LECTURE NOTES ON POLYNOMIAL IDENTITIES

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1 LECTURE NOTES ON POLYNOMIAL IDENTITIES A. S. GORDIENKO 1. Definition of a polynomial identity Let F X be the free associative algebra on the countable set X = {x 1, x 2,...} over a field F, i.e. the algebra of polynomials without a constant term in the noncommuting variables from X. Let A be an associative F -algebra and f = f(x 1,..., x n ) F X. We say that f is a polynomial identity of A if f(a 1,..., a n ) = 0 for all a 1,..., a n A. In this case, we write f 0. The set Id(A) of polynomial identities of A is a T -ideal of F X, i.e. ψ(id(a)) Id(A) for all ψ End(F X ). If Id(A) 0, then A is called a PI-algebra. Remark. All endomorphisms of F X are determined by the images of x i. Hence an ideal of F X is a T -ideal, if and only if it is invariant under all substitutions. Many natural properties of algebras can be described in terms of polynomial identities. E.g., if the identity xy yx 0 holds in an algebra, such algebra is commutative. Analogously, if x 1 x 2... x m 0 holds in an algebra for some m N, such algebra is nilpotent. In this sense, the notion of a polynomial identity is even more important for Lie 1 algebras. 2. Lie algebras A Lie algebra over a field F is a vector space L over F endowed with a bilinear map [, ]: L L L, (a, b) [a, b] where a, b L, such that the following conditions hold: (1) (anticommutativity) [a, a] = 0; (2) (Jacobi s identity) [[a, b], c] + [[b, c], a] + [[c, a], b] = 0 for all a, b, c A. Remark. In order to define the notion of a polynomial identity for nonassociative algebras, one uses the absolutely free nonassociative algebra F {X}, i.e. the algebra of nonassociative and noncommutative polynomials in variables from X. In this language, Lie algebras are those algebras that satisfy polynomial identities x 2 0 and (xy)z + (yz)x + (zx)y 0. Remark. If L is a Lie algebra, then [a, b] + [b, a] = [a + b, a + b] [a, a] [b, b] = 0, i.e. [a, b] = [b, a] for all a, b L. Conversely, if char F 2, then [a, a] = 1 ([a, a] + [a, a]), i.e. 2 [a, b] = [b, a] for all a, b L implies [a, a] = 0 for all a L. Example 1. If A is an associative algebra, then we can define on the same vector space a structure of the Lie algebra A ( ) with the commutator [a, b] := ab ba for all a, b A. Example 2. Let V be a vector space. Consider the algebra End F (V ) of all linear operators on V. The Lie algebra End F (V ) ( ) is denoted by gl(v ). Analogously, if M n (F ) is the algebra of n n matrices over a field F, the Lie algebra M n (F ) ( ) is denoted by gl n (F ). Example 3. The subspace sl n (F ) of all matrices from gl n (F ) with zero trace is a Lie subalgebra, i.e. the subspace closed under the commutator. 1 Marius Sophus Lie ( ) was a Norwegian mathematician. 1

2 2 A. S. GORDIENKO Example 4. The subspace o n (F ) of all matrices from gl n (F ) that are skew-symmetric is again a Lie subalgebra. Example 5. Let A be an algebra (not necessarily associative). A map δ gl(a) is a derivation of A if δ(ab) = δ(a)b + aδ(b) for all a, b A. All derivations of A form a Lie subalgebra in the Lie algebra gl(a). This subalgebra is denoted by Der(A). A subspace I of a Lie algebra L is an ideal if [a, b] I for all a L, b I. The ideal L := [L, L] is called the derived subalgebra. Now we can define the derived series L =: L (0) L (1) L (1) L (2)... where L (k) := (L (k 1) ) for k N. A Lie algebra is solvable if L (k) = 0 for some k N. Solvable Lie algebras play a very important in the Lie theory. For example, the following theorem holds (the proof can be found e.g. in [14, Section 4.1]): Theorem 1 (Lie). Let V be a finite dimensional vector space over an algebraically closed field F of characteristic 0 and let L gl(v ) be a solvable subalgebra. Then there exists a nonzero vector v V such that for every a L there exists α F such that av = αv, i.e. v is a common eigenvector for L. In order to describe polynomial identities in Lie algebras, one has to use the free Lie algebra L(X) := F {X}/I where I is the T -ideal of F {X} generated by x 2 and (xy)z +(yz)x+(zx)y. We define [f + I, g + I] := fg + I for all f, g F {X}/I. It is clear that a Lie algebra L is solvable if and only it satisfies a special polynomial identity. 3. Kurosh problem Another important reason to study polynomial identities is that some well known have one answer in the class of all algebras and the opposite in the case of PI-algebras. One of such problems is the Kurosh 2 problem. Let A be an algebra over a field F. An element a A is algebraic over F if there exists a polynomial f F [x]\{0} such that f(a) = 0. Problem (A. G. Kurosh). Does there exist an infinite dimensional finitely generated algebra A over a field F such that every element a A is algebraic over F? It turns out that such algebra cannot be PI (see [13, Theorem 6.4.3]). However there exist such non-pi-algebra ([13, Theorem 8.1.3]). 4. Basis for polynomial identities An important problem is to find a basis of polynomial identities for a given algebra A, i.e. such polynomial identities that generate Id(A) as a T -ideal. For many algebras A, this problem is very hard. Problem. Let M n (F ) be the algebra of n n matrices over a field F. Find the generators of Id(M n (F )) as an T -ideal. This problem is open for n 3. 2 Alexander Gennadyevich Kurosh ( ) was a Russian mathematician.

3 LECTURE NOTES ON POLYNOMIAL IDENTITIES 3 5. Linearization process Before we provide an example of an algebra where we calculate the basis of polynomial identities, we make several important observations. First, we define the multidegree (k 1, k 2,..., k s,...) of a monomial x i1 x i2... x in F X where k j is the multiplicity of j in (i 1, i 2,..., i n ). We say that a polynomial f F X is multihomogeneous if all the monomials that occur in f with nonzero coefficients, have the same multidegree. If all k i are 1 or 0, then f is multilinear. Theorem 2. Let F be an infinite field. Then all T -ideals of F X are generated as T -ideals by multihomogeneous polynomials. Proof. Let I be a T -ideal in F X and f I. Fix some variable x i. Let f = f 0 + f f t where in all the monomials of f l the variable x i occurs in degree l. Then f α := f(x 1,..., x i 1, αx i, x i+1,..., x s ) I for any α F. Note that f α := f 0 + αf α t f t. Recall that the field F is infinite. Therefore, we can use Vandermonde arguments and show that f j I for all 0 j t. Changing i, we prove that every multihomogeneous component of f belongs to I. Therefore, I is generated by multihomogeneous components of its polynomials. Theorem 3. Let F be a field of characteristic 0. Then all T -ideals of F X are generated by multilinear polynomials. Proof. We have already proved that every T -ideal I of F X is generated as T -ideal by multihomogeneous polynomials. Let f be one of such polynomials. Fix some variable x i. Suppose f is of degree t > 1 in x i. Then f α := f(x 1,..., x i 1, x i +y i, x i+1,..., x s ) I. (Here we denote by y i one of the variables that do not occur in f.) Then f α = f 0 + f f t where all f i are multihomogeneous and each f j is of degree j in x i and (t j) in y j. By the previous theorem, all f j I. Moreover, if we substitute x i for y i in f j, we get ( t j) f. Since the characteristic of the base field is 0, we can divide by ( t j) 0 and every fj generates the same T -ideal as f. Now we notice that the degree of f j in x i and y i is less that t for 1 j < t. Hence we can reduce degree and finally get multilinear polynomials. Corollary. If A is an algebra over a field F of characteristic 0, then all polynomial identities of A a generated by its multilinear polynomial identities. Note that this fails in the characteristic p > 0. Let Z p be a field of p elements. Theorem 4. The polynomial identity x p x 0 of the algebra Z p is not a consequence of multilinear polynomial identities of Z p. This follows from the lemma below since any field of characteristic p with more than p elements does not satisfy x p x 0. Lemma 1. Let F be a field of characteristic p. Then any multilinear polynomial identity of F as an algebra over Z p is a consequence of xy yx 0. Proof. Any multilinear polynomial f modulo xy yx 0 is equivalent to αx 1... x n for some α Z p. If f Id(F ), then α = 0, and f is a consequence of xy yx 0. Exercise 1. Using Vandermonde arguments, show that Id(Z p ) is generated as a T -ideal by x p x 0 and xy yx Upper triangular 2 2 matrices Let P n F X be the subspace of all multilinear polynomials in the variables x 1, x 2,..., x n, n N.

4 4 A. S. GORDIENKO Theorem 5. Let UT 2 (F ) be the algebra of upper triangular 2 2 matrices over a field F of characteristic 0. Then all polynomial identities of UT 2 (F ) are consequences of the identity (Here [x, y] := xy yx.) [x 1, x 2 ][x 3, x 4 ] 0. (1) Proof. Denote by e ij the matrix units of M n (F ), i.e. matrices with 1 on the intersection of the ith row and jth column and 0 in all the other cells. Then UT 2 (F ) = e 11, e 12, e 22 F, [UT 2 (F ), UT 2 (F )] = F e 12, [UT 2 (F ), UT 2 (F )][UT 2 (F ), UT 2 (F )] = 0, and (1) indeed holds in UT 2 (F ). Note that Theorem 3 implies that Id(UT 2 (F )) is generated as a T -ideal by multilinear polynomials. Therefore, it is sufficient to prove that all multilinear polynomial identities of UT 2 (F ) are consequences of (1). First we notice that [x, y]w[z, t] = [x, y][w, [z, t]]+[x, y][z, t]w belongs to the T -ideal. Consider an arbitrary monomial x i1 x i2... x in P n. We find the first inversion of indexes i j > i j+1 (where i 1 < i 2 <... < i j 1 ) and rewrite x ij x ij+1 = [x ij, x ij+1 ] + x ij+1 x ij. Using [x, y]w[z, t] 0, we get x i1 x i2... x ij 1 [x ij, x ij+1 ]x ij+2... x in x i1 x i2... x ij 1 [x ij, x ij+1 ]x i j+2... x i n modulo (1) where {i j+2,..., i n } = {i j+2,..., i n} however i j+2 < i j+3 <... < i n. The item x i1 x i2... x ij 1 x ij+1 x ij x ij+2 x ij+3... x in is treated similarly. Finally, we prove that P n is a linear span of polynomials x 1... x n, x i1 x i2... x ij 1 [x ij, x ij+1 ]x ij+2... x in where i 1 <... < i j, i j+1 > i j, i j+2 <... < i n, (2) and consequences of (1). In order to prove that all the polynomial identities of UT 2 (F ) are consequences of (1), it is sufficient to show that if some linear combination f of x 1 x 2... x n and x i1 x i2... x ij 1 [x ij, x ij+1 ]x ij+2... x in is a polynomial identity, then all the coefficients are zero. In order to check that the coefficient of x 1 x 2... x n is zero, it is sufficient to substitute x 1 = x 2 =... = x n = e 11 + e 22 since e 11 + e 22 commutes with all elements of UT 2 (F ) and all the polynomials (2) except x 1 x 2... x n vanish since they contain a commutator. Choose x i1 x i2... x ij 1 [x ij, x ij+1 ]x ij+2... x in with the maximal j such that the coefficient is nonzero. Then we substitute x i1 = x i2 =... = x ij = e 11, x ij+2 = x ij+3 =... = x in = e 11 + e 22, x ij+1 = e 12. Suppose g = x k1 x k2... x kl 1 [x kl, x kl+1 ]x kl+2... x kn does not vanish under this substitution. Then {i 1, i 2,..., i j+1 } {k 1, k 2,..., k l+1 } and i j+1 {k l, k l+1 }. However, if l > j, then the coefficient of g is zero. Suppose l = j. Then {i 1, i 2,..., i j+1 } = {k 1, k 2,..., k l+1 }, {i j+2, i j+3,..., i n } = {k l+2, k l+3,..., k n }, and i t = k t for all j + 2 t n. Note that i j = max{i 1, i 2,..., i j+1 } = max{k 1, k 2,..., k l+1 } = k l. Since i j+1 {k l, k l+1 }, i 1 <... < i j, k 1 <... < k j, we get i t = k t for all 1 t n. Thus g = x i1 x i2... x ij 1 [x ij, x ij+1 ]x ij+2... x in, and its coefficient must be zero, since f Id(UT 2 (F )). We get a contradiction. Hence the polynomials (2) are linearly independent modulo Id(UT 2 (F )), and Id(UT 2 (F )) is generated by (1).

5 LECTURE NOTES ON POLYNOMIAL IDENTITIES 5 Note that we have simultaneously calculated the numbers n 1 P n dim P n Id(UT 2 (F )) = 1 + ( ) n n 1 ( ) n n 1 ( ) n j = 1 + (j + 1) = j + 1 j + 1 j + 1 j=1 that equal the number of polynomials (2). j=1 j=1 n 1 ( ) n n (2 n 1 n) = j j=1 1 + n(2 n 1 1) (2 n 1 n) = (n 2)2 n Exercise 2. Show that polynomial identities of t 2 (F ) := UT 2 (F ) ( ) where F is a field of characteristic 0 are generated by [[x 1, x 2 ], [x 3, x 4 ]] Codimensions of polynomial identities If A is an algebra, then the numbers c n (A) := dim Pn P n Id(A) are called codimensions. As we have already seen, these numbers naturally come into play when we calculate the basis for polynomial identities in concrete examples. As we will see later, the asymptotic behaviour of c n (A) is tightly connected with the structure of A. Another reason to study codimensions is that the problem of classification of all varieties of algebras, i.e. classes of algebras that satisfy all polynomial identities from a fixed set, with respect to such sets, is wild and codimensions are numeric characteristics of polynomial identities that provide a convenient P tool to perform a coarser classification. In addition, n P n Id(A) can be identified with the spaces of all multilinear maps A... A A that can be presented by multilinear polynomials. In the 1980 s, the following conjectures on asymptotic behaviour of codimensions were made: Conjecture (S. A. Amitsur 3 ). Let A be an associative PI-algebra over a field F of characteristic 0. Then there exists PIexp(A) := lim n n c n (A) Z +. The number PIexp(A) (if it exists) is called the PI-exponent of A. Amitsur s conjecture was refined by A. Regev: Conjecture (A. Regev 4 ). Let A be an associative PI-algebra over a field F of characteristic 0. Then either there exists numbers d N, r Z, and C > 0 such that c n (A) Cn r 2 d n as n, or c n (A) = 0 for all sufficiently large n. (We write f g if lim f g = 1.) Of course, one can consider analogs of the conjectures for Lie and other nonassociative algebras. Amitsur s conjecture was proved in 1999 for all associative PI-algebras by A. Giambruno and M. V. Zaicev [6, Theorem 6.5.2]. The analog of Amitsur s conjecture for finite dimensional Lie algebras was proved in 2002 by M. V. Zaicev [17]. In addition, in 2011 A. Giambruno, I. P. Shestakov, and M. V. Zaicev proved the analog of Amitsur s conjecture for finite dimensional Jordan and alternative algebras [7]. Furthermore, for all the cases the authors provide an explicit formula for PIexp(A), which we discuss in the next sections. I.B. Volichenko [16] gave an example of an infinite dimensional Lie algebra L with a nontrivial polynomial identity for which the growth of codimensions c n (L) of ordinary polynomial identities is overexponential. M.V. Zaicev and S.P. Mishchenko [15, 18] gave an example of an infinite dimensional Lie algebra L with a nontrivial polynomial identity such that there exists fractional PIexp(L). 3 Shimshon Avraham Amitsur (born Kaplan, ) was an Israeli mathematician. 4 Amitai Regev (born 1940) is an Israeli mathematician.

6 6 A. S. GORDIENKO Regev s conjecture was proved in 1992 by V.S. Drensky [4] for associative and Lie algebras of polynomial growth, i.e. for such algebras A that c n (A) is bounded by a polynomial in n, and in 2008 by A. Regev and A. Berele for associative algebras with unity [2, 3]. Regev s conjecture for finite-dimensional Lie algebras and for non-unital associative algebras is an open problem. Moreover, both conjectures for char F = p > 0 are still open problems. In addition, an open problem is to prove that for every Lie algebra with exponentially bounded codimensions there exists a PI-exponent (that, as we have already mentioned, can be a non-integer). Also one could try to find a criterion for a Lie algebra to have exponentially bounded codimensions. 8. Wedderburn Mal cev theorem and a formula for the PI-exponent In order to present the formula for the PI-exponent, we need the following useful result from the structure theory of associative algebras: Theorem 6 (J. Wedderburn 5 A. I. Mal cev 6 ). Let A be a finite dimensional associative algebra over a field F of characteristic 0. Then A = B + J(A) (direct sum of subspaces) where J(A) is the Jacobson radical of A and B is a maximal semisimple subalgebra of A. Moreover, if A contains unity 1 A and B is another maximal semisimple subalgebra of A such that A = B + J(A) (direct sum of subspaces), then B = (1 A + j)b(1 A + j) 1 for some j J. Also we recall that by the Wedderburn ( Artin) theorem, B = B 1... B s (direct sum of ideals) where B i are simple algebras, i.e. matrix algebras over division algebras over F. In particular, if F is algebraically closed, B i are matrix algebras over F. Now we can present a formula for PIexp(A). Theorem 7 (A. Giambruno M. V. Zaicev, see [6, Section 6.2]). Let A be a finite dimensional associative algebra over a field F of characteristic 0. Let A = B + J(A) (direct sum of subspaces) be a Wedderburn decomposition of A where B = B 1... B s (direct sum of ideals) and B i are simple algebras. Then either A is nilpotent and there exits n 0 N such that c n (A) = 0 for all n n 0 or for some r 1, r 2 R, C 1, C 2 > 0, where C 1 n r 1 d n c n (A) C 2 n r 2 d n for all n N d := max dim(b i1... B ik B i1 J(A)B i2... J(A)B ik 0, 1 i l s, 0 k s). In the next section we discuss the main ideas of the proof of the theorem. This result can be generalized for graded, G-, H- and differential polynomial identities [11, 12]. There exists a formula for the PI-exponent of Lie algebras too [17] (see also [8, 9, 10]). However this formula is more complicated. One of the reasons is that the analog of the Wedderburn decomposition for Lie algebras, which is called the Levi decomposition, involves the solvable radical instead of the nilpotent one. 9. Representations of the symmetric group and S n -cocharacters One of the main tools in the investigation of polynomial identities is provided by the representation theory of symmetric groups. The symmetric group S n acts on the space by permuting the variables. P n P n Id(A) 5 Joseph Henry Maclagan Wedderburn ( ) was a Scottish mathematician. 6 Anatoly Ivanovich Mal cev (or Maltsev, ) was a Russian mathematician.

7 LECTURE NOTES ON POLYNOMIAL IDENTITIES 7 Irreducible F S n -modules are described by partitions λ = (λ 1,..., λ s ) of the number n where λ 1 + λ λ s = n and λ 1 λ 2... λ s, s N. For such partitions we use the notation λ n. Each partition λ n can be graphically depicted by the Young diagram D λ = (λ 1 ) (λ 2 ) (λ s )... where the ith row contains exactly λ i boxes. In order to define the irreducible F S n -module M(λ) that corresponds to λ n, we fill in some way the Young diagram D λ with numbers from 1 to n to get a Young tableau. For example, T (4,3,1) = is one of Young diagrams that correspond to λ = (4, 3, 1), 7 n = 8. For each T λ we define the subgroup C Tλ that preserves the set of numbers of each column and the subgroup R Tλ that preserves the set of numbers of each row. The elements e Tλ := a Tλ b Tλ and e T λ := b Tλ a Tλ of the group algebra F S n where a Tλ := π R Tλ π and b Tλ := σ C Tλ (sign σ)σ are called the Young symmetrizers corresponding to the Young tableau T λ. Then M(λ) = F Se Tλ = F Se Tλ is the irreducible F S n -module corresponding to a partition λ n. Its dimension dim M(λ) can be calculated by the hook formula dim M(λ) = i,j n! ij is the length of the hook of D ij λ with the edge (i, j). Example 6. Here we calculate dim M(4, 3, 2)., h 11 = 6,, h 12 = 5,, h 13 = 3,, h 14 = 1,, h 21 = 4,, h 22 = 3,, h 23 = 1, and dim M(4, 3, 2) = 9! = = 168. is called the nth cocharacter of polyno- The character χ n (A) of the F S n -module mial identities of A. We can rewrite it as the sum, h 31 = 2, P n P n Id(A) χ n (A) = λ n m(a, λ)χ(λ), h 32 = 1, of irreducible characters χ(λ). In order to prove Theorem 7, one shows that there exists a number m N such that if λ n and λ d+1 > m, i.e. D λ has too many boxes in the (d + 1)th row, then m(λ) = 0, i.e. such diagram does not occur in the decomposition. By the hook formula, this implies the upper bound for c n (A). Conversely, it is possible to prove that for every n there exists λ n with sufficiently great λ d. By the hook formula, this implies the lower bound. We refer the reader to [1, 5, 6] for additional information on S n -representations and their applications to polynomial identities.

8 8 A. S. GORDIENKO References [1] Bakhturin, Yu. A. Identical relations in Lie algebras. VNU Science Press, Utrecht, [2] Berele, A. Properties of hook Schur functions with applications to p.i. algebras, Advances in Applied Math., 41:1 (2008), [3] Berele, A., Regev, A. Asymptotic behaviour of codimensions of p.i. algebras satisfying Capelli identities, Trans. Amer. Math. Soc., 360 (2008), [4] Drensky, V. S. Relations for the cocharacter sequences of T-ideals, Contemp. Math., Proceedings of the International Conference on Algebra Dedicated to the Memory of A.I. Mal cev, part 2 (Novosibirsk, 1989), 131 (1992), Amer. Math. Soc., Providence, RI, [5] Drensky, V. S. Free algebras and PI-algebras: graduate course in algebra. Singapore, Springer-Verlag, [6] Giambruno, A., Zaicev, M. V. Polynomial identities and asymptotic methods. AMS Mathematical Surveys and Monographs, 122, Providence, R.I., [7] Giambruno, A., Shestakov, I.P., Zaicev, M. V. Finite-dimensional non-associative algebras and codimension growth. Adv. Appl. Math. 47 (2011), [8] Gordienko, A. S. Amitsur s conjecture for polynomial H-identities of H-module Lie algebras. Tran. Amer. Math. Soc. (to appear). arxiv: [math.ra] 6 Jul 2012 [9] Gordienko, A. S., Kochetov, M. V. Derivations, gradings, actions of algebraic groups, and codimension growth of polynomial identities. Algebras and Representation Theory, 17:2 (2014), arxiv: [math.ra] 9 Oct 2012 [10] Gordienko, A. S. On a formula for the PI-exponent of Lie algebras. J. Alg. Appl., 13:1 (2013), arxiv: [math.ra] 6 Nov 2012 [11] Gordienko, A. S. Asymptotics of H-identities for associative algebras with an H-invariant radical. J. Algebra, 393 (2013), arxiv: [math.ra] 6 Dec 2012 [12] Gordienko, A. S. Co-stability of radicals and its applications to PI-theory. Algebra Colloqium (to appear). arxiv: [math.ra] 11 Jan 2013 [13] Herstein, I. N. Noncommutative rings. Carus Mathematical Monographs, 15, MAA, [14] Humphreys, J. E. Introduction to Lie algebras and representation theory. New-York, Springer-Verlag, [15] Mishchenko, S.P., Verevkin, A.B., Zaitsev, M.V. A sufficient condition for coincidence of lower and upper exponents of the variety of linear algebras. Mosc. Univ. Math. Bull., 66:2 (2011), [16] Volichenko, I. B. Varieties of Lie algebras with identity [[X 1, X 2, X 3 ], [X 4, X 5, X 6 ]] = 0 over a field of characteristic zero. (Russian) Sibirsk. Mat. Zh. 25:3 (1984), [17] Zaitsev, M. V. Integrality of exponents of growth of identities of finite-dimensional Lie algebras. Izv. Math., 66 (2002), [18] Zaicev, M. V., Mishchenko, S. P. An example of a variety of Lie algebras with a fractional exponent. J. Math. Sci. (New York), 93:6 (1999), Vrije Universiteit Brussel, Belgium address: alexey.gordienko@vub.ac.be

Classification of algebras with minimal quadratic growth of identities

São Paulo Journal of Mathematical Sciences 1, 1 (2007), 97 109 Classification of algebras with minimal quadratic growth of identities A. C. Vieira Departamento de Matemática, Instituto de Ciências Exatas,