II.2.a Finite basing of sets of identities. II.2.b Commutator identities. C. Detailed description of the proposed research. We follow the outline of the previous part.
Detailed description of the research program Polynomial equations and identities in groups, associative algebras and Lie algebras Application No. 1207/12, PIs: A. Kanel-Belov, B. Kunyavskii, L. Rowen, U. Vishne
A. Scientific background The proposal has at least two starting points, going back to works both of Armand Borel and of Irving Kaplansky. Borel studied the question of dominance of the word map w : Gd → G on a group G, which is defined by evaluating a word w(x1 , . . . xd ) on a d-tuple of elements of G. In [Bo] he proved that such a map is dominant whenever G is a connected semisimple linear algebraic group and w ̸= 1 is an arbitrary word. In naive terms, the word equation w(x1 , . . . , xd ) = g in a simple matrix group has a solution provided the right-hand side is “generic”. It is worth mentioning recent activity related to solving word equations in finite simple groups. Many fascinating results have been obtained here, culminating in solving a long-standing problem of Ore for commutators [LOST]. Much activity over the past years was related to similar problems for power maps [MZ], [SW] and more general word maps; an excellent survey of this area can be found, e.g., in [Sh1] and more recent publications [LaS], [Sh2], [LaST]. Profinite analogues are not less interesting, see the recent monograph [Se] for a survey. A profinite version of Borel’s theorem [NS] relies on some deep group-theoretic considerations, including the solution of the restricted Burnside problem [Ze1], [Ze2] and the classification of finite simple groups. Concurrently, people working with associative algebras have been studying similar questions (attributed to Kaplansky) concerning the possible images of a polynomial map on a matrix algebra R = Mn (k). It follows easily from [Her] that the additive subgroup generated by this image must either contain the additive commutator group [R, R] or be contained in the center of R. However, apart from the discovery of central polynomials [Fo1], [Ra1], not much more was known until recently. Apparently, the results for associative algebras differ from those for groups, essentially because of the existence of nontrivial identities for simple matrix algebras, which cannot occur for simple matrix groups in light of Tits’ alternative. This observation gives a hint to the existence of strong ties between equations and identities in matrix algebras, which we intend to pursue in detail. The first part of the project is focused around these interrelations, with emphasis on the “intermediate” case of Lie algebras (being of its own interest) which may be viewed as a “bridge” between the theories of Borel and Kaplansky. We are planning a comprehensive investigation encompassing algebraic, geometric, and arithmetic properties of polynomial equations in associative algebras. The second part of this proposal is devoted to a more thorough investigation of the structure of identities in various classes of algebras, focusing on problems of Specht type, by which we mean whether every T-ideal of the free algebra is finitely generated, or, perhaps more intuitively, if every set of identities is a consequence of a finite set of identities. Kemer’s celebrated solution [Ke2], for associative algebras over a field of characteristic 0, involved two major breakthroughs:
• A reduction to the affine case (i.e., finitely generated over a field) using the so-called Grassmann envelope; • the theorem that every affine (associative) PI-algebra satisfies the same identities as a finite dimensional algebra. The first step fails in nonzero characteristic, but Kemer [Ke1] showed that the second step carries over to algebras over arbitrary infinite fields, and Belov (in his second dissertation in 2002) extended this to algebras over arbitrary commutative Noetherian rings, using the new technique of full quivers of representations [BeRV3, BeRV4]. Aljadeff and Belov [AB] extended Kemer’s result to graded identities of associative algebras, and one can also ask Specht-type questions for other classes of algebras and identities along these lines. Zelmanov discovered a relationship between identities of Lie algebras and group prop-identities, described in Part C. More recently, he announced the existence of a finite base for systems of pro-p-identities in this situation. Another intriguing application of (nonassociative) polynomial identities is to the famous Jacobian Conjecture and its analogs in various algebraic varieties. Recall that the Jacobian Conjecture in dimension n ≥ 1 asserts that for any field k of characteristic zero, any polynomial endomorphism ϕ of the n-dimensional affine space Spec k[x1 , . . . , xn ] over k, with Jacobian 1, is an automorphism. This has been tied to the Dixmier Conjecture (that every endomorphism of the Weyl algebra is an automorphism), and Zariski’s Cancellation Conjecture (suitably formulated). These questions can be formulated more generally in the theory of universal algebras for algebras of arbitrary signature linked to polynomial identities, as described below. B. Research objectives and expected significance In the proposed research we shall focus on the following major problems. I Fine structure of the image of a polynomial map I.1 Dominance of polynomial maps that are not identities: I.1.a Associative algebras: I.1.a.(1) Description of the image of a polynomial map Mn (k)d → Mn (k); I.1.a.(2) Criteria for dominance of polynomial maps Mn (k)d → Mn (k)m , m > 1; I.1.b Lie algebras: I.1.b.(1) Dominance of polynomial maps Ld → L of Chevalley algebras; I.1.b.(2) Dominance of polynomial maps of Lie algebras of Cartan type. I.1.b.(3) Criteria for dominance of polynomial maps Ld → Lm , m > 1; I.2 Surjectivity: I.2.a Surjectivity of multilinear polynomial maps. I.2.b Surjectivity of Engel maps on Lie algebras of Cartan type. I.3 Generalizations of Makar-Limanov’s results: I.3.a Construction of algebraically closed skew fields in characteristic > 0. I.3.b Freiheitsatz for associative algebras in positive characteristic. I.4 Arithmetic. I.4.a When do polynomial matrix equations over integers have many solutions modulo (almost) all primes?
I.4.b When are such solutions (almost) equidistributed? I.5 Applications to automorphisms of algebras and the Jacobian Conjecture II Identities in algebraic structures II.1 Minimal identities of algebras II.1.a Associative algebras with involution II.1.b Pro-finite identities II.2 Lattices of identities II.2.a Finite basing of sets of identities II.2.b Commutator identities C. Detailed description of the proposed research We follow the outline of the previous part. I. Fine structure of the image of a polynomial map. I.1 Dominance: I.1.a Associative algebras: Question 1. What is the image V := f (Mn (k)) of the polynomial map of matrix algebras P : Mn (k)d → Mn (k) induced by an associative noncommutative polynomial f in d variables? We want to study V in detail, focusing on specific types of elements (semisimple, unipotent, etc.). Working hypothesis: We expect the answer to be that V is Zariski dense in Mn (k) under suitable conditions on f . Methods and preliminary results: V clearly could be 0 (when f is a polynomial identity, otherwise called PI), the field of scalars k (when f is a multilinear central polynomial), the subspace [Mn (k), Mn (k)] of trace zero matrices (when f = x1 x2 − x2 x1 ), or all of Mn (k) (e.g. when f = x1 ). We ask what other possibilities exist. As indicated earlier, the problem becomes difficult since V need not be a vector space, although it is a cone invariant under matrix conjugation. One can proceed by considering specific kinds of polynomials whose images satisfy additivity properties. For example, Belov, Malev, and Rowen [BeMR] indeed proved that these are the only possibilities for n = 2 when f is multilinear. The main ideas in the proof involve some basic algebraic geometry and calculating the ratio of the two eigenvalues of a 2 × 2 matrix. However, this approach does not suffice to resolve the problem for n ≥ 3. L’vov conjectured that in general V must be a vector space when f is multilinear (i.e., linear in each variable), but we suspect there are counterexamples even for n = 3. For more general classes of polynomials such as homogeneous polynomials, we have counterexamples even for n = 2. More in line with the geometric methods is the following question: Question 2. When non-scalar, is V dense (with respect to Zariski topology) in the subspace sln (k) of trace 0 matrices? One basic lemma in [BeMR], really a corollary of a theorem of Amitsur, is that the generic matrix algebra with characteristic coefficients adjoined
does not contain zero divisors. This enables us to study eigenvalues of elements of V . The following result of [BeMR, Theorem 1] is sharp for n = 2: Theorem 1. If f is a homogeneous polynomial evaluated on the algebra M2 (k), then V is either {0}, k, the set of all non-nilpotent matrices having trace zero, sl2 (k), or a dense subset of M2 (k). The method of comparing eigenvalues begins to break down for larger n, but we have managed to prove the following result for n = 3 in characteristic ̸= 2, 3. Theorem 2. Let P be a homogeneous (non-central) polynomial on M3 (k), whose image V is contained in sl3 (k). Then V is either dense in sl3 (k), or is the set of matrices with eigenvalues (c, cε, cε2 ), where ε is a cube root of 1. Let us show the existence of a unipotent element in the image V of a multilinear polynomial P in the space of 2 × 2 matrices. Thus we assume on the contrary that V has no unipotent elements. First we assume that P has a noncentral value P (a1 , . . . , an ) with nonzero trace. Fix all variables except x1 . Then Pa2 ,...,an (x1 ) is a function linear in x1 , meeting the discriminant surface at points of total multiplicity 2. By assumption, the value at one point, say x01 , is a nonzero scalar, and at the other, x11 , is nilpotent. At the point x01 + x11 we obtain a matrix proportional to a unipotent one with some coefficient λ ̸= 0, and are done. Thus we may assume that any element of V either is scalar or has zero trace. While moving any coordinate we observe coincidence of the intersection points with the discriminant surface, i.e. tangency to the discriminant surface. Taking the square root of the discriminant (λ1 − λ2 )2 , one can construct zero divisors to the generic algebra with characteristic coefficients, again contradicting our corollary to Amitsur’s theorem. I.1.b Lie algebras: We are also interested in the following analog of the Borel dominance theorem for semisimple Lie algebras: Question 3. Let g be a semisimple Lie algebra over an algebraically closed field k, and let P (X1 , . . . , Xd ) be an element of the free Lie k-algebra Ld over k, which is not an identity of g. Is the map P : gd → g necessarily dominant? Working hypothesis. We believe that the question can be answered affirmatively. Methods and preliminary results: The cases of classical Lie algebras (i.e. the Lie algebras of semisimple algebraic groups, which we call Chevalley algebras) and Lie algebras of Cartan type (in positive characteristics) are to be treated separately. The restriction of Borel’s theorem to the case of SLn , namely that the image of any word is dense, can be proved via ring theoretic methods by
first establishing the result for n = 2. For the inductive step, one combines a trick of Deligne with our corollary to Amitsur’s theorem, to show that a noncentral element of the algebra of generic matrices cannot be unipotent. We intend to apply the same ideas to Lie algebras. For Chevalley algebras we have the following partial answer [BGKP]: Theorem 3. Let L(R, k) be a Chevalley Lie algebra. If char(k) = 2, assume that R does not contain irreducible components of type Cr , r ≥ 1 (here C1 = A1 , C2 = B2 ). Suppose P (X1 , . . . , Xd ) is not an identity of the Lie algebra sl(2, k). Then the induced map P : L(R, k)d → L(R, k) is dominant. In order to bypass the assumption that P is not an identity of sl(2), one can use an explicit description of those identities [Ra2], [Fi], [Va]. The next step is to go over from solvability of generic equations in Lie algebras to solvability of generic systems of equations. More precisely, we are interested in the following generalization of Question 3: Question 4. For a given collection of elements P1 , . . . , Pm of the finitely generated free Lie k-algebra Ld over a given algebraically closed field k, and a given semisimple Lie algebra g over k, is the corresponding map gd → gm dominant under the condition that none of Pi (X1 , . . . , Xd ) is an identity of g? Working hypothesis. We believe that the question can be answered affirmatively. Methods and preliminary results: As above, the cases of Chevalley algebras and Lie algebras of Cartan type are treated separately. For Chevalley algebras we have an answer in a particular case of multi-commutator maps [GoR]. We also propose to obtain analogs of results of [GoR] for the image of polynomial maps Mn (k)d → Mn (k)m , m > 1. I.2 Surjectivity: I.2.a Surjectivity of multilinear polynomial maps. We already described the question of surjectivity in I.1.a, but let us state it explicitly here Conjecture 1 (Kaplansky-L’vov). A noncentral multilinear map of matrix algebras whose image contains at least one matrix with nonzero trace is surjective. Working hypothesis: Conjecture 1 is false, but the image is dense. Methods: It is natural to focus on a dense subset of matrices, such as those with distinct eigenvalues. But one must be careful, since there could conceivably be a polynomial all of whose images have eigenvalues whose ratios are all roots of 1. Such a dichotomy is common in quantum theory, and is known to be possible when the polynomial P is not multilinear. This conjecture is related to problems on discrete Fourier transforms. Suppose P (ei1 j1 , . . . , eim jm ) = diag(λ1 , . . . , λn ), where eij are matrix units. To prove that the image of P is dense, it is enough to show that the diagonal
matrix D(λ1 , . . . , λn ), together with all its cyclic shifts, generates a vector space of dimension n. This leads to the next conjecture. ∑ Conjecture 2. For n ≥ 5, either λi = 0, or λi = λj for all i, j, or there is a permutation of the λi for which all of the coefficients of the discrete Fourier transform of a periodic sequence with period (λ1 , . . . , λn ) are nonzero. I.2.b Surjectivity of Engel maps on Lie algebras of Cartan type. Looking at the maps as above for particular polynomials one can expect more than dominance. The starting point here is the following fact established in [BGKP]: Definition 1. An Engel polynomial of degree (m + 1) is a polynomial of the form Em (X, Y ) = [[. . . [ X, Y ], Y ], . . . , Y ] ∈ L2 . A generalized Engel | {z } m times ∑ polynomial has the form m i=1 ai Ei (X, Y ) ∈ L2 , where ai ∈ k. Theorem 4. Let P (X, Y ) ∈ L2 be a generalized Engel polynomial of degree m+1, and let P : L(R, k)2 → L(R, k) be the corresponding map of Chevalley algebras. If R does not contain irreducible components of the following types: R = A1 , Br , Cr , F4 for char(k) = 2, R = G2 for char(k) = 3, and |K| > m|R|, then the image of P is (L(R, k) \ Z(L(R, k)) ∪ {0}. Moreover, if P is an Engel polynomial, then the same is true under the assumption |K| > |R+ |. We want to extend this theorem to Lie algebras of Cartan type. Working hypothesis: The statement of Theorem 4 remains true for all finite-dimensional simple Lie algebras defined over a field of characteristic different from 2 and 3, under suitable assumptions on the size of the field. In order to treat the other classical Lie algebras, one can study polynomials on associative matrix rings with involution, incorporating the involution into the language. I.3 Generalizations of Makar-Limanov’s results: I.3.a Construction of algebraically closed skew fields in positive characteristic. A seminal paper of Makar-Limanov [ML3] provides a construction of algebraically closed skew fields in characteristic zero (see also [Ko1], [Ko2] for an alternate exposition of Makar-Limanov’s construction). We hope to apply our results of previous sections for extending it to positive characteristic. I.3.b The Freiheitsatz for associative algebras in nonzero characteristic. Freiheitsatz. For P ∈ A = k⟨x1 , . . . , xn , t⟩ which involves t non-trivially, the algebra k⟨x1 , . . . , xn ⟩ can be naturally embedded into the quotient k⟨x1 , . . . , xn , t⟩/ Id(P ). Working hypothesis: The Freiheitsatz is true in arbitrary characteristic.
Methods: The Freiheitsatz for associative algebras was established in the case char(k) = 0 by Makar-Limanov, as a consequence of his construction of an algebraically closed skew-field. Later on, P. Kolesnikov developed Makar-Limanov’s ideas and, in particular, improved his exposition. The proof is based on solvability of equations in the algebra of Malcev-von Neumann series which are related to differential operators. The construction of the algebra in which the corresponding equations were solved is none other than the ∗-operation related to Kontsevich’s formal quantization. Makar-Limanov has proposed proving that the co-rank of an arbitrary polynomial P on Mn (k) is bounded by some reasonable function of n. This would yield the Freiheitsatz almost immediately. The difficulty is that P need not be homogeneous. We have some information obtained by examining generic matrices, and the hope is that they will be amenable to new geometric techniques. Images of polynomial maps and matrix equations. Let P be a noncommutative polynomial. Suppose that for all sufficiently large n we can solve the system (1)
{Pi (x1 , . . . , xs , t) = 0}. Let ξi , i = 1, . . . , s, ν be a solution of (1). Then after substituting xi 7→ ai + ξi , t 7→ t + ν, we obtain an equation without a free term. Further, assume that after such a substitution there appears a term B(t) linear in t. Let us regard B as an element of an operator algebra. If it turns out that the operator B is invertible, we will be able to solve (1) using the method of consecutive approximations in the product of the matrix algebra and the free algebra, and thus see that this equation does not impose any restrictions on the original x1 , . . . , xs . It is worth noting that it is enough to know how to solve equations (and prove invertibility of operators) modulo matrices of bounded rank and as n → ∞. I.4 Arithmetic. I.4.a When do polynomial matrix equations over integers have many solutions modulo (almost) all primes? Our starting point here is the following observation [La]. Theorem 5. Let w be a non-identity word. Denote by fw the corresponding word map. If {Γi } is an infinite sequence of finite simple groups, no two |Γi | isomorphic to one another, then limi→∞ loglog = 1. |fw (Γd )| i
This fact, a consequence of Borel’s dominance theorem, shows that even in finite simple groups the image of the word map is, in a sense, large. We want to obtain some analogue of this result for polynomial maps of matrix algebras. Conjecture 3. Let P ∈ Z ⟨x1 , . . . , xd ⟩ be an associative noncommutative polynomial in d variables, and let P : Mn (Z)d → Mn (Z) be the corresponding map. Suppose that the induced map P : Mn (Q)d → Mn (Q) is dominant for all n ≥ 2. Then the for almost all primes p and all q = pt , t ≥ 1, the
maps Pq : Mn (Fq )d → Mn (Fq ) satisfy: log(n2 q) = 1. n→∞ log |P (Mn (Fq )d )| lim
Methods: As in [La], we plan to use algebraic-geometric techniques developed in [LaP]. I.4.b When are such solutions (almost) equidistributed? We want to compare the numbers of solutions of equations of the form Pq (x1 , . . . , xd ) = A when A runs over Mn (Fq ). Problem 1. For a fixed d, describe polynomials P ∈ Z ⟨x1 , . . . , xd ⟩ with the following property: given q, the number of solutions of the equation Pq (x1 , . . . , xd ) = A in Mn (Fq )d does not depend on A. A weaker version is Problem 2. For a fixed d, describe polynomials P ∈ Z ⟨x1 , . . . , xd ⟩ with the following property: for almost all p, the number of solutions f (A, q) of the equation Pq (x1 , . . . , xd ) = A in Mn (Fq )d is q d(n−1) (1 + o(1)) (n is fixed, q = pt tends to infinity). Note that group-theoretic analogues of these problems have been intensely investigated over the past few years (see, e.g., [GaS], [BGG], [BG], [Pu]). Working hypothesis. The class of polynomials satisfying the condition of Problem 1 should be narrow enough. For n = 2 one can expect that this class is not too far from the class of primitive monomials (cf. [Pu] for the group-theoretic setting). On the contrary, weaker conditions for Problem 2 may be satisfied by a wide class of polynomials (expectedly, all but those representable as a composition of “degenerate” polynomials; cf. [BK] for group-theoretic analogues). Methods and preliminary results: We are going to combine general techniques of [LaP] for estimating the size of fibres of polynomial maps with more concrete approaches developed in [BK] for the group case. I.5 Applications to automorphisms of algebras and the Jacobian Conjecture Methods and preliminary results: The brilliant deceased mathematician Yagzev worked for twenty years [Y]–[Y7] on automorphisms of algebras and the Jacobian conjecture, translating this conjecture (and related questions concerning tameness of automorphisms) to certain classes of polynomial identities, called Engel identities, defined on arbitrary classes of (not necessarily associative) algebras. Recently Belov has gone over Yagzev’s work and formulated it in a broader, uniform setting, as exposed in [BBR]. Thus, the Jacobian conjecture for a given class of algebras is equivalent to a statement about Engelian identities of algebras satisfying the Capelli polynomial identities. Let M = Mi be a set of varieties of algebras of arbitrary signature. We say that M satisfies the packing property, if for any natural n ∈ N there exists a prime algebra A of rank n in some Mj such that any prime algebra in any Mi of rank n can be embedded into some central extension of A. One result from [BBR]:
Theorem 6. If the set of varieties of Engel algebras (of arbitrary fixed order) satisfying a system of Capelli identities of some order also satisfies the packing property, then the Jacobian Conjecture has a positive solution. Although the Jacobian conjecture is notoriously treacherous, we feel that this novel approach could yield significant new information, and is worth pursuing. Here is a related question Question 5. Is the automorphism group of the Weyl algebra over C isomorphic to the group of polynomial symplectomorphisms? This problem, which was posed in [BeK1], [BeK2], can be reduced to the problem on solvability of the following system of polynomial equations in the Weyl algebra modulo p: zip = Pi (x1 , . . . , xn , y1 , . . . , yn ), ξip = Qi (x1 , . . . , xn , y1 , . . . , yn ) where (P1 , Q1 , . . . , Pn , Qn ) is a ∑ polynomial symplectomorphism preserving the standard symplectic form ω = dxi ∧ dyi . II. Identities in algebraic structures. II.1 Minimal identities of algebras Since the PIs are an obstruction to surjectivity results, it is reasonable to study the PIs of an arbitrary matrix algebra, which starts with the identities of minimal degree. The celebrated Amitsur-Levitzki theorem says that the standard identity of degree 2n is the minimal identity for Mn (k), but when one digs a bit deeper one finds several intriguing problems that remain. II.1.a Associative algebras with involution Working hypothesis. The minimal ∗-identity of Mn (k) with respect to the symplectic involution has degree 32 n. Methods: Kostant proved that the standard identity s2n−2 vanishes for antisymmetric matrices under the transpose involution, and Rowen [Ro1] showed that s2n−2 vanishes for symmetric matrices under the symplectic involution ([DaH] improves on this by finding a ∗-identity of degree 2n − 3 when n is even). Although these results are sharp for the standard polynomial, other identities of lower degree have been found for small n, [DR]. Helling, Procesi, and Razmyslov proved that every PI is a consequence of the Hamilton-Cayley theorem, and Razmyslov provided a lovely proof of the Amitsur-Levitzki theorem using these methods. Likewise, in principle, every ∗-identity should be a consequence of the Hamilton-Cayley theorem. This approach was used in [Ro2] in order to reprove Kostant’s theorem. We plan to apply some sophisticated combinatorial tools to lower further the degree of a ∗-PI. II.1.b Pro-finite identities Zelmanov [Ze3] developed an approach allowing one to go over from polynomials in Lie algebras to pro-p-identities. Utilizing spaces related to the values of polynomials, he proved that in n × n-matrices, for p >> n, there is a pro-p-identity. Thus the free pro-p-group cannot be embedded into
n × n-matrices over a topological ring. (For 2 × 2-matrices see earlier works by Zubkov [Zu1]–[Zu4].) We would like to find algebraic versions of this theory. II.2 Lattices of identities II.2.a Finite basing of sets of identities Problem 3. Does Specht’s problem have a positive solution for affine Lie algebras? For affine Jordan algebras? Problem 4. Are relatively free Lie algebras representable? Working hypothesis. We aim for affirmative answers to these questions Methods and preliminary results: Belov’s approach to Specht’s problem was based on the application of quivers of representations of algebras. Although the idea is quite intuitive, its implementation involves several technical issues, and the main goal of the previous ISF grant of Belov and Rowen, in collaboration with Vishne, was to write a full proof in detail, which finally required four papers [BeRV1]– [BeRV4] totalling about 200 pages. These tools are quite powerful, and we think they yield the representability of arbitrary affine associative PI-algebras; we are in the process of writing down the full proof. The situation for Lie (and Jordan) algebras remains largely open, but we expect to obtain an affirmative answer of Specht’s problem also in the Lie and Jordan cases. II.2.b Commutator identities. For a group G, let GG denote the smallest family of subgroups of G closed under commutators and multiplication. Commutator identities are encoded by inclusion relations in GG , where G is sufficiently generic. Since the inclusion structure of GG is not a-priori obvious (is [G′ , [G, [G, G′ ]]] ⊆ b whose [G, [G′ , [G, G′ ]]] · [G′ , G′′ ]?), one may study a formal semi-lattice G elements are formal products of groups obtained by commutation, subject only to the trivial properties of the commutator operation, including the three-subgroup lemma, which is enough to prove the basic inclusion properties of commutator subgroups (e.g. every commutator of weight n is in Gn = [G, Gn−1 ]). b for G the countably generated free group? Question 6. Is GG equal to G Question 7. Are the lattices GFn equal for every n ≥ 3? (For example, G = F2 satisfies [G′ , [G, G′ ]] = G′′ , so GF2 is smaller.) Working hypothesis: We expect the answer to the first question to be positive; if this is proved by the method sketched below, it will provide a positive answer to the second question as well. Methods and preliminary results: Clearly every GG is a quotient semib To show that there are no extra inclusions, one constructs lattice of G. sufficiently many counterexamples; these can conveniently be taken to be solvable groups, which leads to Lie algebra techniques.
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