Representations of simple anti-Jordan triple systems ...

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Journal of Algebra and Its Applications Vol. 16, No. 5 (2017) 1750093 (28 pages) c World Scientific Publishing Company DOI: 10.1142/S0219498817500931

Representations of simple anti-Jordan triple systems of m × n matrices

Hader A. Elgendy

J. Algebra Appl. 2017.16. Downloaded from www.worldscientific.com by WSPC on 07/18/17. For personal use only.

Department of Mathematics, Faculty of Science Damietta University, Egypt [email protected] Received 21 April 2014 Accepted 28 April 2016 Published 9 June 2016 Communicated by L. Bokut We show that the universal associative envelope of the simple anti-Jordan triple system of all m×n (m is even, m, n ≥ 2) matrices over an algebraically closed field of characteristic 0 is finite-dimensional. The monomial basis and the center of the universal envelope are determined. The explicit decomposition of the universal envelope into matrix algebras is given. The classification of finite-dimensional irreducible representations of an antiJordan triple system is obtained. The semi-simplicity of the universal envelope is shown. We also show that the universal associative envelope of the simple polarized anti-Jordan triple system of (n + 1) × (n + 1)(n > 2) matrices is infinite-dimensional. Keywords: Anti-Jordan triple systems; noncommutative Gr¨ obner–Shirshov bases; representation theory; universal enveloping algebras; free associative algebras. Mathematics Subject Classification: 17C55, 13P10, 16S30, 17B35, 17A40

1. Introduction An anti-Jordan triple system is vector space V over a field F of characteristic = 2 endowed with a trilinear operation V × V × V → V , (a, b, c) → a, b, c that satisfies for all a, b, c, d, e ∈ V a, b, c = −c, b, a, a, b, c, d, e = a, b, c, d, e + c, b, a, d, e + c, d, a, b, e. An anti-Jordan triple system J is called polarized if J = J+ ⊕ J− and the triple product satisfies J , J , J = 0 = J, J , J and J , J− , J ⊂ J , = ±. A representation of an anti-Jordan triple system J is a homomorphism ρ : J → (End V )− from J to the anti-Jordan triple system of endomorphisms of a vector

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space V . In other words, ρ is a linear mapping that satisfies


ρ(a, b, c) = ρ(a)ρ(b)ρ(c) − ρ(c)ρ(b)ρ(a)

for all a, b, c ∈ J.

A subspace W of V is called invariant under ρ if ρ(x)W ⊆ W for all x ∈ J. A representation ρ is called irreducible if the only invariant subspaces of V under ρ are the trivial ones, {0} and V . Finite-dimensional simple anti-Jordan triple systems over algebraically closed fields of characteristic 0 were classified [3] based on the classification of simple anti-Jordan pairs [11]. The classification provides seven types of simple anti-Jordan triple system, none of them is exceptional. Anti-Jordan triple system is a special case of more general (, δ)-Freudenthal– Kantor triple systems [15], which have the potential for a wide range of applications to various branches of mathematics and physics. They have been appearing in physical applications, especially in connection to Maxwell–Einstein Supergravity Theories, U-duality, and black hole entropy and in studying gauge theories [6, 13]. Also, they have been used to find solutions of the Yang–Baxter equation [12]. Moreover, anti-Jordan triple systems are the 3-algebras in [2] which are used to reformulate the N = 6 theories of [1]. In [9] we showed that the universal associative envelope of the simple antiJordan triple system of all n × n (n ≥ 2) matrices is finite-dimensional. We also classified all finite-dimensional irreducible representations and showed that up to equivalence there exist only five finite-dimensional irreducible representations of this anti-Jordan triple system. In general, the universal associative envelopes of antiJordan triple systems are not necessary finite-dimensional, for examples of simple and nonsimple anti-Jordan triple systems with infinite-dimensional envelopes see [8–10, 14]. The aim of the present paper is to extend our results on the simple anti-Jordan triple system of all n × n (n ≥ 2) matrices with the triple product x, y, z = xyz − zyx [9] to the simple anti-Jordan triple system of all m × n (m, n ≥ 2)

matrices with the triple product x, y, z = xy t az − zy t ax, where a = −I0 r I0r , and 2r = m, and to prove our conjecture on the semi-simplicity of finite-dimensional envelopes of finite-dimensional simple anti-Jordan triple systems in this case. We also aim to show that the universal associative envelope of the simple polarized anti-Jordan triple system of (n + 1) × (n + 1) (n > 2) matrices with the triple product x, y, z = xyz − zyx is infinite-dimensional. This paper is organized as follows. In Sec. 2, we recall basic results on noncommutative Gr¨ obner–Shirshov bases developed in [4] and give examples of anti-Jordan triple systems. In Sec. 3, we use the theory of noncommutative Gr¨ obner–Shirshov bases to prove that the universal enveloping algebra of the simple anti-Jordan triple system of m × n (m is even, m, n ≥ 2) matrices over an algebraically closed field is finite-dimensional (Theorem 3.13). We also determine the complete decomposition of the universal enveloping algebra into a direct sum of matrix algebras (Theorem 1750093-2

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Anti-Jordan triple system

3.14) and the center (Lemma 3.15). Finally, in Sec. 4 we construct the universal associative envelope of the simple polarized anti-Jordan triple system of (n+ 1)× (n+ 1) (n > 2) matrices and show that it has infinite dimension (Theorem 4.6). We assume throughout that all vector spaces are over an algebraically closed field F of characteristic 0. 2. Preliminaries 2.1. Gr¨ obner–Shirshov bases in free associative algebras


We recall basic results about noncommutative Gr¨ obner–Shirshov bases for ideals in free associative algebras following [7, 9]. For a comprehensive survey on Gr¨ obner– Shirshov bases and their calculations, see [5]. Definition 2.1. Let X = {x1 , . . . , xn } be a set of symbols with the total order xi < xj if and only if i < j. The free monoid generated by X is the set X ∗ of all (possibly empty) words w = xi1 · · · xik (k ≥ 0) with the (associative) operation of concatenation. For w = xi1 · · · xik ∈ X ∗ the degree is deg(w) = k. The degreelexicographical (deglex ) order < on X ∗ is defined as follows: u < v if and only if either (i) deg(u) < deg(v) or (ii) deg(u) = deg(v) and u = wxi u , v = wxj v where xi < xj (w, u , v ∈ X ∗ ). The free (unital) associative algebra generated by X is the vector space F X with basis X ∗ and multiplication extended bilinearly from concatenation in X ∗ . Definition 2.2. The support of a noncommutative polynomial f ∈ F X is the set of all monomials w ∈ X ∗ that occur in f with nonzero coefficient. The leading monomial of f ∈ F X, denoted LM(f ), is the highest element of the support of f with respect to deglex order. If I is any ideal of F X then the set of normal words modulo I is defined by N (I) = {u ∈ X ∗ | u = LM(f ) for any f ∈ I}. We write C(I) for the subspace of F X spanned by N (I). Proposition 2.3. We have F X = C(I) ⊕ I. Definition 2.4. Let G ⊂ F X be a subset generating an ideal I. An element f ∈ F X is in normal form modulo G if no monomial in the support of f has LM(g) as a factor for any g ∈ G. Definition 2.5. A subset G ⊂ I is a Gr¨ obner–Shirshov basis of I if for every f ∈ I there exists g ∈ G such that LM(g) is a factor of LM(f ). A subset G ⊂ F X is self-reduced if every g ∈ G is in normal form modulo G\{g} and every g ∈ G is monic: the coefficient of LM(g) is 1. Definition 2.6. Consider elements g, h ∈ F X such that LM(g) is not a factor of LM(h) and LM(h) is not a factor of LM(g). Assume that u, v ∈ X ∗ satisfy LM(g) u = v LM(h), u is a proper right factor of LM(h), and v is a proper left factor of LM(g). Then gu − vh is called a composition of g and h. 1750093-3

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Theorem 2.7. If I ⊂ F X is an ideal generated by a self-reduced set G, then G is a Gr¨ obner–Shirshov basis of I if and only if for all compositions f of the elements of G the normal form of f modulo G is zero (all compositions f of the elements of G vanish upon the elimination of the leading terms LM(g) for g ∈ G). 2.2. Examples of anti-Jordan triple systems We recall some examples of finite-dimensional anti-Jordan triple systems. The next three examples are simple anti-Jordan triple systems (see [3]).


Example 2.8. Let V be the vector space of all n × n matrices over an algebraically closed field of characteristic 0. Then V is an anti-Jordan triple system with respect to the triple product x, y, z = xyz − zyx. Example 2.9. Let V be the vector space of all m×n matrices over an algebraically closed field of characteristic 0. Then V is an anti-Jordan triple system with respect t t t to the tripleproduct x, y, z = xy az − zy ax, where y is the transpose of y, a = −I0 r I0r , Ir is the r × r unit matrix and 2r = m. Example 2.10. Let V be the vector space of all m×n matrices over an algebraically closed field of characteristic 0. Then V is an anti-Jordan triple system with respect t t t to the triple product x, y, z = xby z − zby x, where y is the transpose of y, b=

0

Ir

−Ir

0

, Ir is the r × r unit matrix and 2r = n.

The next example is a nonsimple anti-Jordan triple system (see [8, 10]). Example 2.11. Let V be the vector space of 2 × 2 matrices A = (aij ) with a11 = a22 = 0 over an algebraically closed field of characteristic 0. Then V is an anti-Jordan triple system with respect to the triple product a, b, c = abc − cba. Universal associative envelopes for the anti-Jordan triple systems given in Examples 2.8 and 2.11 have been constructed (see [8–10]). 3. The Universal Associative Envelope of the Anti-Jordan Triple System of m × n Matrices Definition 3.1. Let J be the anti-Jordan triple system of all m × n matrices over an algebraically closed field F of characteristic 0 with product x, y, z = the triple xy t az − zy t ax, where y t is the transpose of y, a = matrix and 2r = m.

0

Ir

−Ir

0

, Ir is the r × r unit

Definition 3.2. Let C = {1, 2, . . . , r}, D = {r + 1, r + 2, . . . , m}, H = {m + 1, . . . , m + n}, Ω = C ∪ D and Ω = {1, 2, . . . , n} be finite index sets. Let B = {Ei,j }i∈Ω, j∈Ω be a basis of J, where Ei,j denote the m×n matrix whose (i, j)-entry (resp. Ei,j ) denote the (m + n) × (m + n) is 1 and all the other entries are 0. Let Ei,j (resp. m × m) matrix whose (i, j)-entry is 1 and all the other entries are 0. 1750093-4

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Notation 3.3. We use the following notation throughout the paper: • δi,j for the Kronecker delta, and δi,j = 1 − δi,j , • ∆i,S = 1 if i ∈ S, and 0 otherwise. The structure constants for J can be determined by the following lemma. Lemma 3.4. The structure constants of J are given by Ei,j , Ek, , Es,d = δj, (δk,s−r − δk,s+r )Ei,d + δ,d (−δk,i−r + δk,i+r )Es,j ,


for all i, k, s ∈ Ω and j, , d ∈ Ω . Proof. For all i, k, s ∈ Ω and j, , d ∈ Ω , we have Ei,j , Ek, , Es,d = Ei,j (Ek, )t

r

(Et,t+r − Et+r,t )Es,d − Es,d (Ek, )t

r

t=1

=

δj, Ei,k (∆s,D Es−r,d

(Et,t+r − Et+r,t )Ei,j

t=1

− ∆s,C Es+r,d ) −

δ,d Es,k (∆i,D Ei−r,j

− ∆i,C Ei+r,j )

= δj, (∆s,D δk,s−r Ei,d − ∆s,C δk,s+r Ei,d ) − δ,d (∆i,D δk,i−r Es,j − ∆i,C δk,i+r Es,j ). Consider the bijection φ : B → X = {ei,j }i∈Ω, j∈Ω defined by φ(Ei,j ) = ei,j . We extend φ to a linear map φ : J → F X. Throughout this paper we use the deglex order < where ei,j < ek, if either i < k, or i = k and j < . Definition 3.5. Let G ⊂ F X consist of these elements: (i,j,s,d)

R1

= ei,j es−r,j es,d − es,d es−r,j ei,j − ei,d (r + 1 ≤ s < i ≤ m; 1 ≤ j, d ≤ n),

(s,d,j,i)

R2

= es,d es−r,j ei,j − ei,j es−r,j es,d + ei,d (1 ≤ i < s ≤ m; r + 1 ≤ s; 1 ≤ j, d ≤ n),

(i,j,d) R3

= ei,j ei−r,j ei,d − ei,d ei−r,j ei,j − ei,d

(1 ≤ d < j ≤ n; r + 1 ≤ i ≤ m),

(i,d,j)

= ei,d ei−r,j ei,j − ei,j ei−r,j ei,d + ei,d

(1 ≤ j < d ≤ n; r + 1 ≤ i ≤ m),

R4

(i,j,s,d)

R5

= ei,j es+r,j es,d − es,d es+r,j ei,j + ei,d (1 ≤ s < i ≤ m; s ≤ r; 1 ≤ j, d ≤ n),

(s,d,j,i)

R6

(i,j,d)

R7

= es,d es+r,j ei,j − ei,j es+r,j es,d − ei,d = ei,j ei+r,j ei,d − ei,d ei+r,j ei,j + ei,d 1750093-5

(1 ≤ i < s ≤ r; 1 ≤ j, d ≤ n), (1 ≤ d < j ≤ n; 1 ≤ i ≤ r),

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R8

(i,j,k,,s,d)

R9

= ei,d ei+r,j ei,j − ei,j ei+r,j ei,d − ei,d

(1 ≤ j < d ≤ n; 1 ≤ i ≤ r),

= ei,j ek, es,d − es,d ek, ei,j (1 ≤ s < i ≤ m; 1 ≤ k ≤ m; 1 ≤ j, , d ≤ n),

and either (1)

(j, d = ) or

(2) (k = s ± r, i ± r; j = or d = ) or

(3)

(k = i ± r; k = s − r or s + r; j = , d = ) or

(4)

(k = s ± r; k = i − r or i + r; d = ; j = )

(i,j,k,,d)


R10 = ei,j ek, ei,d − ei,d ek, ei,j (1 ≤ d < j ≤ n; 1 ≤ ≤ n; 1 ≤ i, k ≤ m), and either (1) (j, d = ) or (2) (k = i ± r; j = or d = ). Let I ⊂ F X be the ideal generated by G. We write A(J) = F X/I with surjection π : F X → A(J) sending f to f + I, and ı = π ◦ φ for the natural map ı : J → A(J). Lemma 3.6. The unital associative algebra A(J) and the linear map ı form the universal associative envelope of the anti-Jordan triple system J. 3.1. Gr¨ obner–Shirshov basis and finite-dimensionality Our goal in this subsection is to derive a Gr¨ obner–Shirshov basis for the ideal I from the set G of generators. Notation 3.7. Throughout the proofs we write ≡L to indicate congruence modulo L. We also use R9,t ; t = 1, . . . , 4 (resp. R10,s ; s = 1, 2) to indicate the relation R9 (resp. R10 ) of Definition 3.5 with case number t (resp. s). Lemma 3.8. The set of all normal forms modulo G of nontrivial compositions among elements of G includes the set G1 which consists of the elements: (,d,t)

G1

(,d,t)

G2

= e+r,d e,t + e1,d e1+r,t

(1 ≤ ≤ r; 1 ≤ d, t ≤ n; d = t),

= e,d e+r,t − e1,d e1+r,t

(1 < ≤ r; 1 ≤ d, t ≤ n; d = t),

(,d)

= e,d e+r,d − e,1 e+r,1 + e1,1 e1+r,1 − e1,d e1+r,d

(1 < ≤ r; 1 < d ≤ n),

(,d)

= e+r,d e,d − e+r,1 e,1 + e1,d e1+r,d − e1,1 e1+r,1

(1 ≤ ≤ r; 1 < d ≤ n),

G3 G4

(,d,i) G5 (i,d,k,t) G6

= e,d ei,d − e,1 ei,1 = ei,d ek,t

(1 ≤ , i ≤ m; = i ± r; 1 < d ≤ n),

(1 ≤ i, k ≤ m; i = k ± r; 1 ≤ d, t ≤ n; d = t).

Proof. We consider the composition: (i,d,1,d)

S1 = R5

(1+r,d,1,d,,t)

e,t − ei,d R9,4

(1 < i ≤ m; 1 ≤ < 1 + r; 1 ≤ d, t ≤ n; d = t). 1750093-6

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We eliminate from S1 all occurrences of the leading monomials of G: S1 = −e1,d e1+r,d ei,d e,t + ei,d e,t + ei,d e,t e1,d e1+r,d ≡G −e1,d (e,t ei,d e1+r,d − δi,+r e1+r,t ) + ei,d e,t + e1,d e,t ei,d e1+r,d = δi,+r e1,d e1+r,t + ei,d e,t , (1 + r,d,,t)

using R5 with

(,d,t) G1 .

(1 + r,d,i,d,,t)

, R9,2

(i,d,,t,1,d)

and R9,1

. For i = + r, the result coincides

For i = + r, the normal form of S1 is (1 < i ≤ m; 1 ≤ < 1 + r; i = + r; 1 ≤ d, t ≤ n; d = t).

ei,d e,t

(1)


We consider the composition: (s,d,1,1)

S2 = R6

(s+r,1,1+r,q)

e1+r,q − es,d R1

(1 < s ≤ r; 1 ≤ d, q ≤ n).

We eliminate from S2 all occurrences of the leading monomials of G: S2 = −e1,1 es+r,1 es,d e1+r,q − e1,d e1+r,q + es,d e1+r,q e1,1 es+r,1 + es,d es+r,q ≡G −e1,1 [δd,q (δd,1 + δd,1 )e1+r,q es,d es+r,1 + δd,q (e1+r,d es,d es+r,1 − e1+r,1 )] − e1,d e1+r,q + [δd,q (δq,1 + δq,1 )e1,1 e1+r,q es,d + δd,q (e1,1 e1+r,d es,d − es,1 )] × es+r,1 + es,d es+r,q = δd,q e1,1 e1+r,1 − e1,d e1+r,q − δd,q es,1 es+r,1 + es,d es+r,q , using

(s+r,1,s,d,1+r,q)

R9,1

(s,d,1+r,1,1,1) R9,3

and

(s+r,1,s,1,1+r,q)

,

(s,d,1,1) R5 .

R9,4

(k,j,1,1)

,

(s,d,1+r,q,1,1)

R9,1

,

Clearly, if d = q = 1 then S2 ≡G 0. For d = q (resp. d = (s,d,q)

q = 1), the result coincides with G2 S3 = R5

(s+r,1,d,1+r)

R2

,

(1+r,j,1,i)

ei,1 − ek,j R2

(s,d)

(resp. G3

). We consider the composition:

(1 < k ≤ m; 1 ≤ i < 1 + r; 1 ≤ j ≤ n).

We eliminate from S3 all occurrences of the leading monomials of G: S3 = −e1,1 e1+r,j ek,j ei,1 + ek,1 ei,1 + ek,j ei,1 e1,1 e1+r,j − ek,j ei,j ≡G −e1,1 (ei,1 ek,j e1+r,j − δk,i+r e1+r,1 ) + ek,1 ei,1 + (e1,1 ei,1 ek,j − δk,i+r e1,j )e1+r,j − ek,j ei,j = δk,i+r e1,1 e1+r,1 + ek,1 ei,1 − δk,i+r e1,j e1+r,j − ek,j ei,j , (1+r,j,i,1)

(1+r,j,k,j,i,1)

(i+r,j,1,1)

(k,j,i,1,1,1)

, R9,2 , R2 and R9,2 . Clearly, if j = 1 then using R5 S3 ≡G 0. For j = 1 and k = i + r the (monic) normal form of S3 coincides with (i,j) G4 . For j = 1 and k = i + r, the (monic) normal form of S3 is ek,j ei,j − ek,1 ei,1

(1 < k ≤ m; 1 ≤ i < 1 + r; k = i + r; 1 < j ≤ n). 1750093-7

(2)

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We consider the composition: (k,j,,1)

S4 = R5

(+r,j,1)

e+r,1 − ek,j R4

(1 ≤ < k ≤ m; ≤ r; 1 < j ≤ n).

We eliminate from S4 all occurrences of the leading monomials of G: S4 = −e,1 e+r,j ek,j e+r,1 + ek,1 e+r,1 + ek,j e+r,1 e,1 e+r,j − ek,j e+r,j ≡G −e,1 e+r,1 ek,j e+r,j + ek,1 e+r,1 + e,1 e+r,1 ek,j e+r,j − ek,j e+r,j = ek,1 e+r,1 − ek,j e+r,j , (+r,j,k,j,1)

using R10,2

(k,j,+r,1,,1)

and R9,3


ek,j e+r,j − ek,1 e+r,1

. Hence, the (monic) normal form of S4 is

(1 ≤ < k ≤ m; ≤ r; 1 < j ≤ n).

(3)

Setting + r = i in (3) gives ek,j ei,j − ek,1 ei,1

(i − r < k ≤ m; 1 + r ≤ i ≤ m; 1 < j ≤ n).

(4)

We consider the composition: (,1,j,k)

S5 = R6

(+r,j,k,j,1)

e+r,1 − e,1 R10,2

(1 ≤ k < ≤ r; 1 < j ≤ n).

We eliminate from S5 all occurrences of the leading monomials of G: S5 = −ek,j e+r,j e,1 e+r,1 − ek,1 e+r,1 + e,1 e+r,1 ek,j e+r,j ≡G −ek,j (e+r,1 e,1 e+r,j − e+r,j ) − ek,1 e+r,1 + ek,j e+r,1 e,1 e+r,j , (+r,j,1)

using R4

(,1,+r,1,k,j)

and R9,4

. Hence, the normal form of S5 is

ek,j e+r,j − ek,1 e+r,1

(1 ≤ k < ≤ r; 1 < j ≤ n).

(5)

Setting + r = i in (5) gives ek,j ei,j − ek,1 ei,1

(1 ≤ k < i − r; 1 + r < i ≤ m; 1 < j ≤ n).

(6)

We consider the composition: (1,j,1)

S6 = R8

(1+r,1,1,1,i,j)

ei,j − e1,j R9,4

(1 ≤ i < 1 + r; 1 < j ≤ n).

We eliminate from S6 all occurrences of the leading monomials of G: S6 = −e1,1 e1+r,1 e1,j ei,j − e1,j ei,j + e1,j ei,j e1,1 e1+r,1 ≡G −e1,1 (ei,j e1,j e1+r,1 − ei,1 ) − e1,j ei,j + e1,1 ei,j e1,j e1+r,1 = e1,1 ei,1 − e1,j ei,j , (1+r,1,j,i)

using R2

(1,j,i,j,1)

and R10,2

. Hence the (monic) normal form of S6 is

e1,j ei,j − e1,1 ei,1

(1 ≤ i < 1 + r; 1 < j ≤ n).

(7)

(1 ≤ k ≤ m; 1 ≤ i < 1 + r; k = i + r; 1 < j ≤ n).

(8)

Combining (2) and (7) gives ek,j ei,j − ek,1 ei,1

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Combining (4) and (6) gives ek,j ei,j − ek,1 ei,1

(1 ≤ k ≤ m; r + 1 ≤ i ≤ m; k = i − r; 1 < j ≤ n). (k,j,i)

Combining (8) and (9) gives all elements of the form G5 the next two compositions: (i,j,k,,s,j)

S7 = R9,1

(k,,s,j,t,j)

et,j − ei,j R9,2

(9)

. For r ≥ 2, we consider

(1 ≤ s < i ≤ m; 1 ≤ t < k ≤ m;

s = t ± r, k ± r; j = ; 1 ≤ j, ≤ n), (i,j,k,,s,j)

S8 = R9,1

(k,,s,j,t,j)

et,j − ei,j R9,3

(1 ≤ s < i ≤ m; 1 ≤ t < k ≤ m; s = k ± r;


s = t + r or t − r; j = ; 1 ≤ j, ≤ n). We eliminate from S7 and S8 all occurrences of the leading monomials of G. For S7 , we have S7 = −es,j ek, ei,j et,j + ei,j et,j es,j ek, ≡G −es,j et,j ei,j ek, + (δi,k−r − δi,k+r )es,j et, + es,j et,j ei,j ek, + (−δt,i−r + δt,i+r )es,j ek, = (δi,k−r − δi,k+r )es,j et, + (−δt,i−r + δt,i+r )es,j ek, , (k,,j,t)

(k,,j,t)

(k,,i,j,t,j)

(k,,i,j,t,j)

(i,j,j,s)

(i,j,j,s)

(i,j,t,j,s,j)

using R2 , R6 , R9,2 , R9,3 , R2 , R6 and R9,2 . We first observe that if i = k − r then t = i ± r, for if i = k − r and t = i − r (resp. t = i + r) then t = k − 2r = k − m ≤ 0 (resp. t = k). This contradicts our assumption that 1 ≤ t (resp. t < k). Similarly we can show that if i = k + r then t = i ± r. For i = k − r, i = k + r, t = i − r, or t = i + r, the (monic) normal forms of S7 are es,j et,

(1 ≤ s < r; 1 ≤ t < m; s = t − r; j = ),

(10)

es,j et,

(1 ≤ s < m; 1 ≤ t < r; s = t + r; j = ),

(11)

es,j ek,

(1 ≤ s < m; max{1, s − r + 1} < k ≤ m; s = k ± r; j = ),

es,j ek,

(1 ≤ s < (k − 1) − r; 2 + r < k ≤ m; s = k ± r; j = ),

or

(12) (13)

respectively. For S8 , we have S8 = −es,j ek, ei,j et,j + ei,j et,j es,j ek, ≡G −es,j et,j ei,j ek, + (δi,k−r − δi,k+r )es,j et, + es,j et,j ei,j ek, − (δt,s+r − δt,s−r )ei,j ek, = (δi,k−r − δi,k+r )es,j et, − (δt,s+r − δt,s−r )ei,j ek, , (k,,i,j,t,j)

(k,,j,t)

(k,,j,t)

(i,j,s,j)

(i,j,s,j)

using R9,2 , R2 , R6 , R5 and R1 . We observe that if i = k − r (resp. i = k + r) then t = s − r (resp. t = s + r), for if i = k − r (resp. i = k + r) and t = s − r (resp. t = s + r) then i ≤ r (resp. k ≤ r) and s > r (resp. 1750093-9

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t > r). This contradicts our assumption that i > s (k > t). Three cases need to be considered. Case 1: i = k − r and t = s + r, then S8 ≡G es,j es+r, − ek−r,j ek, , and hence S8 ≡{G (s,j,) ,G (k−r,j,) } e1,j e1+r, − e1,j e1+r, = 0. Case 2: i = k + r and 2 2 t = s − r, then S8 ≡G −es,j es−r, + ek+r,j ek, , and hence S8 ≡{G (s−r,j,) ,G (k,j,) } 1 1 e1,j e1+r, − e1,j e1+r, = 0. Case 3: i = k ± r. For t = s − r or t = s + r the (monic) normal forms of S8 are ei,j ek,

(1 + r < i ≤ m; 1 < k ≤ m; i = k ± r; j = ),

ei,j ek,

(1 < i ≤ m; 1 + r < k ≤ m; i = k ± r; j = ),

or

(14) (15)

respectively. Combining (11) and (15) gives the elements of the form J. Algebra Appl. 2017.16. Downloaded from www.worldscientific.com by WSPC on 07/18/17. For personal use only.

(i,j,k,)

G6

(1 < i < m; 1 ≤ k ≤ m; k = i ± r; k = r, 1 + r; j = ). (16)

= ei,j ek,

From (14), we have (m,j,k,)

G6

(1 < k ≤ m; k = r; j = ).

= em,j ek,

(17)

From (10) and (12), we have (1,j,k,)

G6

= e1,j ek,

(1 ≤ k ≤ m; k = 1 + r; j = ).

(18)

From (11) and (12), we have (1+r,j,k,)

G6

(1+r,j,2,)

G6

(r,j,k,)

G6

= e1+r,j ek,

(2 < k ≤ m; j = ),

(19)

= e1+r,j e2,

(2 < r; j = ),

(20)

(1 ≤ k ≤ m; j = ).

(21)

= er,j ek,

(i,j,k,)

= ei,j ek, Combining (17)–(21) with (16) gives all elements of the form G6 (1+r,j,2,) (m,j,1,) (1 ≤ i, k ≤ m, i = k ± r, j = ) except G6 (for r = 2) and G6 (for r ≥ 2). But these forms can be obtained from (1). Thus from the compositions S1 , (i,j,k,) (for r ≥ 2). To complete S7 and S8 , we obtain all elements of the form G6 the proof of the lemma, it remains to show that for r = 1, all elements of the (i,j,k,) are normal forms of nontrivial compositions among elements of G. form G6 For r = 1, we consider the following compositions (1,j,k)

S9 = R7

(2,j,j,1)

S10 = R2

(k < j; 1 ≤ k, j ≤ n),

(1,j,1,j,k)

(k < j; 1 ≤ k, j ≤ n),

e1,k − e2,j R10,2

(2,j,2,j,k)

S11 = R10,2

(2,j,1,j)

S12 = R5

(2,j,1,k,1,j)

e1,j − e1,j R9,1

(2,j,2,k,1,j)

e1,j − e2,j R9,1

(2,j,k)

e2,k − e2,j R3

(k < j; 1 ≤ k, j ≤ n),

(k < j; 1 ≤ k, j ≤ n).

For S9 , we have S9 = −e1,k e2,j e1,j e1,j + e1,k e1,j + e1,j e1,j e1,k e2,j ≡G −e1,k (e1,j e1,j e2,j − e1,j ) + e1,k e1,j + e1,k e1,j e1,j e2,j = 2e1,k e1,j , 1750093-10

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Anti-Jordan triple system (2,j,j,1)

using R2

(1,j,1,j,k)

and R10,2

. For S10 , we have

S10 = −e1,j e1,j e2,j e1,k + e1,j e1,k + e2,j e1,k e1,j e1,j ≡G −e1,j (e1,k e2,j e1,j − e1,k ) + e1,j e1,k + e1,j e1,k e2,j e1,j = 2e1,j e1,k , (1,j,k)

using R7

(2,j,1,k,1,j)

and R9,1

. For S11 , we have

S11 = −e2,k e2,j e2,j e1,j + e2,j e1,j e2,k e2,j ≡G −e2,k (e1,j e2,j e2,j − e2,j ) + (e2,k e1,j e2,j + e2,k )e2,j = 2e2,k e2,j , using

(2,j,1,j) R5

(2,j,k)

and R3

. For S12 , we have


S12 = −e1,j e2,j e2,j e2,k + e2,j e2,k + e2,j e2,k e1,j e2,j + e2,j e2,k ≡G −e1,j e2,k e2,j e2,j + e2,j e2,k + e1,j e2,k e2,j e2,j + e2,j e2,k = 2e2,j e2,k , (2,j,2,j,k)

(2,j,2,k,1,j)

using R10,2 and R9,1 . Combining the (monic) normal forms of S9 –S12 gives the required elements. This completes the proof. Lemma 3.9. The set of all normal forms modulo G∪G1 of nontrivial compositions among elements of G ∪ G1 includes the set G2 which consists of the elements: (k,,s)

G7

(1 ≤ k, , s ≤ m; k = ± r; = s ± r),

= ek,1 e,1 es,1

()

G8 = e1,1 e,1 e+r,1 − e21,1 e1+r,1 (,k)

G9

(k,)

G10

= e+r,1 e,1 ek,1 + e1,1 e1+r,1 ek,1

()

(t,)

( + r < k ≤ m; 1 ≤ ≤ r),

= ek,1 e,1 e+r,1 + e1,1 e1+r,1 ek,1 − ek,1

G11 = e1,1 e+r,1 e,1 + e21,1 e1+r,1 G12

(1 < ≤ r),

(1 < k < + r; 1 ≤ ≤ r),

(1 < ≤ r),

= et,1 e+r,1 e,1 − e1,1 e1+r,1 et,1 + et,1

(1 < t < ; 2 < ≤ r),

G13 = e1+r,1 e1,1 e1+r,1 + 2e1,1 e1+r,1 e1+r,1 − e1+r,1 , (,t) G14

= e,1 e+r,1 et,1 − e1,1 e1+r,1 et,1

( < t ≤ m; 1 < ≤ r).

Proof. We consider the compositions: (k,1,t,j)

T1 = G6

(,j)

T2 = G4

(,j,k,1)

ek,1 − e+r,j G6

(t,1,+r,j)

T3 = G6

(t,j,s)

es,j − ek,1 G5

(,j)

e,j − et,1 G4

(k = t ± r; t = s ± r; 1 ≤ k, t, s ≤ m; 1 < j ≤ n), ( = k ± r; 1 ≤ k ≤ m; 1 ≤ ≤ r; 1 < j ≤ n), (t = ; 1 ≤ t ≤ m; 1 ≤ ≤ r; 1 < j ≤ n). (k,t,s)

. We now eliminate from T2 Clearly, T1 = ek,1 et,1 es,1 , which coincides with G7 and T3 all occurrences of the leading monomials of G ∪ G1 . For T2 , we have T2 = −e+r,1 e,1 ek,1 + e1,j e1+r,j ek,1 − e1,1 e1+r,1 ek,1 ≡G1 −e+r,1 e,1 ek,1 + δk,1 e1,j e1+r,j e1,1 − e1,1 e1+r,1 ek,1 1750093-11

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≡G −(e+r,1 e,1 ek,1 + e1,1 e1+r,1 ek,1 ) + δk,1 (e1,1 e1+r,j e1,j − e1,1 ) ≡G∪G1 −[δk,1 (e1,1 e,1 e+r,1 − e1,1 ) + δk,1 e+r,1 e,1 ek,1 + e1,1 e1+r,1 ek,1 ] + δk,1 (e1,1 e1+r,1 e1,1 + e21,1 e1+r,1 − e1,1 ), (1+r,j,k,1)

(1,j,1)

(1,j)

(+r,1,1,1)

(1,1,1,j)

using G6 , R7 , G4 , R2 and G6 . For k = 1, the monic form () of the result coincides with G8 . For k = 1 and + r < k (resp. k < + r), the result (,k) (k,) (resp. G10 , because e+r,1 e,1 ek,1 ≡G ek,1 e,1 e+r,1 − ek,1 , coincides with G9 (+r,1,1,k) using R2 ). For T3 , we have T3 = et,1 e+r,1 e,1 − et,1 e1,j e1+r,j + et,1 e1,1 e1+r,1 J. Algebra Appl. 2017.16. Downloaded from www.worldscientific.com by WSPC on 07/18/17. For personal use only.

≡G1 et,1 e+r,1 e,1 − δt,1+r e1+r,1 e1,j e1+r,j + et,1 e1,1 e1+r,1 ≡G∪G1 ∪{G (t,1) ,G (1,t) } et,1 e+r,1 e,1 + δt,1+r e1,1 e1+r,1 e1+r,1 10

+

9

δt,1 e21,1 e1+r,1

+ δt,1 δt,r+1 (−e1,1 e1+r,1 et,1 + et,1 )

+ δt,1+r e1+r,1 e1,1 e1+r,1 , (t,1,1,j)

(1,1,j)

(1+r,j,1+r)

(22)

(t,1)

(t,1,1+r,1)

(1,t)

, G1 , G5 , G10 , R1 and G9 . Two cases need to using G6 () be considered. Case 1. t = 1: the result coincides with G11 . Case 2. t = 1: If t < then t = 1 + r (because ≤ r by the assumption) and the result coincides with (t,) (t,1,,1) gives G12 . If < t then eliminating from (22) the leading of R5 T3 ≡G e,1 e+r,1 et,1 − et,1 + δt,1+r (e1,1 e1+r,1 e1+r,1 + e1+r,1 e1,1 e1+r,1 ) + δt,r+1 (−e1,1 e1+r,1 et,1 + et,1 ).

(23)

Clearly, for (, t) = (1, 1 + r), the result coincides with G13 . For t = 1 + r and = 1, (,1+r) . For t = 1 + r and we eliminate from (23) the leading of G13 and obtain G14 (,t) = 1 (resp. = 1) the result coincides with G14 (resp. 0). Lemma 3.10. The set of all normal forms modulo G ∪ G1 ∪ G2 of nontrivial compositions among elements of G ∪ G1 ∪ G2 includes the set G3 which consists of the elements: ()

G15 = e21,1 e1+r,1 e,1 − e1,1 e,1

(1 < ≤ m; = 1 + r),

()

G16 = e,1 e+r,1 e,1 − 2e1,1 e1+r,1 e,1 + e,1

(1 < ≤ r),

()

G17 = e+r,1 e,1 e+r,1 + 2e1,1 e1+r,1 e+r,1 − e+r,1

(1 < ≤ r).

Proof. We consider the following compositions: (s,1,1,1)

P1 = R6

(,1,j)

P2 = G2

(,j)

P3 = G4

(s+r,1,)

e,1 − es,1 G7

(,j)

e,j − e,1 G4

(1 < ≤ r; 1 < j ≤ n),

(,j,1)

e+r,1 − e+r,j G2

(1 ≤ ≤ m; = 1 + r; 1 < s ≤ r),

(1 < ≤ r; 1 < j ≤ n).

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We eliminate from P1 , P2 and P3 all occurrences of the leading monomials of G ∪ G1 ∪ G2 , and we have P1 = −e1,1 es+r,1 es,1 e,1 − e1,1 e,1 ≡G2 e21,1 e1+r,1 e,1 − e1,1 e,1 , (s)

()

using G11 . For = 1, the result coincides with G15 . For = 1, the result can be (1,1,1) , so reduced further to 0, using the relations G20 (of the next lemma) and G7 we ignore this case. For P2 , we have P2 = −e1,1 e1+r,j e,j + e,1 e+r,1 e,1 − e,1 e1,j e1+r,j + e,1 e1,1 e1+r,1


≡G1 ∪G2 −e1,1 e1+r,1 e,1 + e,1 e+r,1 e,1 − e1,1 e1+r,1 e,1 + e,1 = e,1 e+r,1 e,1 − 2e1,1 e1+r,1 e,1 + e,1 , (1+r,j,)

using G5

(,1,1,j)

, G6

(,1)

()

and G10 . The result coincides with G16 . Finally,

P3 = −e+r,1 e,1 e+r,1 + e1,j e1+r,j e+r,1 − e1,1 e1+r,1 e+r,1 + e+r,j e1,j e1+r,1 ≡G1 −e+r,1 e,1 e+r,1 − e1,1 e1+r,1 e+r,1 + e+r,1 e1,1 e1+r,1 ≡G −e+r,1 e,1 e+r,1 − e1,1 e1+r,1 e+r,1 + e1+r,1 e1,1 e+r,1 + e+r,1 ≡G2 −e+r,1 e,1 e+r,1 − e1,1 e1+r,1 e+r,1 − e1,1 e1+r,1 e+r,1 + e+r,1 , (1+r,j,+r,1)

(+r,j,1)

using G6 , G5 () result coincides with G17 .

(+r,1,1+r,1)

, R1

(1,+r)

and G9

. The (monic) form of the

Lemma 3.11. The self-reduced form G of the set G ∪ elements: (,d,t)

G1

(,d,t)

G2

3 i=1

Gi consists of the

= e+r,d e,t + e1,d e1+r,t

(1 ≤ ≤ r; 1 ≤ d, t ≤ n; d = t),

= e,d e+r,t − e1,d e1+r,t

(1 < ≤ r; 1 ≤ d, t ≤ n; d = t),

(,d)

= e,d e+r,d − e,1 e+r,1 + e1,1 e1+r,1 − e1,d e1+r,d

(1 < ≤ r; 1 < d ≤ n),

G4

(,d)

= e+r,d e,d − e+r,1 e,1 + e1,d e1+r,d − e1,1 e1+r,1

(1 ≤ ≤ r; 1 < d ≤ n),

(,d,i)

= e,d ei,d − e,1 ei,1

G3

G5

(i,d,k,t)

G6

(k,,s)

G7

()

= ei,d ek,t

(1 ≤ , i ≤ m; = i ± r; 1 < d ≤ n),

(1 ≤ i, k ≤ m; i = k ± r; 1 ≤ d, t ≤ n; d = t),

= ek,1 e,1 es,1

(1 ≤ k, , s ≤ m; k = ± r; = s ± r),

G8 = e1,1 e,1 e+r,1 − e21,1 e1+r,1 (,k)

G9

(k,)

G10

(1 < ≤ r),

= e+r,1 e,1 ek,1 + e1,1 e1+r,1 ek,1

(1 < k ≤ m; 1 ≤ ≤ r; k = + r),

= ek,1 e,1 e+r,1 + e1,1 e1+r,1 ek,1 − ek,1 1750093-13

(1 < k ≤ m; 1 ≤ ≤ r; k = + r),

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G11 = e1,1 e+r,1 e,1 + e21,1 e1+r,1 (t,)

G12

(1 < ≤ r),

= et,1 e+r,1 e,1 − e1,1 e1+r,1 et,1 + et,1

(1 < t ≤ m; 1 ≤ ≤ r; t = ),

()

G13 = e+r,1 e,1 e+r,1 + 2e1,1 e1+r,1 e+r,1 − e+r,1 (,t) G14

= e,1 e+r,1 et,1 − e1,1 e1+r,1 et,1

()

G15 = e21,1 e1+r,1 e,1 − e1,1 e,1

(1 < t ≤ m; 1 < ≤ r; t = ),

(1 < ≤ m; = 1 + r),

()

G16 = e,1 e+r,1 e,1 − 2e1,1 e1+r,1 e,1 + e,1 (i,d)

G17

= ei,1 e1,1 e1+r,d + e1,d e1+r,1 ei,1 − ei,d

(d)


G18 = e1,d e1+r,1 e1,1 + e21,1 e1+r,d − e1,d ()

G19 = e+r,1 e,1 e1,1 − e21,1 e1+r,1 + e1,1 G20 = e1,1 e1+r,1 e1,1 +

(1 ≤ ≤ r),

2e21,1 e1+r,1

(1 < ≤ r), (1 < i ≤ m; 1 < d ≤ n),

(1 < d ≤ n), (1 ≤ ≤ r),

− e1,1 ,

(s)

G21 = es,1 es+r,1 e1,1 + e21,1 e1+r,1 − e1,1

(1 < s ≤ r).

3 3 Proof. We eliminate from G ∪ i=1 Gi all occurrences of {LM(u) : u ∈ G ∪ i=1 3 Gi } as a subword of any element of G ∪ i=1 Gi . We note that any element g ∈ 3 3 () i=1 Gi is in normal form modulo G ∪ i=1 Gi \{g} and the relation G13 of the () present lemma is a combination of the relations G13 and G17 of Lemmas 3.9 and 3.10. So we only consider elements of G. For all r + 1 ≤ s < i ≤ m and 1 ≤ j, d ≤ n, we have (i,j,s,d)

R1

= ei,j es−r,j es,d − es,d es−r,j ei,j − ei,d ≡G1 δj,d [ei,d (es−r,1 es,1 + δd,1 (e1,d e1+r,d − e1,1 e1+r,1 )) − (es,1 es−r,1 + δd,1 (−e1,d e1+r,d + e1,1 e1+r,1 ))ei,d ] + δj,d (ei,j e1,j e1+r,d + e1,d e1+r,j ei,j ) − ei,d ,

(s−r,j,d)

(s−r,d,j)

(s−r,d)

(s−r,d)

using G2 , G1 , G3 and G4 (,i) and obtain s − r = in (24) and use G9 (i,1,s,1)

R1

(24)

. Case I. j = d: For d = 1, we set

= ei,1 e,1 e+r,1 − e+r,1 e,1 ei,1 − ei,1 ≡G2 ei,1 e,1 e+r,1 + e1,1 e1+r,1 ei,1 − ei,1 . (i,)

Combining the result with the set {G10 | 1 < i < + r, 1 ≤ ≤ r} ⊂ G2 gives the (i,) (i,d,s−r,1) (i,d,1,1) set {G10 | 1 < i ≤ m, 1 ≤ ≤ r, i = + r}. For d = 1, using G6 , G6 , (s−r,1,i,d) (1+r,1,i,d) and G6 gives G6 (i,d,s,d)

R1

≡G1 ei,d e1,d e1+r,d + e1,d e1+r,d ei,d − ei,d . 1750093-14

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Anti-Jordan triple system (i,d,1)

(1+r,d,i)

Using G5

and G5

(i,d,s,d)

R1

, we get

≡G1 ei,1 e1,1 e1+r,d + e1,d e1+r,1 ei,1 − ei,d . (i,d)

(i,j,1)

The result coincides with G17 . Case II. j = d: Using G5 implies (i,j,s,d)

R1

(i,d)

(s,d,j,i)


in (24)

≡G1 ei,1 e1,1 e1+r,d + e1,d e1+r,1 ei,1 − ei,d .

For d = 1 (resp. d = 1) the result coincides with G17 s ≤ m, r + 1 ≤ s and 1 ≤ j, d ≤ n, we have R2

(1+r,j,i)

and G5

(i,1)

(resp. G10 ). For all 1 ≤ i