ITEP-TH-43/08 Algebraic properties of Manin matrices 1. A. Chervov 1 G. Falqui 2 V. Rubtsov 3 1 Institute for Theoretical and Experimental Physics, Moscow - Russia 9 0 2Universit´a di Milano - Bicocca, Milano - Italy 0 3Universit´e D’Angers, Angers, France 2 n a J Abstract 2 ] A We study a class of matrices with noncommutative entries, which were first considered by Yu. Q I. Manin in 1988 in relation with quantum group theory. They are defined as “noncommutative . hendomorphisms” of a polynomial algebra. More explicitly their defining conditions read: 1) elements t ain the same column commute; 2) commutators of the cross terms are equal: [M ,M ] = [M ,M ] ij kl kj il m (e.g. [M ,M ] = [M ,M ]). The basic claim is that despite noncommutativity many theorems 11 22 21 12 [ of linear algebra hold true for Manin matrices in a form identical to that of the commutative case. 1Moreover in some examples the converse is also true, that is, Manin matrices are the most general v class of matrices such that linear algebra holds true for them. The present paper gives a complete list 5 3anddetailed proofsof algebraicpropertiesofManin matrices known uptothemoment; manyof them 2 are new. In particular we present the formulation of Manin matrices in terms of matrix (Leningrad) 0 .notations; provide complete proofs that an inverse to a Manin matrix is again a Manin matrix and 1 0for the Schur formula for the determinant of a block matrix; we generalize the noncommutative 9 Cauchy-Binet formulasdiscovered recently [arXiv:0809.3516], which includes theclassical Capelliand 0 :related identities. We also discuss many other properties, such as the Cramer formula for the inverse v imatrix, the Cayley-Hamilton theorem, Newton and MacMahon-Wronski identities, Plu¨cker relations, X Sylvester’s theorem, the Lagrange-Desnanot-Lewis Caroll formula, the Weinstein-Aronszajn formula, r asome multiplicativity properties for the determinant, relations with quasideterminants, calculation of the determinant via Gauss decomposition, conjugation to the second normal (Frobenius) form, and so on and so forth. Finally several examples and open question are discussed. We refer to [CF07, RST08] for some applications in the realm of quantum integrable systems. 1E-mail: [email protected] 2E-mail: [email protected] 3E-mail: [email protected] Contents 1 Introduction 2 1.1 Results and organization of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Context, history and related works . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Warm-up 2×2 examples: Manin matrices everywhere 6 2.1 Further properties in 2×2 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Manin matrices. Definitions and elementary properties 9 3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.1.1 Poisson version of Manin matrices . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Characterization via coaction. Manin’s construction . . . . . . . . . . . . . . . . . . 10 3.3 q-analogs and RTT=TTR quantum group matrices . . . . . . . . . . . . . . . . . . . 11 3.4 The determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.5 The permanent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.6 Elementary properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.6.1 Some No-Go facts for Manin matrices. . . . . . . . . . . . . . . . . . . . . . . 17 3.7 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.8 Hopf structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4 Inverse of a Manin matrix 21 4.1 Cramer’s formula and quasideterminants . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.1.1 Relation with quasideterminants . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.2 Lagrange-Desnanot-Jacobi-Lewis Caroll formula . . . . . . . . . . . . . . . . . . . . . 24 4.3 The inverse of a Manin matrix is again a Manin matrix . . . . . . . . . . . . . . . . 25 4.4 On left and right inverses of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5 Schur’s complement and Jacobi’s ratio theorem 27 5.1 Multiplicativity of the Determinant for special matrices of block form . . . . . . . . . 27 5.2 Block matrices, Schur’s formula and Jacobi’s ratio theorem . . . . . . . . . . . . . . 29 5.2.1 Proof of the Schur’s complement theorem. . . . . . . . . . . . . . . . . . . . . 32 5.2.2 Proof 2. (Proof of the Jacobi Ratio Theorem via quasideterminants) . . . . . 33 5.3 The Weinstein-Aronszajn formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 5.4 Sylvester’s determinantal identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.5 Application to numeric matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 6 Cauchy-Binet formulae and Capelli-type identities 37 6.1 Grassman algebra condition for Cauchy-Binet formulae . . . . . . . . . . . . . . . . . 38 6.2 No correction case and new Manin matrices . . . . . . . . . . . . . . . . . . . . . . . 42 6.3 Capelli-Caracciolo-Sportiello-Sokal case . . . . . . . . . . . . . . . . . . . . . . . . . 43 6.4 Turnbull-Caracciolo-Sportiello-Sokal case . . . . . . . . . . . . . . . . . . . . . . . . . 45 6.5 Generalization to permanents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 6.5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 6.5.2 Cauchy-Binet type formulas for permanents . . . . . . . . . . . . . . . . . . . 49 6.5.3 Toy model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 1 7 Further properties 52 7.1 Cayley-Hamilton theorem and the second normal (Frobenius) form . . . . . . . . . . 52 7.2 Newton and MacMahon-Wronski identities . . . . . . . . . . . . . . . . . . . . . . . 54 7.2.1 Newton identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 7.2.2 MacMahon-Wronski relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 7.2.3 Second Newton identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 7.3 Plu¨cker relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 7.4 Gauss decomposition and the determinant . . . . . . . . . . . . . . . . . . . . . . . . 61 7.5 Bibliographic notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 8 Matrix (Leningrad) form of the defining relations for Manin matrices 63 8.1 A brief account of matrix (Leningrad) notations . . . . . . . . . . . . . . . . . . . . . 64 8.1.1 Matrix (Leningrad) notations in 2×2 case . . . . . . . . . . . . . . . . . . . 64 8.2 Manin’s relations in the matrix form . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 8.3 Matrix (Leningrad) notations in Poisson case . . . . . . . . . . . . . . . . . . . . . . 67 9 Conclusion and open questions 68 9.1 Tridiagonal matrices and duality in Toda system . . . . . . . . . . . . . . . . . . . . . 68 9.2 Fredholm type formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 9.3 Tensor operations, immanants, Schur functions ... . . . . . . . . . . . . . . . . . . . . 70 9.4 Other questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 1 Introduction It is well-known that matrices with generically noncommutative elements do not admit a natural construction of the determinant with values in a ground ring and basic theorems of the linear algebra fail to hold true. On the other hand, matrices with noncommutative entries play a basic role in the theory of quantum integrability (see, e.g., [FTS79]), in Manin’s theory of ”noncommutative symmetries” [Man88], and so on and so forth. Further we prove that many results of commutative linear algebra can be applied with minor modifications in the case of ”Manin matrices”. We will consider the simplest case of those considered by Manin, namely - in the present paper - we will restrict ourselves to the case of commutators, and not of (super)-q-commutators, etc. Let us mention that Manin matrices are defined, roughly speaking, by half of the relations of the corresponding quantum group Fun (GL(n)) and taking q = 1 (see section 3.3 page 11). q Definition 1 Let M be an n×n′ matrix with elements M in (not necessarily commutative) ring ij R. We will call M a Manin matrix if the following two conditions hold: 1. Elements in the same column commute between themselves. 2. Commutators of cross terms of 2×2 submatrices of M are equal: [M ,M ] = [M ,M ], ∀i,j,k,l e. g. [M ,M ] = [M ,M ]. (1) ij kl kj il 11 22 21 12 AmoreintrinsicdefinitionofManinmatricesviacoactiononpolynomialandGrassmannalgebraswill be recalled in Proposition 1 page 10 below. (Roughly speaking variables x˜ = M x commute i j ij j among themselves if and only if M is a Manin matrix, where x are commuting variables, also j P commuting with elements of M). 2 In the previous paper [CF07] we have shown that Manin matrices have various applications in quantum integrability and outlined some of their basic properties. This paper is devoted solely to algebraic properties providing a complete amount of facts known up to the moment. Quite probably the properties established here can be transferred to some other classes of matrices with noncommutative entries, for example, to super-q-Manin matrices (see [CFRS]) and quantum Lax matrices of most of the integrable systems. Such questions seems to be quite important for quantum integrability, quantum and Lie groups, as well as in the geometric Langlands correspondence theory [CT06, CF07]. But before studying these complicated issues, it seems to be worth to understand the simplest case in depth, this is one of the main motivations for us. The other one is that many statements are so simple and natural extension of the classical results, that can be of some interest just out of curiosity or pedagogical reasons for wide range of mathematicians. 1.1 Results and organization of the paper The main aim of the paper is to argue the following claim: linear algebra statements hold true for Manin matrices in a form identical to the commutative case. Let us give a list of main properties discussed below. Some of these results are new, some - can be found in a previousliterature: Yu. Maninhasdefined thedeterminant andhasproven aCramer’sinversion rule, Laplace formulas, as well as Plu¨cker identities; in (S. Garoufalidis, T. Le, D. Zeilberger) [GLZ03] the MacMahon-Wronski formula was proved; (M. Konvalinka) [Ko07A, Ko07B] found the Sylvester’s identity and the Jacobi ratio’s theorem, along with partial results on an inverse matrix and block matrices. 4 Some other results were announced in [CF07], where applications to integrable systems, quantum and Lie groups can be found. • Section3.4: Determinant canbedefined inthestandardwayanditsatisfies standardproperties e.g. it is completely antisymmetric function of columns and rows: y detM = detcolumnM = (−1)σ M . (2) σ(i),i σX∈Sn i=Y1,...,n • Proposition 4 page 14 : let M be a Manin matrix and N satisfies: ∀i,j,k,l : [M ,N ] = 0: ij kl det(MN) = det(M)det(N). (3) Let N be additionally a Manin matrix, then MN and M ±N are Manin matrices. Moreover in case [M ,N ] 6= 0, but obeys certain conditions we prove (theorem 6 page 39 ): ij kl detcol(MY +Q diag(n−1,n−2,...,1,0)) = detcol(M)detcol(Y). (4) This generalizes [CSS08] and the classical Capelli identity [Ca1887]. • Section 4.1: Cramer’s rule: M−1 is a Manin matrix and M−1 = (−1)i+jdet(M)−1det(M ). (5) ij ij Here as usually M is the (n−1)×(n−1) submatrix of Mdobtained removing the l-th row lk and the k-th column. c 4These authors actually considered more general classes of matrices 3 • Section 7.1: the Cayley-Hamilton theorem: det(t−M)| = 0. t=M • Section 5.2: the formula for the determinant of block matrices: A B det = det(A)det(D −CA−1B) = det(D)det(A−BD−1C). (6) C D (cid:18) (cid:19) Also, we show that D − CA−1B, A− BD−1C are Manin matrices. This is equivalent to the so-called Jacobi ratio theorem, stating that any minor of M−1 equals, up to a sign, to the product of (detcolM)−1 and the corresponding complementary minor of the transpose of M. • Section 7.2: the Newton and MacMahon-Wronski identities between TrMk, coefficients of det(1−tM) and Tr(SkM). Denote by σ(t),S(t),T(t) the following generating functions: σ(t) = det(1−tM), S(t) = tkTrSkM, T(t) = Tr M , (7) k=0,...,∞ 1−tM Then: 1 = σ(t)S(t), −∂ σ(t) = σ(t)T(t), ∂ S(t) = T(t)S(t). (8) Pt t • Other facts are also discussed: Plu¨cker relations (section 7.3), Sylvester’s theorem (section 5.4), Lagrange-Desnanot-Lewis Caroll formula (section 4.2), Weinstein-Aronszajn formula (section 5.3), calculation of the determinant via Gauss decomposition (section 7.4), conjugation to the second normal (Frobenius) form (section 7.1), some multiplicativity properties for the determinant (proposition 8 page 27 ), etc. • Section 8.2: matrix (Leningrad) notations form of the definition: M is a Manin matrix ⇔ (9) ⇔ [M ⊗1,1⊗M] = P[M ⊗1,1⊗M] ⇔ (10) (1−P) (1−P) (1−P) ⇔ (M ⊗1) (1⊗M) = (M ⊗1) (1⊗M). (11) 2 2 2 • No-gofacts: Mk isnotaManinmatrix; elementsTr(M),det(M), etc. arenotcentral, moreover [Tr(M),det(M)] 6= 0(section3.6.1); det(eM) 6= eTr(M),det(1+M) 6= eTr(ln(1+M)) (section7.2.1). We also discuss relations with the quantum groups (section 3.3) and mention some examples which are related to integrable systems, Lie algebras and quantum groups (section 3.7). Organization of the paper: section 3 contains main definitions and properties. This material is crucial for what follows. The other sections can be read in an arbitrary order. We tried to make the exposition in each section independent of the others at least at formulations of the main theorems and notations. Though the proofs sometimes use the results from the previous sections. A short section 2 is a kind if warm-up, which gives some simple examples to get the reader interested and to demonstrate some of results of the paper on the simplest examples. The content of each section can be seen from the table of contents. 1.2 Context, history and related works Manin matrices appeared first in [Man88], see also [Man87, Man91, Man92, MMDZ], where some basic facts like the determinant, Cramer’s rule, Plu¨cker identities etc. were established. The lectures [Man88] is the main source on the subject. Actually Manin’s construction defines ”noncommutative endomorphisms” of an arbitrary ring (and in principle of any algebraic structure). Here we restrict 4 ourselves with the simplest case of the commutative polynomial ring C[x ,...,x ], its ”noncommuta- 1 n tive endomorphisms” will be called Manin matrices. Some linear algebra facts were also established for a class of ”good” rings (we hope to develop this in future). The main attention and application of the original works were on quantum groups, which are defined by ”doubling” the set of the relations. The literature on ”quantum matrices” (N. Reshetikhin, L. Takhtadzhyan, L. Faddeev) [FRT90] is enormous - let us only mention [KL94], [DL03] and especially D. Gurevich, A. Isaev, O. Ogievetsky, P. Pyatov, P. Saponov papers [GIOPS95]-[GIOPS05], where related linear algebra facts has been established for various quantum matrices. Concerning ”not-doubled” case let us mention papers (S. Wang) [Wang98], (T. Banica, J. Bichon, B. Collins) [BBC06]. They investigate Manin’s construction applied to finite-dimensional algebras for example to Cn (quantum permutation group). Such algebras appears to be C∗-algebras and are related to various questions in operator algebras. Linear algebra of such matrices is not known, (may be it does not exist). The simplest case which is considered here was somehow forgotten for many years after Manin’s work (see however [RRT02, RRT05]). The situation changed after (S. Garoufalidis, T. Le, D. Zeilberger) [GLZ03], who discovered MacMahon-Wronski relations for (q)-Manin matrices. This result was followed by a flow of papers [EP06, HL06, KPak06, FH06] etc.; let us in particular mention (M. Konvalinka) [Ko07A, Ko07B] which contains Sylvester’s iden- tity and Jacobi’s ratio theorem, along with partial results on an inverse matrix and block matrices. We came to this story from the other direction. In [CF07] we observed that some examples of quantum Lax matrices in quantum integrability are exactly examples of Manin matrices. Moreover linear algebra of Manin matrices appears to have various applications in quantum integrability, quantum and Lie group theories. Numerous other quantum integrable systems provide various ex- amples of matrices with noncommutative elements - quantum Lax matrices. This rises the question how to define the proper determinant and to develop linear algebra for such matrices. And more generally proper determinant exists or not? If yes, how to develop the linear algebra? Such questions seems to be quite important for quantum integrability they arerelated to a ”quantum spectral curve” which promises to be the key concept in the theory [CT06], [CM08, CFRy, CFRS, RST08]. Manin’s approach applied for more general rings provides classes of matrices where such questions can be possibly resolved, however many examples from integrable systems do not fit in this approach. From the more general point of view we deal with the question of the linear algebra of matrices with noncommutative entries. It should be remarked that the first appearance of a determinant for matrices with noncommutative entries goes back to A. Cayley. He was the first person who had applied the notion of what we call a column determinant in the non-commutative setting. (We are thankful to V. Retakh for this remark). Let us mention the initial significative difference between our situation and the work I. Gelfand, S. Gelfand, V. Retakh, R. Wilson [GGRW02], where generic ma- trices with noncommutative entries are considered. There is no natural definition of the determinant (n2 of ”quasi-determinants” instead) in the ”general non-commutative case” and their analogs of the linear algebra propositions are sometimes quite different from the commutative ones. Nevertheless results of loc.cit. can be fruitfully applied to some questions here. Our approach is also different from the classical theory of J. Dieudonn´e [Dieud43] (see also [Adj93]), since in this theory the determinant is an element of the K∗/[K∗,K∗], where K is basic ring, while for Manin matrices the determinant is an element of the ring K itself. We provide more detailed bibliographic notes in the text but we would like to add, as a general disclaimer, that our bibliographic notes are neither exhaustive nor historically ordered. We simply want to comment those papers and books that are more strongly related to our work. We refer to “The Theory of Determinants in the Historical Order of Development” [Muir1890] 5 for the early history of many results, which generalization to the noncommutative case are discussed below. 1.3 Remarks In [GLZ03, Ko07A, Ko07B] the name ”right quantum matrices” was used, in [RRT02, RRT05, LT07] the names “left” and “right quantum group”, in [CSS08] the name “row-pseudo-commutative”. We prefer to use the name ”Manin matrices”. All the considerations below work for an arbitrary field of characteristic not equal to 2, but we prefer to restrict ourselves with C. In subsequent papers [CFRS] we plan to generalize the constructions below to the case of the Manin matrices related to the more general quadratic algebras as well as there applications to quan- tum integrable systems and some open problems. Acknowledgments G.F. acknowledges support from the ESF programme MISGAM, and the MarieCurie RTN ENIGMA. The work of A.C.has been partiallysupported by theRussian President Grant MK-5056.2007.1, grant of Support for the Scientific Schools NSh-3035.2008.2, RFBR grant 08-02-00287a, the ANR grant GIMP (Geometry and Integrability in Mathematics and Physics). He acknowledges support and hospitality of Angers and Poitiers Universities. The work of V.R. has been partially supported by the grant of Support for the Scientific Schools NSh-3036.2008.2, RFBR grant 06-02-17382, the ANR grant GIMP (Geometry and Integrability in Mathematics and Physics) and INFN-RFBR project ”Einstein”. He acknowledges support and hospitality of SISSA (Trieste). The authors are grateful to D. Talalaev, A. Molev, A. Smirnov, A. Silantiev, V. Retakh and D. Gurevich formultiple stimulating discussions, toP.Pyatov forsharing withushisunpublished results, pointing out to the paper [GLZ03] and for multiple stimulating discussions. To Yu. Manin for his interest in this work and stimulating discussions. 2 Warm-up 2 × 2 examples: Manin matrices everywhere Here we present some examples of the appearance of the Manin property in various very simple and natural questions concerning 2 × 2 matrices. The general idea is the following: we consider well-known facts of linear algebra and look how to relax the commutativity assumption for matrix elements such that the results will be still true. The answer is: if and only if M is a Manin matrix. This section is a kind of warm-up, we hope to get the reader interested in the subsequent material and to demonstrate some results in the simplest examples. The expert reader may wish to skip this section. Let us consider a 2×2 matrix M: a b M = . (12) c d (cid:18) (cid:19) From Definition (1) M is a ”Manin matrix” if the following commutation relations hold true: • column-commutativity: [a,c] = 0, [b,d] = 0; • equality of commutators of the cross-term: [a,d] = [c,b]. The fact below can be considered as Manin’s original idea about the subject. 6 Observation 1 Coaction on a plane. Consider the polynomial ring C[x ,x ], and assume that the 1 2 x˜ a b x matrix elements a,b,c,d commute with x ,x . Define x˜ ,x˜ by 1 = 1 . Then 1 2 1 2 x˜ c d x 2 2 (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) x˜ ,x˜ commute among themselves iff M is a Manin matrix. 1 2 Proof. [x˜ ,x˜ ] = [ax +bx ,cx +dx ] = [a,c]x2 +[b,d]x2 +([a,c]+[b,d])x x .(cid:3) (13) 1 2 1 2 1 2 1 2 1 2 Similar fact holds true for Grassman variables (see proposition 1 page 10 below). Observation 2 Cramer rule. The inverse matrix is given by the standard formula d −b M−1 = 1 iff M is a Manin matrix. ad−cb −c a (cid:18) (cid:19) Proof. d −b a b da−bc db−bd = = (14) −c a c d −ca+ac −cb+ad (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19) ad−cb 0 iff M is a Manin matrix = .(cid:3) (15) 0 ad−cb (cid:18) (cid:19) Observation 3 Cayley-Hamilton. The equality M2 −(a+d)M +(ad−cb)1 = 0 holds iff M is 2×2 a Manin matrix. M2 −(a+d)M +(ad−cb)1 = 2×2 a2 +bc ab+bd a2 +da ab+db ad−cb 0 = − + = ca+dc cb+d2 ac+dc ad+d2 0 ad−cb (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (bc−da)+(ad−cb) bd−db [a,d]−[c,b] [b,d] = = . (16) ca−ac 0 [c,a] 0 (cid:18) (cid:19) (cid:18) (cid:19) This vanishes iff M is a Manin matrix. (cid:3) Let us mention that similar facts can be seen for the Newton identities, but not in such a strict form (see example 12 page 57 ). Observation 4 Multiplicativity of Determinants (Binet Theorem). detcolumn(MN) = detcolumn(M)det(N) holds true for all C-valued matrices N iff M is a Manin matrix. detcol(MN)−detcol(M)det(N) = [M ,M ]N N +[M ,M ]N N +([M ,M ]−[M ,M ])N N (17) 11 21 11 12 12 22 21 22 11 22 21 12 21 12 Here and in the sequel, detcolumn(X) or shortly detcol(X) is: X X − X X , i.e. elements from 11 22 21 12 the first column stand first in each term. 2.1 Further properties in 2 × 2 case Let us also present some other properties of 2×2 Manin matrices. It is well-known that in commutative case a matrix can be conjugated to the so-called Frobenius normal form. Let us show that the same is possible for Manin matrices, see also page 53. 7 Observation 5 Frobenius form of a matrix. −1 1 0 a b 1 0 0 1 = , (18) a b c d a b −(ad−cb) a+d (cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19) iff [d,a] = [b,c] and db = bd. So we got two of three Manin’s relations, to get the last third relation see example 11 page 54 . Let us denote the matrix at the right hand side of (11) by M and the first matrix at the left Frob hand side by D. To see that (11) is true we just write the following. 1 0 a b a b D M = = . (19) a b c d a2 +bc ab+bd (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) 0 1 1 0 a b M D = = = (20) Frob −(ad−cb) a+d a b −(ad−cb)+a2 +da ab+db (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) a b = . (21) a2 +bc+([d,a]−[b,c]) ab+bd+[d,b] (cid:18) (cid:19) Observation 6 Inversion . The two sided inverse of a Manin matrix M is Manin, and det(M−1) = (det M)−1. See theorem 1 page 25 and corollary 2 page 30 . Let us briefly prove this fact. From the Cramer’s rule above, one knows the formula for the left inverse, by assumption it is also right inverse. To prove the theorem one only needs to write explicitly that theright inverse isgiven byCramer rule andthedesired commutation relations appear automatically. Explicitly, from Cramer’s formula (see Observation 2) we see that: 1 d −b a b 1 0 = . (22) ad−cb −c a c d 0 1 (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19) One knows that if both left and right inverses exist then associativity guarantees that they coincide: a−1 = a−1(aa−1) = (a−1a)a−1 = a−1. So assuming that the right inverse to A exists, and denoting l l r l r r ad−cb ≡ δ we have: 1 0 a b d −b a(δ)−1d−b(δ)−1c −a(δ)−1b+b(δ)−1a = (δ−1) = . (23) 0 1 c d −c a c(δ)−1d−d(δ)−1c −c(δ)−1b+d(δ)−1a (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) Let us multiply the identity above by δ−1 on the left. We have: (δ)−1 = (δ)−1a(δ)−1d−(δ)−1b(δ)−1c, Element (1,1), (24) (δ)−1 = −(δ)−1c(δ)−1b+(δ)−1d(δ)−1a, Element (2,2). (25) So we see that (det(M))−1 equals to det(M−1) and the last does not depend on the ordering of columns. Moreover, equating (24) to (25) yields (δ)−1a(δ)−1d−(δ)−1b(δ)−1c = −(δ)−1c(δ)−1b+(δ)−1d(δ)−1a, (26) hence: [(δ)−1a,(δ)−1d] = [(δ)−1b,(δ)−1c]. (27) 8 So the commutators of the cross-terms of M−1 are equal. From the non-diagonal elements of equality 23 multiplied on the left by δ−1 we have: (δ)−1c(δ)−1d−(δ)−1d(δ)−1c = 0, −(δ)−1a(δ)−1b+(δ)−1b(δ)−1a = 0,hence [(δ)−1c,(δ)−1d] = 0 [(δ)−1a,(δ)−1b] = 0. (28) So we also have the column commutativity of the elements of M−1. Hence the proposition is proved in 2×2 case. (cid:3) Observation 7 A puzzle with det(M) = 1. Let M be a 2×2 Manin matrix, suppose that det(M) is central element and it is invertible (for example det(M) = 1). Then all elements of M commute among themselves. From the observation 6 above one gets that 1 d −b M−1 = (29) det(M) −c a (cid:18) (cid:19) (cid:3) is again a Manin matrix. This gives the commutativity. This is quite a surprising fact that: imposing only one condition we “kill” the three commutators: [a,b],[a,d] = [c,b],[c,d]. In the paper we will consider n×n Manin matrices and prove that these and other properties of linear algebra work for them. 3 Manin matrices. Definitions and elementary properties InthissectionwerecallthedefinitionofManinmatrices andgivesomebasicproperties. Thematerial israthersimpleone, butitisnecessary forthesequel. Firstwewillgiveanexplicit definitionofManin matrices(in terms of commutation relations), and then we will provide a more conceptual point of view which defines themby the coactionproperty onthe polynomial andGrassmanalgebras. (This is the original point of view of Manin). We also explain the relation of Manin matrices with quantum groups. As it was shown by Yu. Manin there exists natural definition of the determinant which satisfies most of the properties of commutative determinants; this will be also recalled below. The mainreference forthispartisYu. Manin’sbook[Man88],aswellas[Man87,Man91,Man92,MMDZ]. 3.1 Definition Definition 2 Let us call a matrix M with elements in an associative ring K a ”Manin matrix” if the properties below are satisfied: • elements which belong to the same column of M commute among themselves • commutators of cross terms are equal: ∀p,q,k,l [M ,M ] = [M ,M ], pq kl kq pl e.g. [M ,M ] = [M ,M ], [M ,M ] = [M ,M ]. 11 22 21 12 11 2k 21 1k Remark 1 The second condition for the case q = l obviously implies the first one. Nevertheless we deem it more convenient for the reader to formulate it explicitly. 9