Wildness of the problems of classifying two-dimensional spaces of ...

arXiv:1709.10334v1 [math.RT] 29 Sep 2017

Wildness of the problems of classifying two-dimensional spaces of commuting linear operators and certain Lie algebras Vyacheslav Futornya , Tetiana Klymchukb,c , Anatolii P. Petravchukc , Vladimir V. Sergeichukd,˚ a

Department of Mathematics, University of S˜ ao Paulo, Brazil b Universitat Polit`ecnica de Catalunya, Barcelona, Spain c Faculty of Mechanics and Mathematics, Taras Shevchenko University, Kiev, Ukraine d Institute of Mathematics, Tereshchenkivska 3, Kiev, Ukraine

Abstract For each two-dimensional vector space V of commuting n ˆ n matrices over a field F with at least 3 elements, we denote by Vr the vector space of all pn`1qˆ pn ` 1q matrices of the form r A0 ˚0 s with A P V . We prove the wildness of the problem of classifying Lie algebras Vr with the bracket operation ru, vs :“ uv´ vu. We also prove the wildness of the problem of classifying two-dimensional vector spaces consisting of commuting linear operators on a vector space over a field. Keywords: Spaces of commuting linear operators, Matrix Lie algebras, Wild problems. 2000 MSC: 15A21, 16G60, 17B10. 1. Introduction Let F be a field that is not the field with 2 elements. We prove the wildness of the problems of classifying • two-dimensional vector spaces consisting of commuting linear operators on a vector space over F (see Section 2), and Corresponding author. Published in: Linear Algebra Appl. 536 (2018) 201–209. Email addresses: [email protected] (Vyacheslav Futorny), [email protected] (Tetiana Klymchuk), [email protected] (Anatolii P. Petravchuk), [email protected] (Vladimir V. Sergeichuk) ˚

Preprint submitted to Elsevier

October 2, 2017

• Lie algebras LpV q with bracket ru, vs :“ uv ´ vu of matrices of the form » fi α1 — .. ffi A — . ffi in which A P V, α1 , . . . , αn P F, (1) — ffi, – αn fl 0 ... 0 0 in which V is any two-dimensional vector space of n ˆ n commuting matrices over F (see Section 3). A classification problem is called wild if it contains the problem of classifying pairs of n ˆ n matrices up to similarity transformations pM, Nq ÞÑ S ´1 pM, NqS :“ pS ´1 MS, S ´1 NSq with nonsingular S. This notion was introduced by Donovan and Freislich [8, 9]. Each wild problem is considered as hopeless since it contains the problem of classifying an arbitrary system of linear mappings, that is, representations of an arbitrary quiver (see [13, 5]). Let U be an n-dimensional vector space over F. The problem of classifying linear operators A : U Ñ U is the problem of classifying matrices A P Fnˆn up to similarity transformations A ÞÑ S ´1 AS with nonsingular S P Fnˆn . In the same way, the problem of classifying vector spaces V of linear operators on U is the problem of classifying matrix vector spaces V Ă Fnˆn up to similarity transformations V ÞÑ S ´1 V S :“ tS ´1 AS | A P V u

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with nonsingular S P Fnˆn (the spaces V and S ´1 V S are matrix isomorphic; see [14]). In Theorem 1(a), we prove the wildness of the problem of classifying two-dimensional vector spaces V Ă Fnˆn of commuting matrices up to transformations (2). Each two-dimensional vector space V Ă Fnˆn is given by its basis A, B P V that is “determined up to transformations pA, Bq ÞÑ pαA ` βB, γA ` δBq, α γ‰ 2ˆ2 in which β δ P F is a change-of-basis matrix. Thus, the problem of classifying two-dimensional vector spaces V Ă Fnˆn up to transformations (2) is the problem of classifying pairs of linear independent matrices A, B P Fnˆn up to transformations pA, Bq ÞÑ pA1 , B 1 q :“ S ´1 pαA ` βB, γA ` δBqS, 2

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“ ‰ in which both S P Fnˆn and αγ βδ P F2ˆ2 are nonsingular matrices. We say that the matrix pairs pA, Bq and pA1 , B 1 q from (3) are weakly similar. In Theorem 1(b), we prove that the problem of classifying pairs of commuting matrices up to weak similarity is wild, which ensures Theorem 1(a). The analogous problem of classifying matrix pairs pA, Bq up to weak congruence S T pαA ` βB, γA ` δBqS appears in the problem of classifying finite p-groups of nilpotency class 2 with commutator subgroup of type pp, pq, in the problem of classifying commutative associative algebras with zero cube radical, and in the problem of classifying Lie algebras with central commutator subalgebra; see [3, 4, 6, 18]. The problem of classifying matrix pairs up to weak equivalence RpαA ` βB, γA ` δBqS appears in the theory of tensors [2]. Note that the group of pn ` 1q ˆ pn ` 1q matrices „  A v , in which A P Fnˆn is nonsingular and v P Fn 0 1 is called the general affine group; it is the group of all invertible affine transformations of an affine space; see [15]. If F “ R, then this group is a Lie group, its Lie algebra consists of all pn ` 1q ˆ pn ` 1q matrices „  A v , in which A P Rnˆn is nonsingular and v P Rn , 0 0 and each Lie algebra LpV q of matrices of the form (1) with F “ R is its subalgebra. The abstract version of the construction of Lie algebras LpV q of matrices of the form (1) is the following. Let Frx, ys be the polynomial ring, and let WFrx,ys be a left Frx, ys-module given by a finite dimensional vector space WF and two commuting linear operators P : w ÞÑ xw and Q : w ÞÑ yw on WF that are linearly independent. The p2 ` dimF Wq-dimensional vector space LW :“ Fx ‘F Fy ‘F W is the metabelian Lie algebra with the bracket operation defined by rx, vs :“ P v, ry, vs :“ Qv, and rx, ys “ rv, ws :“ 0 for all v, w P W. If W “ Fn and V is the two-dimensional vector space generated by P and Q, then the Lie algebra LW coincides with the Lie algebra LpV q of all matrices (1). By [16, Corollary 1] and Theorem 1, the problem of classifying metabelian Lie algebras LW is wild. We use the following definition of wild problems (see more formal definitions in [1, 10, 11]). Every matrix problem M is given by a set M1 of tuples 3

of matrices over a field F and a set M2 of admissible transformations with them. A matrix problem M is wild if there exists a t-tuple Mpx, yq “ pM1 px, yq, . . . , Mt px, yqq

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of matrices, whose entries are noncommutative polynomials in x and y over F, such that (i) MpA, Bq P M1 for all A, B P Fnˆn and n “ 1, 2, . . . (in particular, each scalar entry α of Mi px, yq is replaced by αIn ), (ii) MpA, Bq is reduced to MpA1 , B 1 q by transformations M2 if and only if pA, Bq is similar to pA1 , B 1 q. 2. Spaces of linear operators Theorem 1. (a) The problem of classifying up to similarity (2) of twodimensional vector spaces of commuting matrices over a field F is wild. If F is not the field of two elements, then the problem of classifying up to similarity of two-dimensional vector spaces of commuting matrices over F that contain nonsingular matrices is wild. (b) The problem of classifying up to weak similarity (3) of pairs of commuting matrices over a field F is wild. If F is not the field of two elements, then the problem of classifying up to weak similarity of pairs pA, Bq of commuting matrices over F such that αA`βB is nonsingular for some α, β P F is wild. Proof. (a) This statement follows from statement (b) since each twodimensional vector space V Ă Fnˆn determined up to similarity is given by its basis A, B P V that is determined up to transformations (3). (b) Step 1: We prove that the problem of classifying pairs of commuting and nilpotent matrices up to similarity is wild. This statement was proved by Gelfand and Ponomarev [13]; it was extended in [7] to matrix pairs under consimilarity. By analogy with [7, Section 3], we consider two commuting and nilpotent 5n ˆ 5n matrices » » fi fi 0 In 0 0 0 0 0 X 0 Y — 0 0 In 0 0 ffi — 0 0 0 X 0 ffi — — ffi ffi — 0 0 0 0 0 ffi (5) ffi, 0 0 0 I 0 J :“ — K :“ n XY — — ffi ffi – 0 0 0 0 0 fl – 0 0 0 0 0 fl 0 0 0 0 0 0 0 0 In 0 4

that are partitioned into n ˆ n blocks, in which X, Y P Fnˆn are arbitrary. Let us prove that two pairs pX, Y q and pX 1 , Y 1 q of n ˆ n matrices are similar ðñ two pairs of commuting and nilpotent matrices pJ, KXY q and pJ, KX 1 Y 1 q are similar.

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ùñ. If pX, Y qS “ SpX 1 , Y 1 q, then pJ, KXY qR “ RpJ, KX 1 Y 1 q with R :“ S ‘ S ‘ S ‘ S ‘ S. ðù. Let pJ, KXY qR “ RpJ, KX 1 Y 1 q with nonsingular R. All matrices commuting with a given Jordan matrix are described in [12, Section VIII, § 2]. Since R commutes with J, we analogously find that » fi C C1 C2 C3 D — 0 C C1 C2 0 ffi — ffi ffi. 0 0 C C 0 R“— 1 — ffi – 0 0 0 C 0 fl 0 0 0 E F The equality KXY R “ RKX 1 Y 1 implies that fi » » 0 0 0 XC XC1 ` Y E Y F ffi — 0 — 0 0 0 XC 0 ffi — — ffi“— 0 — 0 0 0 0 0 ffi — — – 0 0 0 0 0 fl – 0 0 0 0 C 0 0

fi 0 CX 1 C1 X 1 ` D CY 1 0 0 CX 1 0 ffi ffi 0 0 0 0 ffi ffi, 0 0 0 0 fl 0 0 F 0

and so pX, Y qC “ CpX 1 , Y 1 q. Step 2: We prove that the problem of classifying matrix pairs up to weak similarity is wild. If the field F has at least 3 elements, we fix any λ P F such that λ ‰ 0 and λ ‰ ´1. If F consists of two elements, we take λ “ 1. For each pair pA, Bq of m ˆ m matrices with m ě 1 over F, define the matrix pair pM1 pAq, M2 pBqq as follows: M1 pAq :“ I2m`2 ‘ 03m`3 ‘ Im`1 ‘ A, M2 pBq :“ 02m`2 ‘ I3m`3 ‘ λIm`1 ‘ B. (Analogous constructions are used in [3, 4].) 5

Let us prove that pM1 pAq, M2 pBqq can be used in (4) in order to prove the wildness of the problem of classifying matrix pairs up to weak similarity. We should prove that arbitrary pairs pA, Bq and pA1 , B 1 q of mˆ m matrices are similar ðñ pM1 pAq, M2 pBqq and pM1 pA1 q, M2 pB 1 qq are weakly similar.

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ùñ. If S ´1 pA, BqS “ pA1 , B 1 q, then pI6m`6 ‘ Sq´1 pM1 pAq, M2 pBqqpI6m`6 ‘ Sq “ pM1 pA1 q, M2 pB 1 qq. ðù. Let S ´1 pαM1 pAq ` βM2 pBq, γM1 pAq ` δM2 pBqqS “ pM1 pA1 q, M2 pB 1 qq “ ‰ with a nonsingular αγ βδ . Then rankpαM1 pAq ` βM2 pBqq “ rank M1 pA1 q, rankpγM1 pAq ` δM2 pBqq “ rank M2 pB 1 q.

If β ‰ 0, then rankpαM1 pAq ` βM2 pBqq ą 4m ` 3 ě rank M1 pA1 q. “ ‰ Hence β “ 0. Since αγ βδ is nonsingular, δ ‰ 0. If γ ‰ 0, then rankpγM1 pAq ` δM2 pBqq ą 5m ` 4 ě rank M2 pB 1 q.

Hence γ “ 0. Thus S ´1 pαM1 pAq, δM2 pBqqS “ pM1 pA1 q, M2 pB 1 qq, and so the pairs pαM1 pAq, δM2 pBqq “ pαI2m`2 , 02m`2 q ‘ p03m`3 , δI3m`3 q ‘ pαIm`1 , δλIm`1 q ‘ pαA, δBq, 1 1 pM1 pA q, M2 pB qq “ pI2m`2 , 02m`2 q ‘ p03m`3 , I3m`3 q ‘ pIm`1 , λIm`1 q ‘ pA1 , B 1 q

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give isomorphic representations of the quiver ü rý. By the Krull–Schmidt theorem for quiver representations (see [17, Theorem 1.2]), every representation of a quiver is isomorphic to a direct sum of indecomposable representations, and this sum is uniquely determined up to replacements of direct summands by isomorphic representations and permutations of direct summands. If we delete in (8) the summands pαI2m`2 , 02m`2 q and p03m`3 , δI3m`3 q of pαM1 pAq, δM2 pBqq and the corresponding isomorphic summands pI2m`2 , 02m`2 q and p03m`3 , I3m`3 q of pM1 pA1 q, M2 pB 1 qq, we find that the remaining pairs pαIm`1 , δλIm`1 q ‘ pαA, δBq,

pIm`1 , λIm`1 q ‘ pA1 , B 1 q

give isomorphic representations of the quiver ü rý. The first pair has m ` 1 direct summands pα, δλq and the second pair has m ` 1 direct summands p1, λq. By the Krull–Schmidt theorem, these summands give isomorphic representations, hence α “ δ “ 1, and so the pairs pA, Bq and pA1 , B 1 q give isomorphic representations too. Therefore, the pairs pA, Bq and pA1 , B 1 q are similar. Step 3. By Steps 1 and 2, the following equivalences hold for arbitrary pairs pX, Y q and pX 1 , Y 1 q of n ˆ n matrices over F: pX, Y q and pX 1 , Y 1 q are similar ðñ pJ, KXY q and pJ, KX 1 Y 1 q are similar ðñ pλI ` J, KXY q and pλI ` J, KX 1 Y 1 q are similar ðñ pM1 pλI ` Jq, M2 pKXY qq and pM1 pλI ` Jq, M2 pKX 1 Y 1 qq are weakly similar. Note that pM1 pλI ` Jq, M2 pKXY qq and pM1 pλI ` Jq, M2 pKX 1 Y 1 qq are pairs of commuting matrices. If F has at least 3 elements, then the matrix M1 pλI ` Jq ` M2 pKXY q is nonsingular.

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3. Lie algebras For each vector space V Ă Fnˆn of commuting matrices over a field F, we denote by Vr the vector space of all pn ` 1q ˆ pn ` 1q matrices of the form » fi » fi α1 α1 — .. ffi A — ffi — .. ffi . pA|aq :“ — in which A P V and a :“ – . fl P Fn . ffi, – fl αn αn 0 ... 0 0 We consider the space Vr as the Lie algebra LpV q with the Lie bracket operation rpA|aq, pB|bqs :“ pA|aqpB|bq ´ pB|bqpA|aq “ p0|Ab ´ Baq.

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Theorem 2. Let a field F be not the field with 2 elements. 1

1

(a) Let V Ă Fnˆn and V 1 Ă Fn ˆn be two vector spaces of commuting matrices that contain nonsingular matrices. Then the following statements are equivalent: (i) The Lie algebras LpV q and LpV 1 q are isomorphic. (ii) n “ n1 and V is similar to V 1 pi.e., SV S ´1 “ V 1 for some nonsingular S P Fnˆn q, (iii) n “ n1 and Vr is similar to Vr 1 .

(b) The problem of classifying Lie algebras LpV q with dimF V “ 2 up to isomorphism is wild.

Proof. (a) Let us prove the equivalence of (i)–(iii). 1 (i)ñ(ii) Let ϕ : LpV qÑLpV r q be an isomorphism of Lie algebras. Then 1 r1 r r r r r ϕrV , V s “ rV , V s. By (9), rV , V s Ă p0|Fn q. Since V contains a nonsingular matrix A, rpA|0q, p0|Fnqs “ p0|Fn q, and so rVr , Vr s “ p0|Fn q. Hence ϕp0|Fn q “ 1 p0|Fn q and n “ n1 . Let e1 , . . . , en be the standard basis of Fn , and let p0|fi q :“ ϕp0|ei q. Since ϕp0|Fn q “ p0|Fn q, f1 , . . . , fn is also a basis of Fn . Denote by S the nonsingular matrix whose columns are f1 , . . . , fn . Then fi “ Sei . 8

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n ř Let A P V and write pB|bq :“ ϕpA|0q. Let A “ rαij si,j“1, i.e., Aej “ i αij ei . Then

p0|Bfj q “ rpB|bq, p0|fj qs “ rϕpA|0q, ϕp0|ej qs “ ϕrpA|0q, p0|ej qs ř ř “ ϕp0|Aej q “ ϕp0, i αij ei q “ ϕ i αij p0|ei q ř ř ř “ i αij ϕp0|ei q “ i αij p0|fi q “ p0| i αij fi q ř and so Bfj “ i αij fi . By (10), ř ř BSej “ i αij Sei “ S i αij ei “ SAej . Therefore, BS “ SA and so V 1 S “ SV .

(ii)ñ(iii) If V and V 1 are similar via S, then Vr and Vr 1 are similar via S ‘ I1 . (iii)ñ(i) If RVr R´1 “ Vr 1 for some nonsingular R P Fpn`1qˆpn`1q , then 1 X ÞÑ RXR´1 is an isomorphism LpV qÑLpV r q. (b) This statement follows from the equivalence (i)ô(ii) and Theorem 1(a).

Acknowledgements V. Futorny was supported by CNPq grant 301320/2013-6 and by FAPESP grant 2014/09310-5. V.V. Sergeichuk was supported by FAPESP grant 2015/05864-9. References [1] M. Barot, Introduction to the Representation Theory of Algebras, Springer, Cham, 2015. [2] G. Belitskii, M. Bershadsky, V.V. Sergeichuk, Normal form of m-by-nby-2 matrices for equivalence, J. Algebra 319 (2008) 2259–2270. [3] G. Belitskii, A.R. Dmytryshyn, R. Lipyanski, V.V. Sergeichuk, A. Tsurkov, Problems of classifying associative or Lie algebras over a field of characteristic not two and finite metabelian groups are wild, Electr. J. Linear Algebra 18 (2009) 516–529.

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