Gr\" obner-Shirshov bases for some Lie algebras

Gröbner-Shirshov bases for some Lie algebras∗ Yuqun Chen, Yu Li and Qingyan Tang School of Mathematical Sciences, South China Normal University

arXiv:1305.4546v1 [math.RA] 15 May 2013

Guangzhou 510631, P. R. China [email protected] [email protected] [email protected]

Abstract: We give Gröbner-Shirshov bases for Drinfeld-Kohno Lie algebra Ln in [27] and Kukin Lie algebra AP in [32], where P is a semigroup. As applications, we show that as Z-module Ln is free and a Z-basis of Ln is given. We give another proof of Kukin Theorem: if semigroup P has the undecidable word problem then the Lie algebra AP has the same property. Key words: Gröbner-Shirshov basis, Lie algebra, Drinfeld-Kohno Lie algebra, word problem, semigroup. AMS 2000 Subject Classification: 17B01, 16S15, 3P10, 20M05, 03D15

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Introduction

Gröbner bases and Gröbner-Shirshov bases were invented independently by A.I. Shirshov for ideals of free (commutative, anti-commutative) non-associative algebras [40, 41], free Lie algebras [39, 41] and implicitly free associative algebras [39, 41] (see also [2, 4]), by H. Hironaka [29] for ideals of the power series algebras (both formal and convergent), and by B. Buchberger [20] for ideals of the polynomial algebras. Gröbner bases and Gröbner-Shirshov bases theories have been proved to be very useful in different branches of mathematics, including commutative algebra and combinatorial algebra, see, for example, the books [1, 18, 21, 22, 24, 25], the papers [2, 3, 4, 7, 9, 10, 11, 12, 13, 20, 23, 30, 35], and the surveys [5, 6, 14, 15, 16, 17]. A.A. Markov [34], E. Post [37], A. Turing [42], P.S. Novikov [36] and W.W. Boone [19] constructed finitely presented semigroups and groups with the undecidable word problem. This result also follows from the Higman theorem [28] that any recursive presented group is embeddable into finitely presented group. A weak analogy of Higman theorem for Lie ∗ Supported by the NNSF of China (11171118), the Research Fund for the Doctoral Program of Higher Education of China (20114407110007), the NSF of Guangdong Province (S2011010003374) and the Program on International Cooperation and Innovation, Department of Education, Guangdong Province (2012gjhz0007).

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algebras was proved in [3] that was enough for existence of a finitely presented Lie algebra with the undecidable word problem. In [32], Kukin constructed the Lie algebra AP for a semigroup P such that if P has the undecidable word problem then AP has the same property. In this paper we find a Gröbner-Shirshov basis for AP and then give another proof of the above result. The Drinfeld-Kohno Lie algebra Ln appears in [26, 31] as the holonomy Lie algebra of the complement of the union of the diagonals zi = zj , i < j. The universal KnizhnikZamolodchikov connection takes values in this Lie algebra. In this paper, we give a Gröbner-Shirshov basis for the Drinfeld-Kohno Lie algebra Ln over Z and a Z-basis of Ln . As an application we get a simple proof that Ln is an iterated semidirect product of free Lie algebras. We are very grateful to Professor L.A. Bokut for his guidance and useful discussions.

2

Composition-Diamond lemma for Lie algebras over a field

For the completeness of the paper, we formulate the Composition-Diamond lemma for Lie algebras over a field in this section, see [6, 33, 38, 41] for details. Let k be a filed, I a well ordered index set, X = {xi |i ∈ I} a set, X ∗ the free monoid generated by X and Lie(X) the free Lie algebra over k generated by X. We order X = {xi |i ∈ I} by xi > xt if i > t for any i, t ∈ I. We use two linear orders on X ∗ : for any u, v ∈ X ∗ , (i) (lex order) 1 ≻ t if t 6= 1 and, by induction, if u = xi ui and v = xj vj then u ≻ v if and only if xi > xj , or xi = xj and ui ≻ vj ; (ii) (deg-lex order) u > v if and only if deg(u) > deg(v), or deg(u) = deg(v) and u ≻ v, where deg(u) is the length of u. We regard Lie(X) as the Lie subalgebra of the free associative algebra khXi, which is generated by X under the Lie bracket [u, v] = uv − vu. Given f ∈ khXi, denote f¯ the leading word of f with respect to the deg-lex order. f is monic if the coefficient of f¯ is 1. Definition 2.1 An associative word w ∈ X ∗ \{1} is an associative Lyndon-Shirshov word (ALSW for short) if (∀u, v ∈ X ∗ , u, v 6= 1) w = uv ⇒ w > vu. A non-associative word (u) in X is a non-associative Lyndon-Shirshov word (NLSW for short), denoted by [u], if (i) u is an ALSW; (ii) if [u] = [(u1 )(u2 )] then both (u1 ) and (u2 ) are NLSW’s; (iii) if [u] = [[[u11 ][u12 ]][u2 ]] then u12 u2 . We denote the set of all ALSW’s and NLSW’s in X by ALSW (X) and NLSW (X) respectively. 2

For an ALSW w, there is a unique bracketing [w] such that [w] is NLSW: [w] = [[u][v]] if deg(w) > 1, where v is the longest proper associative Lyndon-Shirshov end of w. Shirshov Lemma Suppose that w = aub, where w, u ∈ ALSW (X). Then (i) [w] = [a[uc]d], where b = cd and possibly c = 1. (ii) Represent c in the form c = c1 c2 . . . cn , where c1 , . . . , cn ∈ ALSW (X) and c1 ≤ c2 ≤ . . . ≤ cn . Replacing [uc] by [. . . [[u][c1 ]] . . . [cn ]] we obtain the word [w]u = [a[. . . [[[u][c1 ]][c2 ]] . . . [cn ]]d] which is called the Shirshov special bracketing of w relative to u. (iii) [w]u = w. Definition 2.2 Let S ⊂ Lie(X) with each s ∈ S monic, a, b ∈ X ∗ and s ∈ S. If a¯ sb is an ALSW, then we call [asb]s¯ = [a¯ sb]s¯|[¯s]7→s a special normal S-word, where [a¯ sb]s¯ is defined as in Shirshov Lemma. An S-word (asb) is called a normal S-word if (asb) = asb. Suppose that f, g ∈ S. Then, there are two kinds of compositions: (i) If w = f¯ = a¯ g b for some a, b ∈ X ∗ , then the polynomial (f, g)w = f − [agb]g¯ is called the composition of inclusion of f and g with respect to w. (ii) If w is a word such that w = f¯b = a¯ g for some a, b ∈ X ∗ with deg(f¯) + deg(¯ g) > deg(w), then the polynomial (f, g)w = [f b]f¯ − [ag]g¯ is called the composition of intersection of f and g with respect to w. In (i) and (ii), w is called an ambiguity. Let h be a Lie polynomial and w ∈ X ∗ .PWe shall say that h is trivial modulo (S, w), denoted by h ≡Lie 0 mod(S, w), if h = i αi (ai si bi ), where each (ai si bi ) is a normal S-word and ai s¯i bi < w. The set S is called a Gröbner-Shirshov basis in Lie(X) if any composition in S is trivial modulo S and corresponding w. Theorem 2.3 (Composition-Diamond lemma for Lie algebras over a field) Let S ⊂ Lie(X) ⊂ khXi be nonempty set of monic Lie polynomials. Let Id(S) be the ideal of Lie(X) generated by S. Then the following statements are equivalent. (i) S is a Gröbner-Shirshov basis in Lie(X). (ii) f ∈ Id(S) ⇒ f¯ = a¯ sb for some s ∈ S and a, b ∈ X ∗ . (iii) Irr(S) = {[u] ∈ NLSW (X) | u 6= a¯ sb, s ∈ S, a, b ∈ X ∗ } is a linear basis for Lie(X|S) = Lie(X)/Id(S). Remark: The above Composition-Diamond lemma is also valid if we replace the base field k by an arbitrary commutative ring K with identity. If this is the case, as K-module, Lie(X|S) is free with a K-basis Irr(S). 3

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Kukin’s construction of a Lie algebra with unsolvable word problem

Let P = sgphx, y|ui = vi , i ∈ Ii be a semigroup. Consider the Lie algebra AP = Lie(x, xˆ, y, yˆ, z|S), where S consists of the following relations: (1) [ˆ xx] = 0, [ˆ xy] = 0, [ˆ y x] = 0, [ˆ y y] = 0, (2) [ˆ xz] = −[zx], [ˆ y z] = −[zy], (3) ⌊zui ⌋ = ⌊zvi ⌋, i ∈ I. Here, ⌊zu⌋ means the left normed bracketing. In this section, we give a Gröbner-Shirshov basis for Lie algebra AP and by using this result we give another proof for Kukin’s theorem, see Corollary 3.2. Let the order xˆ > yˆ > z > x > y and > the deg-lex order on {ˆ x, yˆ, x, y, z}∗. Let ρ be the congruence on {x, y}∗ generated by {(ui , vi ), i ∈ I}. Let (3)′ ⌊zu⌋ = ⌊zv⌋, (u, v) ∈ ρ with u > v. Theorem 3.1 With the above notation, the set S1 = {(1), (2), (3)′} is a Gröbner-Shirshov basis in Lie(ˆ x, yˆ, x, y, z). Proof: For any u ∈ {x, y}∗, by induction on |u|, ⌊zu⌋ = zu. All possible compositions in S1 are intersection of (2) and (3)′ , and inclusion of (3)′ and (3)′ . For (2) ∧ (3)′ , w = xˆzu, (u, v) ∈ ρ, u > v, f = [ˆ xz] + [zx], g = ⌊zu⌋ − ⌊zv⌋. We have xg]g¯ ([ˆ xz] + [zx], ⌊zu⌋ − ⌊zv⌋)w = [f u]f¯ − [ˆ ≡ ⌊([ˆ xz] + [zx])u⌋ − [ˆ x(⌊zu⌋ − ⌊zv⌋)] ≡ ⌊zxu⌋ + ⌊ˆ xzv⌋ ≡ ⌊zxu⌋ − ⌊zxv⌋ ≡ 0 mod(S1 , w). For (3)′ ∧ (3)′ , w = zu1 = zu2 e, e ∈ {x, y}∗, (ui , vi ) ∈ ρ, ui > vi , i = 1, 2. We have (⌊zu1 ⌋ − ⌊zv1 ⌋, ⌊zu2 ⌋ − ⌊zv2 ⌋)w ≡ (⌊zu1 ⌋ − ⌊zv1 ⌋) − ⌊(⌊zu2 ⌋ − ⌊zv2 ⌋)e⌋ ≡ ⌊⌊zv2 ⌋e⌋ − ⌊zv1 ⌋ ≡ ⌊zv2 e⌋ − ⌊zv1 ⌋ ≡ 0 mod(S1 , w). Thus, the set S1 = {(1), (2), (3)′} is a Gröbner-Shirshov basis in Lie(ˆ x, yˆ, x, y, z). Corollary 3.2 (Kukin [32]) Let u, v ∈ {x, y}∗. Then u = v in the semigroup P ⇔ ⌊zu⌋ = ⌊zv⌋ in the Lie algebra AP .

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Proof: Suppose that u = v in the semigroup P . Without loss of generality, we may assume that u = au1 b, v = av1 b for some a, b ∈ {x, y}∗ and (u1, v1 ) ∈ ρ. For any r ∈ {x, y}, by the relations (1), we have [ˆ xr] = 0 and so ⌊zxc⌋ = ⌊[zˆ x]c⌋ = [⌊zc⌋ˆ x], ⌊zyc⌋ = [⌊zc⌋ˆ y] ∗ for any c ∈ {x, y} . From this it follows that in AP , ⌊zu⌋ = ⌊zau1 b⌋ = ⌊⌊zau1 ⌋b⌋ = −−−−−−− ← − ← − ← − ⌊⌊zu1 c a ⌋b⌋ = ⌊zu1 c a b⌋ = ⌊zv1 c a b⌋ = ⌊zav1 b⌋ = ⌊zv⌋, where ← xi− 1 xi2 · · · xin = xin xin−1 · · · xi1 ′ c c c and xi1 x\ i1 x i2 · · · x in , xij ∈ {x, y}. Moreover, (3) holds in AP . i2 · · · xin = x Suppose that ⌊zu⌋ = ⌊zv⌋ in the Lie algebra AP . Then both ⌊zu⌋ and ⌊zv⌋ have the same normal form in AP . Since S1 is a Gröbner-Shirshov basis in AP by Theorem 3.1, both ⌊zu⌋ and ⌊zv⌋ can be reduced to the same normal form of the form ⌊zc⌋ for some c ∈ {x, y}∗ only by the relations (3)′ . This implies that in P , u = c = v. By the above corollary, if the semigroup P has the undecidable word problem then so does the Lie algebra AP .

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Gr¨ obner-Shirshov basis for the Drinfeld-Kohno Lie algebra Ln

In this section we give a Gröbner-Shirshov basis for the Drinfeld-Kohno Lie algebra Ln . Definition 4.1 ([27]) Let n > 2 be an integer. The Drinfeld-Kohno Lie algebra Ln over Z is defined by generators tij = tji for distinct indices 1 ≤ i, j ≤ n − 1, and relations tij tkl = 0, tij (tik + tjk ) = 0, where i, j, k, l are distinct. Clearly, Ln has a presentation LieZ (T |S), where T = {tij | 1 ≤ i < j ≤ n − 1} and S consists of the following relations tij tkl = 0, k < i < j, k < l, l 6= i, j tjk tij + tik tij = 0, i < j < k tjk tik − tik tij = 0, i < j < k

(1) (2) (3)

Now we order T : tij < tkl if either i < k or i = k and j < l. Let < be the deg-lex order on T ∗ . Theorem 4.2 Let S = {(1), (2), (3)} be as before, < the deg-lex order on T ∗ . Then S is a Gröbner-Shirshov basis for Ln . Proof. We list all the possible ambiguities. Denote (i) ∧ (j) the composition of the type (i) and type (j). For (1) ∧ (n), 1 ≤ n ≤ 3, the possible ambiguities w’s are: (1) ∧ (1) tij tkl tmr , (k < i < j, k < l, l 6= i, j, m < k < l, m < r, r 6= k, l), (1) ∧ (2) tij tkl tmk , ( k < i < j, k < l, l 6= i, j, m < k < l), (1) ∧ (3) tij tkl tml , (k < i < j, k < l, l 6= i, j, m < k < l).

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For (2) ∧ (n), 1 ≤ n ≤ 3, the possible ambiguities w’s are: (2) ∧ (1) tjk tij tmr , ( m < i < j < k, m < r, r 6= i, j), (2) ∧ (2) tjk tij tmi , ( m < i < j < k), (2) ∧ (3) tjk tij tmj , ( m < i < j < k). For (3) ∧ (n), 1 ≤ n ≤ 3, the possible ambiguities w’s are: (3) ∧ (1) tjk tik tmr , ( m < i < j < k, m < r, r 6= i, k), (3) ∧ (2) tjk tik tmi , ( m < i < j < k), (3) ∧ (3) tjk tik tmk , ( m < i < j < k). We claim that all compositions are trivial relative to S. Here, we only prove cases (1) ∧ (1), (1) ∧ (2), (2) ∧ (1), (2) ∧ (2), and the other cases can be proved similarly. For (1) ∧ (1), let f = tij tkl , g = tkl tmr , k < i < j, k < l, l 6= i, j, m < k < l, m < r, r 6= k, l. Then w = tij tkl tmr and (f, g)w = (tij tkl )tmr − tij (tkl tmr ) = (tij tmr )tkl mod(S, w). There are three subcases to consider: r 6= i, j, r = i, r = j. Subcase 1. If r 6= i, j, then (tij tmr )tkl ≡ 0 mod(S, w). Subcase 2. If r = i, then (tij tmr )tkl = (tij tmi )tkl ≡ tkl (tmj tmi ) ≡ (tkl tmj )tmi + tmj (tkl tmi ) ≡ 0 mod(S, w). Subcase 3. If r = j, then (tij tmr )tkl = (tij tmj )tkl ≡ −tkl (tmj tmi ) ≡ −(tkl tmj )tmi − tmj (tkl tmi ) ≡ 0 mod(S, w). For (1) ∧ (2), let f = tij tkl , g = tkl tmk + tml tmk , k < i < j, k < l, l 6= i, j, m < k < l. Then w = tij tkl tmk and (f, g)w = (tij tkl )tmk − tij (tkl tmk + tml tmk ) = (tij tmk )tkl − tij (tml tmk ) ≡ −tij (tml tmk ) ≡ −(tij tml )tmk − tml (tij tmk ) ≡ 0 mod(S, w). 6

For (2) ∧ (1), let f = tjk tij + tik tij , g = tij tmr , m < i < j < k, m < r, r 6= i, j. Then w = tjk tij tmr and (f, g)w = (tjk tij + tik tij )tmr − tjk (tij tmr ) ≡ (tjk tmr )tij + (tik tmr )tij mod(S, w). There are two subcases to consider: r 6= k, r = k. Subcase 1. If r 6= k, then (tjk tmr )tij + (tik tmr )tij ≡ 0 mod(S, w). Subcase 2. If r = k, then (tjk tmr )tij + (tik tmr )tij = (tjk tmk )tij + (tik tmk )tij ≡ −tij (tmk tmj ) − tij (tmk tmi ) ≡ (tij tmj )tmk + (tij tmi )tmk ≡ −tmk (tmj tmi ) + tmk (tmj tmi ) ≡ 0 mod(S, w). For (2)∧(2), let f = tjk tij +tik tij , g = tij tmi +tmj tmi , m < i < j < k. Then w = tjk tij tmi and (f, g)w = (tjk tij + tik tij )tmi − tjk (tij tmi + tmj tmi ) = (tjk tmi )tij + (tik tmi )tij + tik (tij tmi ) − tjk (tmj tmi ) ≡ tij (tmk tmi ) − tik (tmj tmi ) − tjk (tmj tmi ) ≡ −(tij tmi )tmk + (tik tmi )tmj − (tjk tmj )tmi ≡ −tmk (tmj tmi ) − (tmk tmi )tmj + (tmk tmj )tmi ≡ 0 mod(S, w). So S is a Gröbner-Shirshov basis for Ln .

Let L be a Lie algebra over a commutative ring K, L1 an ideal of L and L2 a subalgebra of L. We call L a semidirect product of L1 and L2 if L = L1 ⊕ L2 as K-modules. By Theorems 2.3 and 4.2, we have immediately the following corollaries. Corollary 4.3 The Drinfeld-Kohno Lie algebra Ln is a free Z-module with a Z-basis Irr(S) = {[tik1 tik2 · · · tikm ] | tik1 tik2 · · · tikm is an ALSW in T ∗ , m ∈ N}. Corollary 4.4 ([27]) Ln is an iterated semidirect product of free Lie algebras. Proof. Let Ai be the free Lie algebra generated by {tij | i < j ≤ n − 1}. Clearly, Ln = A1 ⊕ A2 ⊕ · · · ⊕ An−2 as Z-modules, and from the relations (1), (2), (3), we have Ai ⊳ Ai + Ai+1 + · · · + An−2 . Remark: In this section, if we replace the base ring Z by an arbitrary commutative ring K with identity, then all results hold. 7

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