The central polynomials for the finite dimensional Grassmann algebras

In this note we describe the central polynomials for the finite dimensional unitary Grassmann algebras Gk over an infinite field F of characteristic ≠2. We exhibit a set of generators of C(Gk), the T-space of the central polynomials of Gk in a free associative F-algebra.

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Date:	2009
Main Authors:	Koshlukov, P., Krasilnikov, A., Elida Alves da Silva
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Language:	English
Published:	Інститут прикладної математики і механіки НАН України 2009
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Cite this:	The central polynomials for the finite dimensional Grassmann algebras / P. Koshlukov, A. Krasilnikov, Elida Alves da Silva // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 3. — С. 69–76. — Бібліогр.: 15 назв. — англ.

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spelling	irk-123456789-1544942019-06-16T01:28:07Z The central polynomials for the finite dimensional Grassmann algebras Koshlukov, P. Krasilnikov, A. Elida Alves da Silva In this note we describe the central polynomials for the finite dimensional unitary Grassmann algebras Gk over an infinite field F of characteristic ≠2. We exhibit a set of generators of C(Gk), the T-space of the central polynomials of Gk in a free associative F-algebra. 2009 Article The central polynomials for the finite dimensional Grassmann algebras / P. Koshlukov, A. Krasilnikov, Elida Alves da Silva // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 3. — С. 69–76. — Бібліогр.: 15 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:16R10. http://dspace.nbuv.gov.ua/handle/123456789/154494 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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language	English
description	In this note we describe the central polynomials for the finite dimensional unitary Grassmann algebras Gk over an infinite field F of characteristic ≠2. We exhibit a set of generators of C(Gk), the T-space of the central polynomials of Gk in a free associative F-algebra.
format	Article
author	Koshlukov, P. Krasilnikov, A. Elida Alves da Silva
spellingShingle	Koshlukov, P. Krasilnikov, A. Elida Alves da Silva The central polynomials for the finite dimensional Grassmann algebras Algebra and Discrete Mathematics
author_facet	Koshlukov, P. Krasilnikov, A. Elida Alves da Silva
author_sort	Koshlukov, P.
title	The central polynomials for the finite dimensional Grassmann algebras
title_short	The central polynomials for the finite dimensional Grassmann algebras
title_full	The central polynomials for the finite dimensional Grassmann algebras
title_fullStr	The central polynomials for the finite dimensional Grassmann algebras
title_full_unstemmed	The central polynomials for the finite dimensional Grassmann algebras
title_sort	central polynomials for the finite dimensional grassmann algebras
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2009
url	http://dspace.nbuv.gov.ua/handle/123456789/154494
citation_txt	The central polynomials for the finite dimensional Grassmann algebras / P. Koshlukov, A. Krasilnikov, Elida Alves da Silva // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 3. — С. 69–76. — Бібліогр.: 15 назв. — англ.
series	Algebra and Discrete Mathematics
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first_indexed	2025-07-14T06:35:04Z
last_indexed	2025-07-14T06:35:04Z
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fulltext	Jo u rn al A lg eb ra D is cr et e M at h . Algebra and Discrete Mathematics RESEARCH ARTICLE Number 3. (2009). pp. 69 – 76 c⃝ Journal “Algebra and Discrete Mathematics” The central polynomials for the finite dimensional Grassmann algebras Plamen Koshlukov, Alexei Krasilnikov and Élida Alves da Silva Communicated by guest editors Abstract. In this note we describe the central polynomials for the finite dimensional unitary Grassmann algebras Gk over an infinite field F of characteristic ∕= 2. We exhibit a set of generators of C(Gk), the T-space of the central polynomials of Gk in a free associative F -algebra. Dedicated to Professor Miguel Ferrero on occasion of his 70-th anniversary Introduction Central polynomials of algebras with polynomial identities are of funda- mental importance in PI-theory. The existence of proper central polyno- mials for the matrix algebras Mn(F ) over a field F was conjectured by Kaplansky, and confirmed by means of direct constructions by Formanek [5] and by Razmyslov [14]. One can find further references about central polynomials of PI algebras in [1], [4] and [8]. However, an explicit description of the vector space of all central polynomials was obtained for very few algebras so far (in the results mentioned above some central polynomials for the corresponding algebras P. Koshlukov was partially supported by Grants 302651/2008-0 CNPq, and 2005/60337-2 FAPESP; A. Krasilnikov was partially supported by CNPq, FAPDF and FINATEC; E. Silva was partially supported by CNPq 2000 Mathematics Subject Classification: 16R10. Key words and phrases: polynomial identities, central polynomials, Grassmann algebra. Jo u rn al A lg eb ra D is cr et e M at h .70 Central polynomials for Grassmann algebras were constructed). The module structure of the centre of the generic matrix algebra of order 2 was given by Formanek [6], and generators for the central polynomials for M2(F ) were exhibited by Okhitin in [13]; both results were obtained assuming the base field F of characteristic 0. For an infinite field F , charF = p ∕= 2, generating sets for the central polynomials for M2(F ) were described in [2]. Very recently in [1] the central polynomials of the infinite dimensional Grassmann algebra G over an infinite field F of characteristic ∕= 2 were described. In fact, this is an almost complete list of known results concerning an explicit description of the central polynomials in a given algebra. In this note we describe the central polynomials of the finite dimen- sional Grassmann algebras Gk over an infinite field F , charF ∕= 2. We exhibit a set of generators of the T-space C(Gk) of the central polynomi- als of Gk. Let us give the precise definitions. Let F be a field and let F1⟨X⟩ be the free unitary associative algebra over F on the free generating set X = {x0, x1, x2, . . .}. A polynomial f(x1, . . . , xn) ∈ F1⟨X⟩ is a polynomial identity in an F -algebra A if f(a1, . . . , an) = 0 for all a1, . . . , an ∈ A. An ideal I of F1⟨X⟩ is called a T-ideal if I is closed under all endomorphisms of F1⟨X⟩. If A is an algebra then its polynomial identities form a T-ideal T (A) in F1⟨X⟩; conversely, for every T-ideal I in F1⟨X⟩ there is an algebra A such that I = T (A), that is, I is the ideal of all polynomial identities satisfied in A. We refer to [3], [4], [10] and [15] for the terminology and basic results concerning PI algebras. A vector subspace V of F1⟨X⟩ is called a T-space if V is closed under all (algebra) endomorphisms of F1⟨X⟩. A set S ⊂ V generates V as a T-space if V is the minimal T-space in F1⟨X⟩ containing S. Therefore V is the span of all polynomials f(g1, . . . , gn) where f ∈ S and gi ∈ F1⟨X⟩. Note that if I is a T-ideal in F1⟨X⟩ then T-spaces and T-ideals can be defined in the quotient algebra F1⟨X⟩/I in a natural way. In recent years T-spaces turned out to be objects of intensive study, see [9] for an account. The polynomial f(x1, . . . , xn) is called a central polynomial for A if f(a1, . . . , an) ∈ Z(A), the centre of A, for every ai ∈ A. The central poly- nomials for a given algebra A form a T-space C(A) in F1⟨X⟩. However, not every T-space can be obtained as the T-space of the central polyno- mials for some algebra. In fact the central polynomials for a given algebra A are closed under multiplication, and so they form a T-subalgebra in F1⟨X⟩. Let V be the vector space over a field F of characteristic ∕= 2, with a countable infinite basis e1, e2, . . . and let Vk denote the subspace of V generated by e1, . . . , ek (k = 2, 3, . . .). Let G and Gk denote the Jo u rn al A lg eb ra D is cr et e M at h .P. Koshlukov, A. Krasilnikov, E. A. Silva 71 unitary Grassmann algebras of V and of Vk respectively. Then as a vector space G has a basis that consists of 1 and of all monomials ei1ei2 . . . eik , i1 < i2 < ⋅ ⋅ ⋅ < ik, k ≥ 1. The multiplication in G is induced by eiej = −ejei for all i and j. The algebra Gk is the subalgebra of G generated by e1, . . . , ek, and dimGk = 2k. Let a, b, c ∈ A, we denote by [a, b] = ab − ba the commutator of a and b, and we set [a, b, c] = [[a, b], c]. Krakowski and Regev [11] described the polynomial identities of G when charF = 0, and several authors described the generators of T (G) in the general case. Let T be the T-ideal in F1⟨X⟩ generated by the triple commutator [x1, x2, x3]. Proposition 1 ([7, 11, 12], see also [3, 4, 8, 10]). Let F be an infinite field of characteristic ∕= 2. Then T (G) = T . The description of the polynomial identities of Gk can be obtained easily from the proof of Proposition 1, see for instance [3, 4] if charF = 0, and [7] if charF ∕= 2. Let T (Gk) be the T-ideal of the polynomial identities of Gk and let Tn be the T-ideal generated by the polynomi- als [x1, x2] . . . [x2n−1, x2n] and [x1, x2, x3]. Proposition 2 ([7]). Let F be an infinite field of characteristic ∕= 2. Then T (Gk) = Tn where n = [k/2] + 1, [a] being the integer part of the rational number a. Very recently the central polynomials for the infinite dimensional Grass- mann algebra G were described in [1]. Let q(x1, x2) = xp−1 1 [x1, x2]x p−1 2 and let, for each s ≥ 1, qs = qs(x1, . . . , x2s) = q(x1, x2)q(x3, x4) . . . q(x2s−1, x2s). Theorem 3 ([1]). Over an infinite field F of characteristic p > 2, the vector space C(G) of the central polynomials of G is generated (as a T- space in F1⟨X⟩) by the polynomial x0[x1, x2, x3] and by the polynomials xp0 , xp0 q1 , xp0 q2 , . . . , xp0 qn , . . . . Proposition 4 ([1]). If charF = 0 then the T-space C(G) is generated by 1, x0[x1, x2, x3] and [x1, x2]. In this note we deal with the central polynomials for the finite dimen- sional Grassmann algebras Gk. Our main results are as follows. Jo u rn al A lg eb ra D is cr et e M at h .72 Central polynomials for Grassmann algebras Theorem 5. Over an infinite field F of a characteristic p > 2 the vector space C(Gk) of the central polynomials of Gk is generated (as a T-space in F1⟨X⟩) by the polynomials x0[x1, x2, x3], x0[x1, x2] . . . [x2n−3, x2n−2] and by the polynomials xp0 , xp0 q1 , xp0 q2 , . . . , xp0 qn−2, n = [k/2] + 1. Proposition 6. If charF = 0 then the T-space C(Gk) is generated by 1, x0[x1, x2, x3], [x1, x2] and x0[x1, x2] . . . [x2n−3, x2n−2] where n = [k2 ] + 1. We deduce Theorem 5 and Proposition 6 from the following proposi- tion of independent interest. Proposition 7. Let F be an infinite field of characteristic ∕= 2. Then, for each k ≥ 2, C(Gk) = C(G) + Tn−1, where n = [k2 ] + 1. 1. Proof of the main results To prove our results we need the following well-known properties of the T-ideal T (see, for instance, [3, 10, 7]). Lemma 8. Let F be a field. For all g, g1, g2, g3, g4 ∈ F1⟨X⟩ we have the following: (i) [g1, g2] + T is central in F1⟨X⟩/T ; (ii) [g1, g2][g3, g4] + T = −[g1, g3][g2, g4] + T ; (iii) [g1, g2][g3, g4] + T = T if gi = gj for some i and j, i ∕= j. Let B be the set of all polynomials in F1⟨X⟩ of the form xn1 i1 xn2 i2 . . . xns is [xj1 , xj2 ] . . . [xj2r−1 , xj2r ] where s, r ≥ 0, i1 < i2 < . . . < is, j1 < j2 < . . . < j2r, nk > 0 for all k. Note that 1 ∈ B because 1 is of the form above for s = r = 0. Let, for each n ≥ 1, Bn be the subset of B consisting of all elements with 0 ≤ r < n, that is, of elements of B whose “commutator part” [xj1 , xj2 ] . . . [xj2r−1 , xj2r ] contains less than n commutators. The next proposition is well-known. It follows immediately, for instance, from [3, Theorem 4.3.11 (i) and the proof of Theorem 5.1.2 (i)]. Jo u rn al A lg eb ra D is cr et e M at h .P. Koshlukov, A. Krasilnikov, E. A. Silva 73 Proposition 9. Let F be an infinite field of characteristic ∕= 2. Then the F -vector space F1⟨X⟩/T has a basis {b + T ∣ b ∈ B} and the vector space F1⟨X⟩/Tn has a basis {b+ Tn ∣ b ∈ Bn}. First we prove Proposition 7. Note that C(G) + Tn−1 ⊆ C(Gk). Indeed, C(G) ⊂ C(Gk) because T ⊂ Tn and C(G)/Tn and C(Gk)/T are the centres of F1⟨X⟩/Tn and of F1⟨X⟩/T , respectively. On the other hand, Tn−1 ⊂ C(Gk) because the elements of Tn−1/Tn are central in F1⟨X⟩/Tn. Indeed, Tn−1/Tn is spanned by elements of the form ℎ+ Tn, where ℎ = g0[g1, g2] . . . [g2n−3, g2n−2] (gi ∈ F1⟨X⟩). Since [g, g′] + T is central in F1⟨X⟩/T for all g, g′, for each t we have [ℎ, xt] + T = [g0, xt][g1, g2] . . . [g2n−3, g2n−2] + T ∈ Tn/T, that is, [ℎ, xt] ∈ Tn. Hence, ℎ+ Tn is central in F1⟨X⟩/Tn and so is each element of Tn−1/Tn. Thus, to prove Proposition 7 it suffices to check that C(Gk) ⊆ C(G) + Tn−1. Let f be an arbitrary element of C(Gk). By Proposition 9, the set {b+T ∣ b ∈ B} is an F -basis of the algebra F1⟨X⟩/T so f + T = ∑ �ib (1) i + ∑ �ib (2) i + T where, for all i, �i, �i ∈ F , b (1) i ∈ Bn−1 and b (2) i ∈ B∖Bn−1. Equivalently, f = ∑ �ib (1) i + ∑ �ib (2) i + f1 where �i, �i, b (1) i and b (2) i are as above and f1 ∈ T . Note that ∑ �ib (2) i ∈ Tn−1 and f1 ∈ T ⊂ Tn−1 so ( ∑ �ib (2) i + f1) ∈ Tn−1. Hence, to prove that f ∈ C(G) + Tn−1 it suffices to check that g = ∑ �ib (1) i ∈ C(G) or, equivalently, that [g, xt] ∈ T for all t. Let b (1) i = xm1 i1 . . . xms is [xj1 , xj2 ] . . . [xj2r−1 , xj2r ]. Then [b (1) i , xt] + T = [xm1 i1 . . . xms is , xt][xj1 , xj2 ] . . . [xj2r−1 , xj2r ] + T. Note that if A is an associative ring then [v1v2 . . . vl, u] = l∑ i=1 v1 . . . vi−1[vi, u]vi+1 . . . vl. Jo u rn al A lg eb ra D is cr et e M at h .74 Central polynomials for Grassmann algebras Also recall that [g, g′] + T is central in F1⟨X⟩/T for all g, g′. Hence we obtain that [b (1) i , xt] + T equals s∑ l=1 ml x m1 i1 . . . xml−1 il . . . xms is [xil , xt][xj1 , xj2 ] . . . [xj2r−1 , xj2r ] + T. Further, it follows from the items ii) and iii) of Lemma 8 that, for all gi ∈ F1⟨X⟩ and for each permutation � on the set {1, 2, . . . , 2u}, [g1, g2] . . . [g2u−1, g2u] + T = ±[g�(1), g�(2)] . . . [g�(2u−1), g�(2u)] + T and [g1, g2] . . . [g2u−1, g2u] + T = T if gi = gj for some i and j, i ∕= j. Therefore we can rewrite [b (1) i , xt] + T as a linear combination of elements of the form x m′ 1 i1 . . . x m′ s is [xj′ 1 , xj′ 2 ] . . . [xj′ 2r+1 , xj′ 2r+2 ] + T, where j′1 < j′2 < . . . < j′2r+2. Since b (1) i ∈ Bn−1, we have r < n − 1 so each element above belongs to Bn. Thus, for each i, [b (1) i , xt] + T = ∑ ijb (3) ij + T, where ij ∈ F , b (3) ij ∈ Bn. It follows that [g, xt] + T = ∑ �i′bi′ + T (1) where �i′ ∈ F , bi′ ∈ Bn for all i′. Note that g ∈ C(Gk). Indeed, as we observed above, Tn−1 ⊂ C(Gk) so ( ∑ �ib (2) i + f1) ∈ C(Gk). Also f ∈ C(Gk) so g = f − ( ∑ �ib (2) i + f1) ∈ C(Gk). Since g ∈ C(Gk), we have [g, xt] + Tn = Tn. On the other hand, (1) implies [g, xt] + Tn = ∑ �i′bi′ + Tn because T ⊂ Tn. It follows that∑ �i′bi′ + Tn = Tn. Since {b+ Tn ∣ b ∈ Bn} is a basis of F1⟨X⟩/Tn over F , we have �i′ = 0 for all i′. Then, by (1), [g, xt] + T = T for all t, that is, g ∈ C(G). Thus, f = g + ( ∑ �ib (2) i + f1) ∈ C(G) + Tn−1, as required. This completes the proof of Proposition 7. Jo u rn al A lg eb ra D is cr et e M at h .P. Koshlukov, A. Krasilnikov, E. A. Silva 75 Now we prove Theorem 5. Recall that charF = p > 2. By Proposi- tion 7, C(Gk) = C(G) + Tn−1, where n = [k2 ] + 1. It can be easily seen that as a T-space Tn−1 is generated by x0[x1, x2, x3] (2) and x0[x1, x2][x3, x4] . . . [x2n−3, x2n−2]. (3) Since, by Theorem 3, the T-space C(G) is generated by (2) and by the set xp0 , xp0 q1 , . . . , xp0 qs , . . . , (4) the T-space C(Gk) = C(G) + Tn−1 is generated by (2), (3) and the set (4). Notice that xp0 qs ∈ Tn−1 for all s ≥ n− 1 because, by Lemma 8, xp0 qs + T = xp0 xp−1 1 [x1, x2]x p−1 2 . . . xp−1 2s−1[x2s−1, x2s]x p−1 2s + T = xp0x p−1 1 xp−1 2 . . . xp−1 2s [x1, x2] . . . [x2s−1, x2s] + T. It follows that C(Gk) is generated as a T-space by the polynomials (2), (3) and xp0 , xp0 q1 , . . . , xp0 qn−2. The proof of Theorem 5 is completed. Finally, we prove Proposition 6. Here we assume charF = 0. By Proposition 7, C(Gk) = C(G) + Tn−1 where n = [k2 ] + 1. By Proposition 4, the T-space C(G) is generated by 1 and by the polynomials (2) and [x1, x2]. Since the T-space Tn−1 is generated by the polynomials (2) and (3), the T-space C(Gk) is generated by 1 and by the polynomials (2), (3) and [x1, x2], as required. Proposition 6 is proved. References [1] A. P. Brandão Jr., P. Koshlukov, A. Krasilnikov, E. A. da Silva, The central polynomials for the Grassmann algebra, to appear in Israel J. Math. [2] J. Colombo, P. Koshlukov, Central polynomials in the matrix algebra of order two, Linear Algebra Appl. 377 (2004), 53–67. [3] V. Drensky, Free algebras and PI-algebras. Graduate course in algebra, Springer, Singapore, 1999. [4] V. Drensky, E. Formanek, Polynomial identity rings. Advanced Courses in Math- ematics. CRM Barcelona. Birkhäuser Verlag, Basel, 2004. [5] E. Formanek, Central polynomials for matrix rings, J. Algebra 23 (1972), 129– 132. [6] E. Formanek, Invariants and the ring of generic matrices, J. Algebra 89 (1984), 178–223. [7] A. Giambruno, P. Koshlukov, On the identities of the Grassmann algebras in characteristic p > 0, Israel J. Math. 122 (2001), 305–316. Jo u rn al A lg eb ra D is cr et e M at h .76 Central polynomials for Grassmann algebras [8] A. Giambruno, M. Zaicev, Polynomial identities and asymptotic methods. Math- ematical Surveys and Monographs, 122. American Mathematical Society, Provi- dence, RI, 2005. [9] A. V. Grishin, V. V. Shchigolev, T -spaces and their applications., J. Math. Sci. (N. Y.) 134 (2006), 1799–1878. [10] A. Kanel-Belov, L. H. Rowen, Computational aspects of polynomial identities. Research Notes in Mathematics, 9. A K Peters, Ltd., Wellesley, MA, 2005. [11] D. Krakowski, A. Regev, The polynomial identities of the Grassmann algebra, Trans. Amer. Math. Soc. 181 (1973), 429–438. [12] V. N. Latyshev, On the choice of basis in a T-ideal (Russian), Sibirsk. Matem. Zh. 4 (1963), 1122–1126. [13] S. V. Okhitin, Central polynomials of an algebra of second-order matrices (Rus- sian), Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1988, no. 4, 61–63; English translation: Moscow Univ. Math. Bulletin 43 (1988), 49–51. [14] Yu. P. Razmyslov, A certain problem of Kaplansky (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 483–501; English translation: Math. USSR-Izv. 7 (1973) 479–496. [15] L.H. Rowen, Polynomial Identities in Ring Theory. Pure and Applied Mathemat- ics, 84, Acad. Press, New York-London, 1980. Contact information P. Koshlukov IMECC, UNICAMP, P.O.Box 6065,13083- 970 Campinas, SP, Brazil E-Mail: plamen@ime.unicamp.br A. Krasilnikov Departamento de Matemática, Universi- dade de Braśılia, 70910-900 Braśılia, DF, Brazil E-Mail: alexei@unb.br E. A. Silva Departamento de Matemática, Universi- dade Federal de Goiás, Campus de Catalão, 75705-220 Catalão, GO, Brazil E-Mail: elida.alves@ig.com.br Received by the editors: 31.08.2009 and in final form 21.09.2009.

The central polynomials for the finite dimensional Grassmann algebras

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