On the Power of Standard Polynomial to M a,b (E)

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1 International Journal of Algebra, Vol. 10, 2016, no. 4, HIKARI Ltd, On the Power of Standard Polynomial to M a,b (E) Fernanda G. de Paula Departamento de Ciências Exatas e Tecnológicas Universidade Estadual de Santa Cruz Ilhéus, BA, Brasil Sérgio M. Alves Departamento de Ciências Exatas e Tecnológicas Universidade Estadual de Santa Cruz Ilhéus, BA, Brasil Copyright c 2016 Sérgio M. Alves and Fernanda G. de Paula. This article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The algebras M a,b (E) appeared in the list of verbally prime algebras. In this paper we study conditions for that the power polynomial of degree n let an polynomial identity for the algebra M a,b (E ). Also, we study power of standard polynomial that are identity for M a,b (E). Mathematics Subject Classification: 16R10, 16R20, 16R40, 15A75 Keywords: PI Theory, Verbally prime algebra 1 Introduction In characteristic zero the algebra M a,b (E) appear in the list of non-trivial verbally-prime algebras. Verbally prime algebras play a prominent role in the PI theory. Recall that an algebra A is verbally prime if its T -ideal is prime in the class of all T -ideals in the free associative algebra. Most of the known results about verbally prime algebras concern the case when these are over a field of characteristic 0. The structure theory of T -ideals developed by Kemer classified the verbally prime algebras over such fields.

2 172 Fernanda G. de Paula and Sérgio M. Alves Two algebras A and B are PI equivalent, A B, if they satisfy the same polynomial identities. Kemer proved that the tensor product of two verbally prime algebras is PI equivalent to another such algebra. This description holds in characteristic 0 and is known as the: Tensor Product Theorem (Kemer): M a,b (E) E M a+b (E), M a,b (E) M c,d (E) M ac+bd,ad+bc (E), and M 1,1 (E) E E. In our previous research, (see [1]; [4]; [5]), we proved that Tensor Product Theorem of Kemer is no longer valid over a field of characteristic positive feature two. Let X = {x 1, x 2,...} be a countable infinite set of symbols (variables) no commuting, and let K X be the algebra free associative algebra with 1, over k. We denote by (1) s n (x 1,..., x n ) = σ S n ( 1) σ x σ(1)... x σ(n), S n is the symmetric group of order n; (2) P n (x) = x n. In the sequel, we shall refer to (1) and (2) as the standard and power polynomial of degree n, respectively. In 2006 (see [6]), Alves and Koshlukov proved that: (Lemma 24): If a b then M a,b (E) satisfies the identity s k 2a for some k > 1 but satisfies neither s 2a nor identities of the form s k m for any k, when m < 2a. As a consequence they obtained that the algebras M a,b (E) E and M a+b (E) are not pi-equivalents. It is worth mentioning that the authors did not exhibited the value of the power k, they only ensured its existence. In this paper we answer the following question: When char K = p > 2 and a b as worth k so that s k 2a is an identity to M a,b (E)? We hope this work contribute to a better understanding about the verbally prime algebras in positive characteristic.

3 On the power of standard polynomial to M a,b (E) Preliminary Notes We recall some of the main definitions and notations that wil be used in what follows. Unless stated otherwise, we consider associative unitary algebras, K is a fixed infinite field with char K = p 2. The algebras, vector spaces and tensor products will be considered over K. We denote by K X the free associative algebra of infinite rank freely generated over K by the set X = {x 1, x 2,...}. We denote by A B the PI-equivalence of the algebras A and B. We refer to the books of Drensky [10] for background information on PI-algebras. Let V be an vector space over K of countable infinite dimension with basis e 1, e 2,..., and denote by V k the subspace spanned by e 1, e 2,..., e k. The Grassmann algebra E(K) of V is the associative algebra with K-basis consisting of 1 and all products of the form {e i1 e i2... e im i 1 < i 2 <... < i m ; m = 1, 2,...} and with multiplication induced by e 2 i = 0 and e i e j = e j e i. Analogously, one defines the non-unitary Grassmann algebra E (K) as the subalgebra of E(K) generated by e i1 e i2... e im, i 1 < i 2 <... < i m, m 1. Denote by E 0 the subspace of E(K) spanned by 1 and all basic elements of the form e i1 e i2... e i2m, m 1, and let E 1 be the subspace spanned by all elements of the form e i1 e i2... e i2m+1, m 0. Then E 0 is the center of E(K), and ab = ba for every a, b E 1. When char K = p = 2, then obviously E(K) and E (K) are commutative and hence they are not very interesting from the PI point of view. Therefore, we restrict our attention of the case p > 2. Let M n (E) the n n matrix algebras over E. Let 0 be the set of all (i, j) such that either 1 i, j a or a + 1 i, j a + b = n, and let 1 be the set of (i, j) with either 1 i a, a+1 j a+b, or 1 j a, a+1 i a+b. Then M a,b (E) consists of the matrices in M n (E) such that the (i, j)-th entry belongs to E β when (i, j) β. (Teorema de Amitsur-Levitzki, [10]) The matrix algebra M n (K) satisfies the standard identity of degree 2n s 2n (x 1,..., x 2n ) = σ S 2n ( 1) σ x σ(1)... x σ(2n). The following theorem was proved in [8, pag. 343]. Theorem 1 Every associative PI algebra over field of characteristic p > 2 satisfies the symmetric polynomial, for some n.

4 174 Fernanda G. de Paula and Sérgio M. Alves The following lemma was proved in [9, Lemma 1.2]. Lemma 2.1 Let char K = p. Then P p (x) T (E ) and P p (x) T (E). 3 Main Result In this section we study condition for that the polynomial P n (x) let an polynomial identity for the algebra M a,b (E ). Also, we study power standard polynomial in related algebras. Let A = (x ij ) M a,b (E ), we have that x ij = y ij E 0, if (i, j) 0 e x ij = z ij E 1 = E 1, if (i, j) 1. Moreover, by Lemma (2.1) and of the definition of the E, we see that y p ij = 0 and z 2 ij = 0. (1) Theorem 2 Let n = (a 2 + b 2 )(p 1) + 2ab + 1. Then (i) P n (x) is polynomial identity for the algebra M a,b (E ); (ii) P n (x) is not polynomial identity for the algebra M a,b (E). Proof. (i) Let A = (x ij ) M a,b (E ) be as above, the entries of A n will be linear combinations of monomials in y ij and z ij each of then of degree n. Now we use the fact that y ij are central in E and z ij anticommute, and write, up to a sign, every such monomial in the form y aij ij. z bij ij (i,j) 0 (i,j) 1 We observe that, if we obtain that with a ij + b ij = n. (i,j) 0 (i,j) 1 a ij p 1 for all (i, j) 0 e b ij 1 for all (i, j) 1 n = a ij + b ij 0 (p 1) + 1 = (a 2 + b 2 )(p 1) + 2ab, (i,j) 0 (i,j) 1 thus, we have that n < (a 2 + b 2 )(p 1) + 2ab + 1,

5 On the power of standard polynomial to M a,b (E) 175 that is one contradiction with the choice of the n = (a 2 + b 2 )(p 1) + 2ab + 1. Now, at least one of the a ij p or b ij 2. By (1) we see that each monomial is vanish. Therefore, P n (x) T (M a,b (E )) when n = (a 2 + b 2 )(p 1) + 2ab + 1. (ii) Since 1 M a,b (E); P n (x) is not an identity of the algebra M a,b (E). Corollary 3.1 The inclusion T (M a,b (E)) T (M a,b (E )) is proper. In particular, the algebras M a,b (E ) and M a,b (E) are not PI equivalent. Proof. Since E E, we see that M a,b (E ) M a,b (E). Thus, we obtain that, T (M a,b (E)) T (M a,b (E )). By Theorem 2, the result follows. Recall that, according to [7, Corollary 11] we see that, the algebras E E and K M 1,1 (E ) are PI equivalent, this is, T (E E) = T (K M 1,1 (E )). Here K M 1,1 (E ) be the algebra obtain of the M 1,1 (E ) for adjunction of the unit. Lemma 3.2 Let s 2 = s 2 (x 1, x 2 ) the standard polynomial of the degree two. Then (i) s 2p+1 2 (x 1, x 2 ) is polynomial identity for the algebra E E; (ii) s p 2(x 1, x 2 ) is polynomial identity for the algebra E; (iii) s 2p+1 2 (x 1, x 2 ) is polynomial identity for the algebra M 1,1 (E). Proof. (i) Since T (E E) = T (K M 1,1 (E )) is sufficient to prove that s 2p+1 2 (x 1, x 2 ) T (K M 1,1 (E )). If a, b K M 1,1 (E ), we obtain that, s 2 (a, b) M 1,1 (E ). Theorem 2, we see that By the s 2p+1 2 (x 1, x 2 ) T (K M 1,1 (E )) = T (E E). (ii) If a, b E, we obtain that, s 2 (a, b) E. By Lemma (2.1), we see that s p 2(x 1, x 2 ) E. (iii) If a, b M 1,1 (E), we obtain that, s 2 (a, b) M 1,1 (E ). By Theorem 2, we see that s 2p+1 2 (x 1, x 2 ) T (M 1,1 (E)).

6 176 Fernanda G. de Paula and Sérgio M. Alves The next result is an generalization of the Lemma 3.2 (iii). Corollary 3.3 Let n = (a 2 + b 2 )(p 1) + 2ab + 1 and a b. Then, ( ) A 0 Proof. Let M a M b = { 0 B observe that (1) M a,b (E) = M a M b M a,b (E ); (2) T (M a M b ) = T (M a (K)). s n 2a(x 1,..., x 2a ) T (M a,b (E)). Using the Amitsur-Levitzik Theorem, we see that A M a (K) and B M b (K)} and s 2a (x 1,..., x 2a ) T (M a (K)) = T (M a M b ). Let A 1,..., A 2a M a,b (E), we write each A i = B i + C i with B i M a M b and C i M a,b (E ). Now, we obtain that By Theorem 2, we have that s 2a (A 1,..., A 2a ) = D M a,b (E ). s n 2a(x 1,..., x 2a ) T (M a,b (E)). References [1] Sergio M. Alves, Fernanda G. de Paula, The Gelfand-Kirillov dimension of M n (E) E in positive characteristic, submitted [2] Sergio M. Alves, K. K. Sartori, V. A. T. Arakawa, The standard identity on M n (E) in characteristic p > 2, International Journal of Algebra, 9 (2015), [3] Sergio M. Alves, The algebras M n,n (E) and M n (E) E in positive characteristic, International Journal of Algebra, 8 (2014), [4] Sergio M. Alves, Fernanda G. de Paula, A conjecture about the Gelfand-Kirillov dimension of the universal algebra of A E in positive characteristic, International Journal of Algebra, 7 (2013),

7 On the power of standard polynomial to M a,b (E) 177 [5] S. M. Alves, F. G. de Paula, M. Fidelis, The Gelfand-Kirillov dimension of the universal algebras of M a,b (E) E in positive characteristic, Rend. Circ. Mat. Palermo, 61 (2011), [6] S.M. Alves, P. Koshlukov, Polynomial Identities of Algebras in Positive Characteristic, J. Algebra, 305 (2006), no. 2, [7] S.S. Azevedo, M. Fidelis and P. Koshlukov, Tensor Product Theorems in positive characteristic, J. Algebra, 276 (2004), no. 2, [8] S.S. Azevedo, M. Fidelis and P. Koshlukov, Graded identities and PI equivalence of algebras in positive characteristic, Commun. Algebra, 33 (2005), no. 4, [9] A. Regev, Grassmann algebras over finite fields, Commun. Algebra, 19 (1991), no. 6, [10] V. Drensky, Free Algebras and PI Algebras: Graduate Course in Algebra, Springer-Verlag, [11] A.R. Kemer, The standard identity in characteristic p: A conjecture of I.B. Volichenko, Israel J. Math., 81 (1993), no. 3, [12] P. Koshlukov and S.S. de Azevedo, Graded identities for T-prime algebras over fields of positive characteristic, Israel J. Math., 128 (2002), no. 1, Received: March 9, 2016; Published: May 10, 2016