2 ), (f 4, g 2 ), which appear when g 1 enters H and (g 3, f 1 ),..., (g 3, f 6 ), (g 3, g 1 ), (g 3, g 2 ), which appear when g 3 enters H (note, that (f 1, g 3 ), (f 4, g 3 ), (g 1, g 3 ) are already included in the latter) are discarded by the V -criterion. Thus ι 1 ({f 1, f 4, g 1, g 3 }) is a truncated Gröbner basis up to degree 5. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

55 Implementation We have developed an implementation of the letterplace algorithm in the computer algebra system Singular: Our implementation consists of kernel part of SINGULAR the library freegb.lib in the SINGULAR language Even is the implementation can be improved much, the comparisons with the best implementations of non-commutative Gröbner bases (classic algorithm) are very encouraging. They show that, in addition to the interesting feature to be portable in any commutative computer algebra system, the proposed method is really feasible. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

56 Examples: Lie algebras Free nilpotent Lie algebra of class c The ideal I K X is generated by all (left-normed) commutators [x i1,..., x ic+1 ] of length c + 1. In particular, we study the case when the number of variables n = 5 and c = 3, 4. We called these examples nilp3 and nilp4. Free metabelian Lie algebra I K X is generated by all the commutators [[x i, x j ], [x k, x l ]]. We fix the dimension to be n = 5 and denote this example as metab5. 2-by-2 upper triangular matrices as PI algebra I K X is generated by polynomials [x i, x j ]w[x k, x l ], where w is an arbitrary word (including 1) of K X. Example tri4: the number of variables n = 4, the degree d = 7. La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

57 Examples: Serre s relations Let n N and A = (a ij ) Z n n be a square integer matrix. Suppose, that we have 1 a ij Z + for i j. The (i, j)-th Serre s relation associated to A is ad 1 a ij x j (x i ) = [x j, [... [x j, x i ]]...] = 0. We consider ideals S, generated by all Serre s relations associated to: Cartan matrices, which lead to simple Lie algebras. S has usually finite Gröbner basis. generalized Cartan matrices, which are used to define Kac-Moody algebras. S often has infinite Gröbner basis (therefore such algebras are very interesting to study). La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

58 Open problems Extend the letterplace method to the computation of non-homogeneous ideals (under development). One-sided Gröbner bases over fin. pres. algebras One- and two-sided syzygies and resolutions over f.p.a. Hilbert functions and dimensions (like Gel fand-kirillov) Homological algebra For ideals that are invariants under the actions of (semi)groups, algebras, etc, to integrate the methods of representation theory to Gröbner bases techniques. Do there exist letterplace analogues of Lie ideals? Of Gröbner-Shirshov bases? La Scala, Levandovskyy (RWTH) Letterplace ideals / 39

59 Thank you for your attention! Please visit the SINGULAR homepage La Scala, Levandovskyy (RWTH) Letterplace ideals / 39