Jordan-Kronecker invariants of finite-dimensional Lie algebra

A. Vorontsov (7159406)

Alexey Bolsinov (1258812)

Anton Izosimov (7159013)

D. Dowell (7159409)

I. Kozlov (7159412)

Pumei Zhang (7158701)

English

Loughborough University Institutional Repository

Jordan-Kronecker invariantsof finite-dimensional Lie algebrasAlexey BolsinovLoughborough University(joint with A. Vorontsov, P. Zhang, D. Dowell, I. Kozlov andA. Izosimov)Integrability, Recursion, Geometry And MechanicsCelebrating Franco Magri’s 70th birthdaySeptember 5-9, 2016 at RISM – Villa Toeplitz – VareseOutlineI Motivation and Generalised Argument Shift ConjectureI Jordan-Kronecker decomposition theoremI Jordan-Kronecker invariants of a finite-dimensional Lie algebraI General facts about JK invariantsI Polynomiality of the algebra of Ad ∗-invariantsI ExamplesMotivationI It is a very natural idea.A Lie algebra is defined by its structure tensor ckij . Too complicated!Classical approach: Take ad ξ =∑i ckij ξi for generic ξ ∈ g andthen study the properties of this operator.Another option: Take a form Ax =∑k ckij xk for generic x ∈ g∗.Unfortunately, the only invariant is the rank of rankAx . But. . .non-trivial invariants appear if we consider a pair of formsAx and Aa, x , a ∈ g∗. Can we get anything interesting?I Many classical facts become more transparent and some new resultscan be derived.I Arbitrary Lie algebras, not necessarily semisimple.I Generalised argument shift conjectureHistorical remarks1. The first and main ingredient is the argument shift methodsuggested by A. Mischenko and A. Fomenko in 1976 as ageneralisation of S. Manakov construction.2. In 1988, I. Gelfand and I. Zakharevich observed a very importantrelationship between compatible Poisson brackets andJordan–Kronecker decomposition theorem and used it to study theso-called micro-Kronecker pencils and their applications.3. Jordan–Kronecker decomposition for a pencil of skew-symmetricforms (C. Jordan, K. Weierstrass, L. Kronecker, F. Gantmacher,G. Gurevich, R. Thompson, . . . ).4. Transition from algebraic canonical forms of pencils in linear algebrato normal forms of compatible Poisson brackets was done byF.-J. Turiel (series of papers since 1989).5. The idea of JK invariants was conceptualised in the framework of aninformal Moscow–Loughborough research seminar for PhD students(A. Izosimov, P. Zhang, A. Konjaev, A. Vorontsov, I. Kozlov,D. Dowell).6. This talk is based on our joint paper with Pumei Zhang.ConjectureIn the two forms Ax Aa, one can easily recognise two famous compatiblePoisson structures on the dual space g∗ of a Lie algebra g.The first is the standard Lie-Poisson bracket:{f , g}(x) = Ax(df (x), dg(x)) =∑(ckij xk) ∂f∂xi∂g∂xj.The second is:{f , g}a(x) = Aa(df (x), dg(x)) =∑(ckij ak) ∂f∂xi∂g∂xj, a ∈ g∗ (fixed).Consider the following family of functions on g∗ (algebra of shifts):Fa = {f (x + λa) | f ∈ I(g), λ ∈ R}.Theorem (Mischenko, Fomenko)1) Fa is commutative w.r.t. both { , } and { , }a.2) If g is semisimple, then Fa is complete.Mischenko-Fomenko conjecture.For any g, there is a complete commutative family G ⊂ P(g) ofpolynomials in involution.ConjectureIn the two forms Ax and Aa, one can easily recognise two famouscompatible Poisson structures on the dual space g∗ of a Lie algebra g.The first is the standard Lie-Poisson bracket:{f , g}(x) = Ax(df (x), dg(x)) =∑(ckij xk) ∂f∂xi∂g∂xj.The second is:{f , g}a(x) = Ax(df (x), dg(x)) =∑(ckij ak) ∂f∂xi∂g∂xj, a ∈ g∗ (fixed).Consider the following family of functions on g∗ (algebra of shifts):Fa = {f (x + λa) | f ∈ I(g), λ ∈ R}.Theorem (Mischenko, Fomenko)1) Fa is commutative w.r.t. both { , } and { , }a.2) If g is semisimple, then Fa is complete.Generalised Argument Shift Conjecture.For any g, there is a complete commutative family Ga ⊂ P(g) ofpolynomials in bi-involution.Jordan-Kronecker decompositionThis theorem gives a classification of skew-symmetric bilinear forms A,Bon a finite-dimensional vector space by reducing them simultaneously toan elegant block-diagonal form.TheoremLet A and B be two skew-symmetric bilinear forms on a complex vectorspace V . Then by an appropriate choice of a basis, their matrices can besimultaneously reduced to the following canonical block-diagonal form:A 7→A1A2. . .Ak , B 7→B1B2. . .Bkwhere the pairs of the corresponding blocks Ai and Bi can be of thefollowing three types:Jordan-Kronecker decompositionAi BiJordan block(λi ∈ C) J(λi )−J>(λi )  −IdIdJordan block(λi =∞) −IdId  J(0)−J>(0)Kroneckerblock1 0. . .. . .1 0−10. . .. . . −100 1. . .. . .0 10−1 . . .. . . 0−1where J(λi ) denotes the standard Jordan block.Some commentsI Characteristic numbers λi play the same role as “eigenvalues” in thecase of linear operators (recursion operators). More precisely, λi arethose numbers for which the rank of Aλ = A+ λB for λ = λi is notmaximal.I If µ 6= λi , then Aµ = A+ µB is called regular (in the pencilP = {Aλ}).I Characteristic numbers are the roots of the characteristic polynomialfP(λ) = gcd {Pf i1...i2k (A+ λB)}, 2k = rankP.I The sizes of Kronecker blocks are odd 2ki − 1, the sizes of Jordanblocks are even 2jm. The numbers ki and jm are called Kroneckerand Jordan indices of the pencil.I The Jordan–Kronecker decomposition theorem implies the existenceof a large subspace which is isotropic simultaneously for all formsfrom the pencil P (bi-Lagrangian subspace).Some more commentsTheoremFor every pencil P = {Aλ}, there is a bi-Lagrangian subspace L ⊂ V .This means that L is isotropic with respect to all Aµ ∈ P and is maximalisotropic for all regular forms Aλ ∈ P. In particular,dim L = 12 (dimV + corankP).There is one very special bi-isotropic subspace of V :Lcan =∑λKerAλ,PropositionI Lcan is contained in any bi-Lagrangian subspace L and isthe intersection of all bi-Lagrangian subspaces.I dim Lcan =∑si=1 ki =12 (dimV + corankP)− deg fPI dim Lcan is bi-Lagrangian if and only if P is of pure Kronecker type.Definition of Jordan-Kronecker invariantsLet g be a Lie algebra and g∗ its dual space. We say that (x , a)∈g∗× g∗is a generic pair if the type of the Jordan-Kronecker decomposition ofAx =∑k ckij xk and Aa =∑k ckij ak is the same for all points in theneighborhoods of (x , a).DefinitionThe type of the Jordan-Kronecker canonical form for the pencil Ax +λAafor a generic pair (x , a)∈g∗, is called the Jordan-Kronecker invariant of g.In particular, a Lie algebra g isI of Kronecker type,I of Jordan (symplectic) type,I of mixed type,if the Jordan–Kronecker decomposition of a (generic) pencil Ax + λAaconsists ofI only Kronecker blocks,I only Jordan blocksI both Jordan and Kronecker blocks.The Kronecker and Jordan indices of a generic pencil {Ax + λAa} aresaid to be the Kronecker and Jordan indices of g.Prerequisites: basic notionsI Finite-dimensional Lie algebra g and its dual space g∗I Lie-Poisson bracket on g∗:{f , g}(x) = 〈x , [df (x), dg(x)]〉I Annihilator of a ∈ g∗:Ann a = {ξ ∈ g | ad ∗ξa = 0}I fg(x) = gcd {Pf i1...i2k (Ax)}, the fundamental semi-invariant of g.I Index of g is the codimension of generic Ad ∗-orbits:ind g = minx∈g∗ dimAnn xIf ind g = 0, then the Lie algebra g is called Frobenius.I Singular setSing = {y ∈ g∗ | dimAnn (x) > ind g}Kronecker caseTheoremThe following properties of a Lie algebra g are equivalent1. g is of Kronecker type, i. e. the Jordan–Kronecker decomposition ofthe (generic) pencil Ax + λAa consists of Kronecker blocks only,2. codimSing ≥ 2, whereSing = {y ∈ g∗ | dimAnn y > ind g} ⊂ g∗is the singular subset of g∗,3. the algebra of shifts Fa is complete.ExampleLet g be semisimple. Then g is of Kronecker type and the Kroneckerindices k1, . . . , ks , s = ind g, are exactly the degrees of the basic Casimirsf1, . . . , fs (invariants of the adjoint representation).This property holds for many other classes of Lie algebras, e. g.,e(n) = so(n) + Rn.Jordan caseTheoremThe following properties of a Lie algebra g are equivalent1. g is of Jordan type, i.e. the Jordan–Kronecker decomposition of thegeneric pencil Ax + λAa consists of Jordan blocks only,2. a generic form Ax is non-degenerate, i.e., ind g = 0 and g is aFrobenius Lie algebra,3. Fa is trivial, i.e., Fa = C.General caseProposition1) The number of Kronecker blocks in the JK decomposition is equal tothe index of g.2) The number of trivial Kronecker blocks is greater or equal to thedimension of the center of g.3) The number of independent functions in the family of shifts Fa isequal to∑ki where ki are the Kronecker indices of g or, equivalently,tr.deg.Fa = 12(dim g + ind g)− deg pg.Theorem (A. Vorontsov, 2011)Let f1(x), f2(x), . . . , fs(x) ∈ P(g) be algebraically independent polynomialAd ∗-invariants of g, s = ind g, and m1 ≤ m2 ≤ · · · ≤ ms be theirdegrees, mi = deg fi . Thenmi ≥ ki ,where k1 ≤ k2 ≤ · · · ≤ ks are Kronecker indices of the Lie algebra g.In the semisimple case (but not only!): mi = ki .Polynomiality of the algebra I(g) of Ad ∗-invariantsLet f1, . . . , fs , s = ind g, be algebraically independent Ad∗-invariantpolynomials. Thens∑i=1deg fi ≥s∑i=1ki =12(dim g + ind g)− deg fg.For many classes of Lie algebras, this estimate becomes an equality(known as a sum rule) which (surprisingly) implies that the differentialsdf1, . . . , dfs are linearly independent at x ∈ g∗ if and only if x /∈ Sing1.TheoremLet k1 ≤ · · · ≤ ks be the Kronecker indices of g and f1, . . . , fs ∈ I(g) bealgebraically independent Ad ∗-invariant polynomials withdeg f1 ≤ deg f2 ≤ · · · ≤ deg fs , s = ind g. Assume that g is unimodularand fg ∈ I(g). Then the following conditions are equivalent:1. ki = deg fi , i = 1. . . . , s;2.∑si=1 deg fi =12 (dim g + ind g)− deg fg;3. I(g) is polynomial on f1, . . . , fs .Characteristic numbers and singular setI The characteristic numbers λi of the Lie algebra g∗ are, bydefinition, the characteristic numbers for a generic pencil Ax + λAa,(x , a) ∈ g∗ × g∗.I The characteristic numbers λi = λi (x , a) are defined by thefollowing algebraic condition x + λia ∈ Sing.I The characteristic numbers exist if and only if codim Sing = 1.I The codimension one component of the singular set Sing is definedby one polynomial equation fg(x) = 0, where fg(x) is thefundamental invariant of g. The rest of Sing will be denoted bySing1.I The characteristic numbers of g are the roots of fg(x + λa) = 0.I The characteristic numbers are in bi-involution.Frobenius Lie algebrasLet g be Frobenius, i.e., ind g = 0. Then Sing is defined by onepolynomial, namely, fg(x) = Pf(Ax) =√det(ckij xk). The degree of thispolynomial is 12 dim g.TheoremLet g be a Frobenius Lie algebra, and the (geometric) degree ofSing ⊂ g∗ be equal to k = 12 dim g.Then a generic pencil Ax + λAa is diagonalisable (i.e. has no Jordanblocks of size greater than 2× 2), all characteristic numbers are distinct,and the coefficients of the “characteristic polynomial” p(λ) = Pf Ax+λaform a complete family of polynomials in bi-involution.Mixed caseConsider the polynomial fg(x), substitute x + λa and consider it as apolynomial in λ:f(λ) = fg(x + λa) = g0(x) + λg1(x) + λ2g2(x) + · · ·+ λmgm(x).The homogeneous polynomials g0(x), . . . , gm(x) are obviously thesymmetric polynomials of characteristic numbers. So they are inbi-involution and, moreover, they are in bi-involution with the algebra ofshifts Fa.Combining the collection of gk ’s with the algebra of shifts Fa, we obtainan extended algebra of functions in bi-involution Ga.Question. Is Ga complete?Theorem (A.Isosimov)The algebra Ga is complete if and only if, the annihilator of a genericsingular element is isomorphic to a2 ⊕ centre, where a2 is thetwo-dimensional non-Abelian Lie algebra.Elashvili conjectureLet Ann a = {ξ ∈ g | ad ∗ξa = 0} be the stationary subalgebra of a ∈ g∗with respect to the coadjoint representation. The following estimate iswell-known:indAnn a ≥ ind gElashvili conjecture: if g is semisimple, then indAnn a = ind g for alla ∈ g = g∗.Interpretation in terms of Jordan-Kronecker decomposition:PropositionLet a ∈ g∗ be fixed and x ∈ g∗ is generic in the sense that the type of theJordan-Kronecker decomposition of Ax and Aa does not change in acertain neighborhood of x . ThenindAnn a = ind gif and only if the Jordan–Kronecker decomposition does not contain anynon-trivial Jordan blocks, i.e., the Jordan part is diagonalisable.Otherwise, i.e. if there are non-trivial Jordan blocks, we have stronginequality:indAnn a > ind gExamplesI Semisimple Lie algebras:Pure Kronecker case, the Kronecker indices ki coincide with thedegrees of mi of basis Casimirs:ki = mi .For classical series these numbers mi are:I An: 2, 3, 4, . . . , n + 1;I Bn: 2, 4, 6, . . . , 2n;I Cn: 2, 4, 6, . . . , 2n;I Dn: 2, 4, 6, . . . , 2n − 2 and n.I Some semidirect sums.I e(n) = so(n) + Rn:JK invariants are the same as those for so(n + 1).I g = sl(n) + Rn:Pure Kronecker type with one single Kronecker block, i.e.,k1 =12(dim g+ 1).I k+ρ V with k simple and ρ irreducible:Pure Kronecker type(F. Knop, P. Littelmann, B. Priwitzer, AB)I Lie algebras of low dimension ≤ 5. The complete list of JKinvariants is obtained by P. Zhang.ExamplesI Affine Lie algebra aff(n) = gl(n) + Rn. The PfaffianPf(Ax) =√det(ckij xk)is irreducible and we can apply one ofabove theorems. The pencil Ax + λAa is of Jordan type,diagonalisable, with 12 dim g distinct characteristic number. In otherwords, the Jordan indices of g are1, 1, . . . , 1︸ ︷︷ ︸k, k =12(n2 + n) =12dim aff(n).I Another interesting example g = gl(n) + Rn2 , where Rn2 is realisedas the space of n × n-matrices, and the action of gl(n) on it is leftmultiplication. The matrix realisation of g is:(A C0 0).This Lie algebra is Frobenius, the singular set is defined bydetC = 0. Hence, we have n distinct characteristic numbers, themultiplicity of each of them is 2n, and the Jordan indices are1, 1, . . . , 1︸ ︷︷ ︸n − 2, 2Triangular Lie algebraLet tn be the Lie algebra of upper triangular n × n matrices.The description of Jordan-Kronecker invariants for tn easily follows fromthe results by A.Arkhangelskii.If n is even, then tn is of mixed type. The coadjoint invariants are rationalfunctions fk =PkQk, k = 1, . . . , n2 with degPk = k and degQk = k − 1.The Kronecker indices are exactly degPk + degQk , namely1, 3, 5, . . . , n − 1.The singular set Sing ⊂ t∗n is defined by an irreducible polynomial f ofdegree n2 . Therefore, tn possessesn2 distinct characteristic numbers, eachof multiplicity one. In particular, the Jordan part of a generic pencilAx+λa is diagonalisable and Jordan indices are 1, . . . , 1 ( n2 times).If n is odd, then tn is of Kronecker type and the Kronecker indices are1, 3, 5, . . . , n.Important: For all these Lie algebras, the Generalised Argument ShiftConjecture holds.Many thanksfor your attention

Jordan-Kronecker invariants of finite-dimensional Lie algebras

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Jordan-Kronecker invariants of finite-dimensional Lie algebras

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