Periodic automorphisms, compatible Poisson brackets, and Gaudin subalgebras
aa r X i v : . [ m a t h . R T ] F e b February 19, 2021
PERIODIC AUTOMORPHISMS, COMPATIBLE POISSON BRACKETS, ANDGAUDIN SUBALGEBRAS
DMITRI I. PANYUSHEV AND OKSANA S. YAKIMOVA
To the memory of our teacher Ernest Borisovich Vinberg C ONTENTS
Introduction 11. Preliminaries on Poisson brackets and polynomial contractions 42. Automorphisms of finite order and compatible Poisson brackets 63. Poisson-commutative subalgebras of S ( g ) : the semisimple case 84. On algebraically independent generators 145. On the maximality problem 196. The case of a cyclic permutation 217. Gaudin subalgebras 238. Fixed-point subalgebras in the infinite dimensional case 25References 28I NTRODUCTION
The ground field k is algebraically closed and char ( k ) = 0 . Let q = ( q , [ , ]) be a finite-dimensional algebraic Lie algebra, i.e., q = Lie Q , where Q is a connected affine algebraicgroup. The dual space q ∗ is a Poisson variety, i.e., the algebra of polynomial functions on q ∗ , k [ q ∗ ] ≃ S ( q ) , is equipped with the Lie–Poisson bracket { , } . Here { x, y } = [ x, y ] for x, y ∈ q . Poisson-commutative subalgebras of k [ q ∗ ] are important tools for the study ofgeometry of the coadjoint representation of Q and representation theory of q .0.1. There is a well-known method, the Lenard–Magri scheme , for constructing “large”Poisson-commutative subalgebras of k [ q ∗ ] , which is related to compatible Poisson brackets ,see e.g. [GZ00]. Two Poisson brackets { , } ′ and { , } ′′ are said to be compatible , if any linearcombination { , } a,b := a { , } ′ + b { , } ′′ with a, b ∈ k is a Poisson bracket. Then one definesa certain dense open subset Ω reg ⊂ k that corresponds to the regular brackets in the pencil Mathematics Subject Classification.
Key words and phrases. index of Lie algebra, contraction, commutative subalgebra, symmetric invariants.The first author is funded by RFBR, project } P = {{ , } a,b | ( a, b ) ∈ k } . If Z a,b ⊂ S ( q ) denotes the Poisson centre of ( S ( q ) , { , } a,b ) ,then the subalgebra Z ⊂ S ( q ) generated by Z a,b with ( a, b ) ∈ Ω reg is Poisson-commutativew.r.t. { , } ′ and { , } ′′ , see Section 2.2 for more details. An obvious first step is to take theinitial Lie–Poisson bracket { , } as { , } ′ . The rest depends on a clever choice of { , } ′′ . Thegoal of this article is to introduce a new class of compatible Poisson brackets, study therespective subalgebras Z , and provide applications.Let us recall some known pencils of compatible Poisson brackets.• For any γ ∈ q ∗ , the Poisson bracket ( x, y )
7→ { x, y } γ := γ ([ x, y ]) , x, y ∈ q , is com-patible with { , } , cf. [DZ05, Example 1.8.16]. This leads to the fabulous Mishchenko–Fomenko (= MF) subalgebras Z = A γ ⊂ S ( q ) , which first appear in [MF78] for semisimple q . To the best of our knowledge, the very term “Mishchenko–Fomenko” is coined by Vin-berg [Vi90].• In [PY], we introduced compatible Poisson brackets related to a Z -grading q = q ⊕ q and studied the respective subalgebra Z = Z ( q , q ) . Here the second bracket isdefined by the relations { x, y } ′′ = { x, y } if x ∈ q and { x, y } ′′ = 0 if x, y ∈ q . As wellas with MF subalgebras, one cannot get too far if q is arbitrary. Assuming that q = g isreductive, we obtained a number of interesting results on Z ( g , g ) . We proved that:– Z ( g , g ) is a Poisson-commutative subalgebra of S ( g ) g having the maximal possibletranscendence degree, which equals tr . deg Z ( g , g ) = (dim g + rk g + rk g ) / ;– with only four exceptions related to exceptional Lie algebras, Z ( g , g ) is a polynomialalgebra whose algebraically independent generators are explicitly described;– if g is a classical Lie algebra and g contains a regular nilpotent element of g , then Z ( g , g ) is a maximal Poisson-commutative subalgebra of S ( g ) g .The proofs exploit numerous invariant-theoretic properties of the adjoint representationof g [K63] and their analogues for the isotropy representation G → GL( g ) [KR71].0.2. Results of the present paper stem from a surprising observation that if q is equippedwith a Z m -grading with any m > , then one can naturally construct a compatible Poissonbracket { , } ′′ (Section 2). In this case, all Poisson brackets in P are linear and there aretwo lines l , l ⊂ k such that Ω = k \ ( l ∪ l ) ⊂ Ω reg and the Lie algebras corresponding to ( a, b ) ∈ Ω are isomorphic to q . The lines l and l give rise to new Lie algebras, denoted q (0) and q ( ∞ ) . These new algebras are different contractions of q . Let ind q denote the index of q (see Section 1). Then ind q ind q ( n ) , n ∈ { , ∞} , and our first task is to realise whetherit is true that ind q = ind q ( n ) . Although basic theory can be developed for arbitrary q ,essential applications require a better class of Lie algebras, and we eventually stick to thesemisimple case. Let ϑ ∈ Aut ( g ) be an automorphism of order m and g = L m − j =0 g j theassociated Z m -grading, i.e., if ζ = m √ is primitive, then g j = { x ∈ g | ϑ ( x ) = ζ j x } . ERIODIC AUTOMORPHISMS AND COMPATIBLE BRACKETS 3
The invariant-theoretic base for our consideration is Vinberg’s theory of “ ϑ -groups” (i.e.,theory of orbits and invariants for representations of reductive groups related to the peri-odic automorphisms of g ), see [Vi76, Vi79].A bad news is that, for m > , the invariant-theoretic picture related to ϑ and propertiesof g (0) become more complicated. For instance, if m = 2 , then g (0) ≃ g ⋉ g ab (semi-directproduct) and it is known that here ind g (0) = ind g = rk g [P07]. For m > , the number ind g (0) remains mysterious. We suspect that it is equal to rk g for any ϑ . Other techni-cal difficulties are discussed in Section 3.3. Nevertheless, we succeeded in computing ind g ( ∞ ) , see Theorem 3.2, and can state that ind g ( ∞ ) = rk g if and only if g is abelian.A good news is that there are still many interesting cases (automorphisms ϑ ), whereanalogues of results of [PY] are valid and also some unexpected applications pop up.Write Z ( g , ϑ ) for the Poisson-commutative subalgebra associated with the pencil of com-patible Poisson brackets related to ϑ . Let N be the set of nilpotent elements of g . Supposethat ϑ has the following properties:(1) g contains a regular semisimple element of g ;(2) each irreducible component of g ∩ N contains a regular nilpotent element of g .Under these assumptions we prove that Z ( g , ϑ ) has the same properties as the above al-gebra Z ( g , g ) , see Sections 4, 5.0.3. Another good news is that there is a special case of ( g , ϑ ) , when Z ( g , ϑ ) is as good aspossible and it has a nice quantisation. Namely, let h be a non-abelian simple Lie algebraand g = h m be the direct sum of m copies of h . If ϑ is the cyclic permutation of thesummands, then g = ∆ h ≃ h and we prove in Section 6 that Z = Z ( h m , ϑ ) is a polynomialring in (( m −
1) dim h + ( m + 1)rk h ) generators. Furthermore, Z is a maximal Poisson-commutative subalgebra of S ( g ) g . The quantisation problem asks for a lift of Z to theenveloping algebra U ( g ) , i.e., for a commutative algebra ˜ Z ⊂ U ( g ) such that gr (˜ Z ) = Z .To describe ˜ Z in this context, we need some preparations.The enveloping algebra U ( h [ t, t − ]) of the loop algebra h [ t, t − ] contains a large com-mutative subalgebra z ( b h ) of infinite transcendence degree, known as the Feigin–Frenkelcentre [FF92]. Actually z ( b h ) ⊂ U ( b h − ) , where b h − = t − h [ t − ] . Having a vector ~z ∈ ( k ⋆ ) m ,one defines a homomorphism ρ ~z : U ( b h − ) → U ( g ) . The image of z ( b h ) under ρ ~z is calledthe Gaudin subalgebra G ( ~z ) [FFR]. For the case, where the entries of ~z are pairwise distinct m -th roots of unity, we provide a simpler construction of G ( ~z ) (Proposition 7.2) and showthat gr( G ( ~z )) = Z ( h m , ϑ ) , see Theorem 7.4.It is worth noting that, for the MF subalgebras A γ ⊂ S ( h ) , the quantisation problemwas posed by Vinberg [Vi90]. A solution given by Rybnikov [R06] states that the imageof z ( b h ) under a certain homomorphism ̺ γ : U ( b h − ) → U ( h ) , depending on γ ∈ h ∗ , is the quantum MF subalgebra ˜ A γ and one has gr ( ˜ A γ ) = A γ in many cases. D. PANYUSHEV AND O. YAKIMOVA
Let us identify b h − with the quotient space h [ t, t − ] / h [ t ] . Then h [ t ] acts on t − h [ t − ] andcorrespondingly on S ( b h − ) . By [FF92, F07], one has(0 · gr( z ( b h )) = S ( b h − ) h [ t ] . Yet another property of the Poisson-commutative algebra gr( z ( b h )) is that it is a polyno-mial ring in infinitely many variables, by a direct generalisations of a Ra¨ıs–Tauvel theo-rem [RT92].Suppose that ϑ ∈ Aut ( h ) . In Section 8, we consider the ϑ -twisted version of gr( z ( b h )) ,a certain subalgebra Z ( b h − , ϑ ) of S (( b h − ) ϑ ) of infinite transcendence degree. Assuming theequality ind h (0) = rk h , we prove that Z ( b h − , ϑ ) is Poisson-commutative, see Theorem 8.2.In many cases, Z ( b h − , ϑ ) is a polynomial ring.Our general reference for semisimple Lie groups and algebras is [Lie3].1. P RELIMINARIES ON P OISSON BRACKETS AND POLYNOMIAL CONTRACTIONS
Let Q be a connected affine algebraic group with Lie algebra q . The symmetric algebra of q over k is N -graded, i.e., S ( q ) = L i > S i ( q ) . It is identified with the algebra of polynomialfunctions on the dual space q ∗ , and we also write k [ q ∗ ] = L i > k [ q ∗ ] i for it.1.1. The coadjoint representation.
The group Q acts on q ∗ via the coadjoint represen-tation and then ad ∗ : q → GL( q ∗ ) is the coadjoint representation of q . The algebra of Q -invariant polynomial functions on q ∗ is denoted by S ( q ) Q or k [ q ∗ ] Q . Write k ( q ∗ ) Q for thefield of Q -invariant rational functions on q ∗ .Let q ξ = { x ∈ q | ad ∗ ( x ) · ξ = 0 } be the stabiliser in q of ξ ∈ q ∗ . The index of q , ind q , is the minimal codimension of Q -orbits in q ∗ . Equivalently, ind q = min ξ ∈ q ∗ dim q ξ .By the Rosenlicht theorem (see [Sp89, IV.2]), one also has ind q = tr . deg k ( q ∗ ) Q . Set b ( q ) = (dim q + ind q ) / . Since the Q -orbits in q ∗ are even-dimensional, b ( q ) is an integer.If q is reductive, then ind q = rk q and b ( q ) equals the dimension of a Borel subalgebra.The Lie–Poisson bracket in S ( q ) is defined on S ( q ) = q by { x, y } := [ x, y ] . It is thenextended to higher degrees via the Leibniz rule. Hence S ( q ) has the usual associative-commutative structure and additional Poisson structure. Whenever we refer to subalge-bras of S ( q ) , we always mean the associative-commutative structure. Then a subalgebra A ⊂ S ( q ) is said to be Poisson-commutative , if { H, F } = 0 for all H, F ∈ A . It is well knownthat if A is Poisson-commutative, then tr . deg A b ( q ) , see e.g. [Vi90, 0.2]. More gener-ally, suppose that h ⊂ q is a Lie subalgebra and A ⊂ S ( q ) h is Poisson-commutative. Then tr . deg A b ( q ) − b ( h ) + ind h , see [MY19, Prop. 1.1].The centre of the Poisson algebra ( S ( q ) , { , } ) is Z ( q ) := { H ∈ S ( q ) | { H, F } = 0 ∀ F ∈ S ( q ) } . ERIODIC AUTOMORPHISMS AND COMPATIBLE BRACKETS 5
Using the Leibniz rule, we obtain that Z ( q ) is a graded Poisson-commutative subalgebraof S ( q ) , which coincides with the algebra of symmetric invariants of q , i.e., Z ( q ) = { H ∈ S ( q ) | { H, x } = 0 ∀ x ∈ q } =: S ( q ) q = k [ q ∗ ] q . As Q is connected, we have S ( q ) q = S ( q ) Q = k [ q ∗ ] Q . Since the quotient field of k [ q ∗ ] Q iscontained in k ( q ∗ ) Q , we deduce from the Rosenlicht theorem that(1 · tr . deg ( S ( q ) q ) ind q . The set of Q - regular elements of q ∗ is(1 · q ∗ reg = { η ∈ q ∗ | dim q η = ind q } = { η ∈ q ∗ | dim Q · η is maximal } . It is a dense open subset of q ∗ . Set q ∗ sing = q ∗ \ q ∗ reg . We say that q has the codim– n propertyif codim q ∗ sing > n . The codim– property is going to be most important for us.For γ ∈ q ∗ , let ˆ γ be the skew-symmetric bilinear form on q defined by ˆ γ ( ξ, η ) = γ ([ ξ, η ]) for ξ, η ∈ q . It follows that ker ˆ γ = q γ . The -form ˆ γ is related to the Poisson tensor (bivector) π of the Lie–Poisson bracket { , } as follows.Let d H denote the differential of H ∈ S ( q ) = k [ q ∗ ] . Then π is defined by the formula π ( d H ∧ d F ) = { H, F } for H, F ∈ S ( q ) . Then π ( γ )( d γ H ∧ d γ F ) = { H, F } ( γ ) and therefore ˆ γ = π ( γ ) . In this terms, ind q = dim q − rk π , where rk π = max γ ∈ q ∗ rk π ( γ ) .1.2. Contractions and invariants.
We refer to [Lie3, Ch. 7, § 2] for basic facts on contrac-tions of Lie algebras. In this article, we consider contractions of the following form. Let k ⋆ = k \ { } be the multiplicative group of k and ϕ : k ⋆ → GL( q ) , s ϕ s , a polynomialrepresentation. That is, the matrix entries of ϕ s : q → q are polynomials in s w.r.t. some(any) basis of q . Define a new Lie algebra structure on the vector space q and associatedLie–Poisson bracket by(1 · [ x, y ] ( s ) = { x, y } ( s ) := ϕ − s [ ϕ s ( x ) , ϕ s ( y )] , x, y ∈ q , s ∈ k ⋆ . The corresponding Lie algebra is denoted by q ( s ) . Then q (1) = q and all these algebras areisomorphic. The induced k ⋆ -action in the variety of structure constants in not necessarilypolynomial, i.e., lim s → [ x, y ] ( s ) may not exist for all x, y ∈ q . Whenever such a limitexists, we obtain a new linear Poisson bracket, denoted { , } , and thereby a new Liealgebra q (0) , which is said to be a contraction of q . If we wish to stress that this constructionis determined by ϕ , then we write { x, y } ( ϕ,s ) for the bracket in (1 ·
3) and say that q (0) isthe ϕ - contraction of q or is the zero limit of q w.r.t. ϕ . A criterion for the existence of q (0) can be given in terms of Lie brackets of the ϕ -eigenspaces in q , see [Y17, Section 4]. Weidentify all algebras q ( s ) and q (0) as vector spaces. The semi-continuity of index impliesthat ind q (0) > ind q .The map ϕ s , s ∈ k ⋆ , is naturally extended to an invertible transformation of S j ( q ) ,which we also denote by ϕ s . The resulting graded map ϕ s : S ( q ) → S ( q ) is nothing but D. PANYUSHEV AND O. YAKIMOVA the comorphism associated with s ∈ k ⋆ and the dual representation ϕ ∗ : k ⋆ → GL( q ∗ ) .Since S j ( q ) has a basis that consists of ϕ ( k ⋆ ) -eigenvectors, any F ∈ S j ( q ) can be written as F = P i > F i , where the sum is finite and ϕ s ( F i ) = s i F i ∈ S j ( q ) . Let F • denote the nonzerocomponent F i with maximal i . Proposition 1.1 ([Y14, Lemma 3.3]) . If F ∈ Z ( q ) and q (0) exists, then F • ∈ Z ( q (0) ) .
2. A
UTOMORPHISMS OF FINITE ORDER AND COMPATIBLE P OISSON BRACKETS
In this section, we associate a pencil of compatible Poisson brackets to any automorphismof finite order of a Lie algebra q , describe the limit algebras q (0) and q ( ∞ ) , and constructthe related Poisson-commutative subalgebra of S ( q ) .2.1. Periodic gradings of Lie algebras.
Let ϑ ∈ Aut ( q ) be a Lie algebra automorphismof finite order m > and ζ = m √ a primitive root of unity. Write also ord ( ϑ ) for theorder of ϑ . If q i is the ζ i -eigenspace of ϑ , i ∈ Z m , then the direct sum q = L i ∈ Z m q i is a periodic grading or Z m - grading of q . The latter means that [ q i , q j ] ⊂ q i + j for all i, j ∈ Z m .Here q = q ϑ is the fixed-point subalgebra for ϑ and each q i is a q -module. We will beprimarily interested in periodic gradings of semisimple Lie algebras, but such a generalsetting is going to be useful, too.We choose { , , . . . , m − } ⊂ Z as a fixed set of representatives for Z m = Z /m Z . Underthis convention, we have q = q ⊕ q ⊕ . . . ⊕ q m − and(2 · [ q i , q j ] ⊂ q i + j , if i + j m − , q i + j − m , if i + j > m. This is needed below, when we consider Z -graded contractions of q associated with ϑ .The presence of ϑ allows us to split the Lie–Poisson bracket on q ∗ into a sum of twocompatible linear Poisson brackets, as follows. Consider the polynomial representation ϕ : k ⋆ → GL( q ) such that ϕ s ( x ) = s j x for x ∈ q j . As in Section 1.2, this defines a family oflinear Poisson brackets in S ( q ) parametrised by s ∈ k ⋆ , see (1 · Proposition 2.1.
For any ϑ ∈ Aut ( q ) of finite order m and ϕ as above, we have (i) The map ϕ − s : q → q provides an isomorphism between ( q , [ , ]) and ( q , [ , ] ( s ) ) . Inparticular, the Poisson brackets { , } ( s ) , s ∈ k ⋆ , are isomorphic; (ii) there is a limit lim s → { x, y } ( s ) =: { x, y } , which is a linear Poisson bracket in S ( q ) ; (iii) the difference { , } − { , } =: { , } ∞ is a linear Poisson bracket on S ( q ) and { , } ( s ) = { , } + s m { , } ∞ for any s ∈ k ⋆ . (iv) The Poisson bracket { , } ∞ is obtained as the zero limit w.r.t. the polynomial representa-tion ψ : k ⋆ → GL( q ) such that ψ s = s m · ϕ s − = s m · ϕ − s , s ∈ k ⋆ . In other words, we have { , } ∞ = lim s → { , } ( ψ,s ) . ERIODIC AUTOMORPHISMS AND COMPATIBLE BRACKETS 7
Proof. (i)
This readily follows from Eq. (1 · (ii) If x ∈ q i and y ∈ q j , then { x, y } ( s ) = [ x, y ] , if i + j < m ; s m [ x, y ] , if i + j > m. Therefore, the limitof { x, y } ( s ) as s tends to zero exists and is given by { x, y } := [ x, y ] , if i + j < m , if i + j > m . Thelimit of Poisson brackets is again a Poisson bracket, hence the Jacobi identity is satisfiedfor { , } , cf. Section 1.2. However, this is easily verified directly. (iii) By the above formula for { x, y } , we have { x, y } ∞ = , if i + j < m ;[ x, y ] , if i + j > m. Therefore { x, y } ( s ) = { x, y } + s m { x, y } ∞ for all x, y ∈ q . It is also easily verified that { , } ∞ satisfies the Jacobi identity. (iv) We have ψ s ( x ) = s m − i x for x ∈ q i . Then an easy calculation shows that the ( ψ, s ) -bracket is given by { x, y } ( ψ,s ) = s m [ x, y ] , if i + j < m ;[ x, y ] , if i + j > m. (cid:3) Two Poisson brackets on the algebra S ( q ) are said to be compatible , if any linear combi-nation of them is again a Poisson bracket. Actually, if { , } and { , } are Poisson brack-ets, then it suffices to check that just { , } + { , } is a Poisson bracket [PY, Lemma 1.1].Anyway, we have Corollary 2.2.
For any ϑ ∈ Aut ( q ) of finite order, the Poisson brackets { , } and { , } ∞ arecompatible, and the corresponding pencil contains the initial Lie–Poisson bracket. All Poisson brackets involved in Proposition 2.1 are linear. Therefore, in place of Pois-son brackets on the symmetric algebra S ( q ) , we can stick to the corresponding Lie algebrastructures on the vector space q . Let q ( s ) be the Lie algebra corresponding to { , } ( s ) . Thenall algebras with s ∈ k ⋆ are isomorphic, whereas the brackets { , } and { , } ∞ give riseto entirely different Lie algebras q (0) and q ( ∞ ) , respectively. Both q (0) and q ( ∞ ) are Lie al-gebra contractions of q in the sense of [Lie3, Ch. 7, § 2]. Therefore ind q (0) > ind q and ind q ( ∞ ) > ind q (the semi-continuity of the index). Proposition 2.3.
The Lie algebras q (0) and q ( ∞ ) are N -graded. More precisely, if r [ i ] stands forthe component of grade i ∈ N in an N -graded Lie algebra r , then q (0) [ i ] = q i for i = 0 , , . . . , m − otherwise , q ( ∞ ) [ i ] = q m − i for i = 1 , , . . . , m otherwise . In particular, q ( ∞ ) is nilpotent and the subspace q , which is the highest grade component of q ( ∞ ) ,belongs to the centre of q ( ∞ ) .Proof. Use the formulae for { x, y } and { x, y } ∞ from the proof of Proposition 2.1. (cid:3) D. PANYUSHEV AND O. YAKIMOVA
Poisson-commutative subalgebras related to compatible Poisson brackets.
Thereis a general method for constructing a Poisson-commutative subalgebra of S ( q ) with“large” transcendence degree that exploits compatible Poisson brackets and the centreof S ( q ) , see e.g. [DZ05, Sect. 1.8.3], [GZ00, Sect. 10], [PY, Sect. 2]. We recall this method inour present setting.By Proposition 2.1, we have { , } = { , } + { , } ∞ and { , } ( s ) = { , } + s m { , } ∞ . Since { , } ( s ) = { , } ( s ′ ) if s m = ( s ′ ) m , it is convenient to replace s m with t and set { , } t = { , } + t { , } ∞ , where t ∈ P := k ∪ {∞} and the value t = ∞ corresponds to the bracket { , } ∞ . But, wewill use the parameter s ∈ k ⋆ , when the multiplicative group ϕ : k ⋆ → GL( q ) is needed.Let q ( t ) stand for the Lie algebra corresponding to { , } t . All these Lie algebras have thesame underlying vector space. Since the algebras q ( t ) with t ∈ k ⋆ are isomorphic, theyhave one and the same index. We say that t ∈ P is regular if ind q ( t ) = ind q and write P reg for the set of regular values. Then P sing := P \ P reg ⊂ { , ∞} is the set of singular values.Let Z t be the centre of the Poisson algebra ( S ( q ) , { , } t ) . In particular, Z = S ( q ) q . If t ∈ k ⋆ , then Z t = ϕ − s ( Z ) , where s m = t . By Eq. (1 · tr . deg Z t ind q ( t ) . Ourmain object is the subalgebra Z ⊂ S ( q ) generated by the centres Z t with t ∈ P reg , i.e., Z = Z ( q , ϑ ) = alg hZ t | t ∈ P reg i . By a general property of compatible brackets, the algebra Z is Poisson-commutative w.r.t. all brackets { , } t with t ∈ P , cf. [PY, Sect. 2]. Note that the Lie subalgebra q ⊂ q = q (1) isalso the same Lie subalgebra in any q ( t ) with t = ∞ (cf. Proposition 2.3 for q (0) ). Therefore,(2 · Z t ⊂ S ( q ) q for t = ∞ . In general, one cannot say much about Z ⊂ S ( q ) . To arrive at more definite conclusionson Z , a lot of extra information on S ( q ) q , q (0) , and q ( ∞ ) is required. In particular, one hasto know whether and/or ∞ belong to P reg . And this is the reason, why we have to stickto semisimple Lie algebras.3. P OISSON - COMMUTATIVE SUBALGEBRAS OF S ( g ) : THE SEMISIMPLE CASE
From now on, G is a connected semisimple algebraic group and g = Lie G . We consider ϑ ∈ Aut ( g ) of order m > and freely use the previous notation and results, with q beingreplaced by g . In particular, g = g ⊕ g ⊕ . . . ⊕ g m − , where { , , . . ., m − } is the fixed set of representatives for Z m , and G is the connectedsubgroup of G with Lie ( G ) = g . Then g ( t ) is a family of Lie algebras parameterised by ERIODIC AUTOMORPHISMS AND COMPATIBLE BRACKETS 9 t ∈ P = k ∪ {∞} , where the algebras g ( t ) with t ∈ k ⋆ are isomorphic to g = g (1) , while g (0) and g ( ∞ ) are different N -graded contractions of g . Next, Z t is the Poisson centre of ( S ( g ) , { , } t ) and the construction of Section 2.2 provides a Poisson-commutative subalge-bra Z = alg hZ t | t ∈ P reg i ⊂ S ( g ) . The connected algebraic group corresponding to g ( t ) isdenoted by G ( t ) .Our goal is to demonstrate that there are many interesting cases, in which Z is a poly-nomial algebra having the maximal possible transcendence degree. Let us recall standardinvariant-theoretic properties of semisimple Lie algebras.The Poisson centre S ( g ) g = S ( g ) G is a polynomial algebra of Krull dimension l = rk g and ind g = l . Hence one has now the equality in Eq. (1 · g is a reductiveLie algebra. Write N for the cone of nilpotent elements of g . Let κ be the Killing form on g .We identify g and g with their duals via κ . Moreover, since κ ( g i , g j ) = 0 if i + j
6∈ { , m } ,the dual space of g j , g ∗ j , can be identified with g m − j . Here g ∗ reg = g reg = { x ∈ g | dim g x = l } .By [K63], g has the codim– property, i.e., codim g ( g \ g reg ) = 3 . Recall also that N ∩ g reg isnon-empty, and it is a sole G -orbit, the regular nilpotent orbit . Convention.
We think of g ∗ as the dual space for any Lie algebra g ( t ) and sometimesomit the subscript ‘ ( t ) ’ in g ∗ ( t ) . However, if ξ ∈ g ∗ , then the stabiliser of ξ with respect tothe coadjoint representation of g ( t ) is denoted by g ξ ( t ) .Each Lie algebra g ( t ) has its own singular set g ∗ ( t ) , sing = g ∗ \ g ∗ ( t ) , reg , which is regarded asa subset of g ∗ . If π t is the Poisson tensor of the bracket { , } t , then g ∗ ( t ) , sing = { ξ ∈ g ∗ | rk π t ( ξ ) < rk π t } , which is the union of the coadjoint G ( t ) -orbits in g ∗ having a non-maximal dimension. Forsimplicity, we write g ∗∞ , sing or g ∗∞ , reg in place of g ∗ ( ∞ ) , sing or g ∗ ( ∞ ) , reg . Proposition 3.1.
The closure of S t =0 , ∞ g ∗ ( t ) , sing in g ∗ is a subset of codimension at least .Proof. Let ξ = ξ + ξ + · · · + ξ m − ∈ g ∗ , where ξ i ∈ g ∗ i . Using Proposition 2.1(i) and thedual representation ϕ ∗ : k ⋆ → GL( g ∗ ) , s ϕ ∗ s , one readily verifies that ξ ∈ g ∗ sing = g ∗ (1) , sing if and only if ϕ ∗ s ( ξ ) ∈ g ∗ ( s ) , sing . Therefore, [ t =0 , ∞ g ∗ ( t ) , sing = [ s ∈ k ⋆ ϕ ∗ s ( g ∗ sing ) = { ξ + s − ξ + . . . + s − m ξ m − | ξ ∈ g ∗ sing , s ∈ k ⋆ } . Since codim g ∗ sing = 3 , the closure of S t =0 , ∞ g ∗ ( t ) , sing is a subset of g ∗ of codimension at least . (cid:3) In order to compute tr . deg Z , we have to elaborate on some relevant properties of thelimit Lie algebras g ( ∞ ) and g (0) .3.1. Properties of g ( ∞ ) . By Proposition 2.3, g ( ∞ ) is a nilpotent N -graded Lie algebra, hence G ( ∞ ) is a unipotent algebraic group. Recall also that the subspace g belongs to the centreof g ( ∞ ) . Theorem 3.2.
For any ϑ ∈ Aut ( g ) of finite order, one has ind g ( ∞ ) = dim g + rk g − rk g .Proof. (1) For µ ∈ g ∗ ⊂ g ∗ ( ∞ ) , the stabiliser g µ ( ∞ ) is also N -graded. Furthermore, if η ∈ g j with j m − , then κ ( µ, [ η, g m − j ] ∞ ) = 0 if and only if [ η, µ ] = 0 . Hence(3 · g µ ( ∞ ) = g µ + g , where g µ is the usual stabiliser of µ ∈ g ∗ (w.r.t. the initial Lie algebra structure on g ). Asis well-known, the reductive subalgebra g = g ϑ contains regular semisimple elements of g , see e.g. [Ka83, §8.8]. These elements form a dense open subset of g , which is denotedby Ω . If h ∈ Ω , then g h is a Cartan subalgebra of g and g h ∩ g is a Cartan subalgebraof g . Using the identification of g and g ∗ , we may think of the subset Ω ∗ of “regularsemisimple” elements of g ∗ ⊂ g ∗ ( ∞ ) . For µ ∈ Ω ∗ , it follows from (3 ·
1) that dim g µ ( ∞ ) =dim g + rk g − rk g and hence ind g ( ∞ ) dim g + rk g − rk g . (2) Let us prove the opposite inequality. We think of g ∗ ( ∞ ) as a graded vector space ofthe form g ∗ ( ∞ ) = g ∗ ⊕ g ∗ ⊕ . . . ⊕ g ∗ m − . Write ad ∗ ( ∞ ) for the coadjoint representation of g ( ∞ ) . The graded structure of g ( ∞ ) describedin Proposition 2.3 implies that ad ∗ ( ∞ ) has the property that(3 · ad ∗ ( ∞ ) ( g j ) · g ∗ i ⊂ g ∗ i + m − j . Take any ξ = P m − j =0 ξ j ∈ g ∗ ( ∞ ) such that ξ ∈ Ω ∗ . Let h ∈ Ω be the regular semisimpleelement of g corresponding to ξ under our identifications. Set t = g h . Then [ g , h ] = t ⊥ isthe orthogonal complement of t with respect to κ and g = [ g , h ] ⊕ t . For t j = t ∩ g j , thisimplies that [ g j , h ] = g j ∩ t ⊥ m − j and [ g j , h ] ⊕ t j = g j . Our goal is to prove that the orbit G ( ∞ ) · ξ contains an element γ such that dim g γ ( ∞ ) > dim g + rk g − rk g . We perform thisstep by step, as follows.For η ∈ g m − , we have ad ∗ ( ∞ ) ( η ) · ξ ∈ g ∗ and exp(ad ∗ ( ∞ ) ( η )) · ξ ∈ ξ + ( ξ + ad ∗ ( ∞ ) ( η ) · ξ ) + ( M j > g ∗ j ) . Since ad ∗ ( ∞ ) ( g m − ) · ξ = g ∗ ∩ Ann( t ) , there is η ∈ g m − such that γ := ξ +ad ∗ ( ∞ ) ( η ) · ξ ∈ t ∗ .Next, applying exp(ad ∗ ( ∞ ) ( η )) with η ∈ g m − to exp(ad ∗ ( ∞ ) ( η )) ξ = ξ + γ + ξ ′ + . . . , we donot affect the summands ξ + γ . In doing so, we can replace ξ ′ with γ ∈ t ∗ . Eventually,we get in G ( ∞ ) · ξ an element of the form γ = ξ + P m − j =1 γ j with γ j ∈ t ∗ j . It is easily seenthat t ⊂ g γ ( ∞ ) , hence dim g γ ( ∞ ) > dim g + rk g − rk g , as required. (cid:3) Corollary 3.3.
One has ∞ ∈ P reg if and only if dim g = rk g , i.e., g is an abelian subalgebraof g . ERIODIC AUTOMORPHISMS AND COMPATIBLE BRACKETS 11
Remark . (1) There is a short proof of the corollary that does not use Theorem 3.2 infull strength. If ∞ ∈ P reg , then Z ∞ ⊂ Z . Since g belongs to the centre of g ( ∞ ) , we have g ⊂ Z ∞ . Hence g has to be abelian in g . Conversely, if g is abelian, then, for µ ∈ Ω ∗ inthe first part of the proof, we obtain dim g µ ( ∞ ) = rk g . Hence ind g ( ∞ ) = rk g and ∞ ∈ P reg .(2) By V.G. Kac’s classification of elements of finite order in G [Ka83, Chap. 8], if g issimple, ϑ is inner, and g = g ϑ is abelian, then ord ( ϑ ) is at least the Coxeter number of g .This means that, for many interesting examples with small ord ( ϑ ) , we have ∞ ∈ P sing .Let us check another technical condition, which is required below. Recall from Sec-tion 1.1 that π ( γ ) = ˆ γ is a skew-symmetric bilinear form on g associated with γ ∈ g ∗ . Wethen need the rank of the restriction π ( ξ ) | V , if ξ ∈ g ∗ is generic and V = ker π ∞ ( ξ ) = g ξ ( ∞ ) .Set g > = L j> g j and g ∗ > = L j> g ∗ j . Lemma 3.5.
For any ϑ and generic ξ ∈ g ∗ , we have rk( π ( ξ ) | V ) = dim V − rk g .Proof. Take any ξ ∈ g ∗ reg ∩ g ∗∞ , reg . Then rk( π ( ξ ) | V ) dim V − rk g by [PY, Appendix] and dim V = dim g + rk g − rk g by Theorem 3.2. Write ξ = ξ + ξ ′ with ξ ∈ g ∗ and ξ ′ ∈ g ∗ > .Assume also that ξ ∈ ( g ) ∗ reg . Note that this is an open condition on ξ , too.As g ⊂ V (cf. Eq. 3 · V = g ⊕ V ′ for some V ′ ⊂ g > . Here we have dim V ′ = rk g − rk g . Let e V be the kernel of π ( ξ ) | V . Then e V ∩ g ⊂ g ξ and rk g > dim( e V ∩ g ) > dim e V − dim V ′ = dim e V + rk g − rk g . Hence dim e V rk g and thereby rk ( π ( ξ ) | V ) > dim V − rk g . This settles the claim. (cid:3) For future use, we record yet another property of g ( ∞ ) . Lemma 3.6. If ξ = ξ + ξ ′ , where ξ ′ ∈ g ∗ > , and ξ ∈ g ∗ is regular in g , then ξ ∈ g ∗∞ , reg .Proof. As in Proposition 2.1 (iv) , consider the invertible linear map ψ s : g → g such that ψ ( x j ) = s m − j x j for x j ∈ g j and s ∈ k ⋆ . It follows from Proposition 2.3 that ψ s is anautomorphism of the graded Lie algebra g ( ∞ ) . Let ψ ∗ s denote the induced action on g ∗ ( ∞ ) ,i.e., ψ ∗ s ( ξ j ) = s j − m ξ j if ξ j ∈ g ∗ j . Then dim g ξ ( ∞ ) = dim g ψ ∗ s ( ξ )( ∞ ) for any ξ ∈ g ∗ . Note that lim s → s m ψ s ( ξ ) = ξ . If ξ is regular in g ∗ ( ∞ ) , then so is ξ . It remains to deal with ξ .In view of (3 · dim g ξ ( ∞ ) = dim g + dim g ξ − dim g ξ . Because ξ is regular in g , it is alsoregular in g . Thus dim g ξ ( ∞ ) = ind g ( ∞ ) , and we are done. (cid:3) Pencils of skew-symmetric matrices and differentials.
Recall that any F ∈ S ( g ) isregarded as a polynomial function on g ∗ . Then d F is the differential of F , which is apolynomial mapping from g ∗ to g . If γ ∈ g ∗ , then d γ F ∈ g stands for the value of d F at γ .For a subalgebra A ⊂ S ( g ) and γ ∈ g ∗ , set d γ A = h d γ F | F ∈ Ai k . Then tr . deg A =max γ ∈ q ∗ dim d γ A . By the definition of Z , we have d γ Z = P t ∈ P reg d γ Z t . Since Z t is the centreof ( S ( g ) , { , } t ) , there is the inclusion d γ Z t ⊂ ker π t ( γ ) for each t ∈ P and each γ ∈ g ∗ . Let { H , . . . , H l } , l = rk g , be a set of homogeneous algebraically independent genera-tors of S ( g ) g . For any H ∈ S ( g ) g , we have d ξ H ∈ g ξ . The Kostant regularity criterion for g ,see [K63, Theorem 9], asserts that(3 · h d ξ H j | j l i k = g ξ if and only if ξ ∈ g ∗ reg .Clearly, this criterion applies to semisimple algebras g ( t ) when t = 0 , ∞ . That is,(3 ·
4) if t = 0 , ∞ , then ξ ∈ g ∗ ( t ) , reg ⇔ d ξ Z t = ker π t ( ξ ) ⇔ dim ker π t ( ξ ) = rk g . As a step towards describing Z , we first consider the smaller algebra Z × = alg hZ t | t ∈ k ⋆ i ⊂ Z . An open subset of an algebraic variety is said to be big , if its complement does not containdivisors. By Proposition 3.1, there is a big open subset U sr ⊂ g ∗ such that ξ ∈ g ∗ ( t ) , reg forany ξ ∈ U sr and any t = 0 , ∞ . Suppose that ξ ∈ U sr . Then(3 ·
5) d ξ Z × = X t ∈ k ⋆ ker π t ( ξ ) ⊂ X t : rk π t ( ξ )= l ker π t ( ξ ) =: L ( ξ ) . Let us recall a method that provides an upper bound on dim L ( ξ ) and thereby on tr . deg Z × ,see [PY20, Section 1.1].Let P ( ξ ) = { cπ t ( ξ ) | c ∈ k , t ∈ P } be a pencil of skew-symmetric -forms on g , which isspanned by π ( ξ ) and π ( ξ ) . A -form in this pencil is said to be regular if rk ( cπ t ( ξ )) = l .Otherwise, it is singular . Set U srr = U sr ∩ g ∗ (0) , reg ∩ g ∗∞ , reg . It is a dense open subset of g ∗ ,which may not be big. Proposition 3.7. If ξ ∈ U srr , then d ξ Z × = d ξ Z = L ( ξ ) .Proof. Suppose that ξ ∈ U sr . The space L ( ξ ) is the sum of kernels over the regular lines inthe pencil. Since P reg \ k ⋆ is finite, we have P t ∈ k ⋆ ker π t ( ξ ) = L ( ξ ) by [PY08, Appendix].Hence d ξ Z × = L ( ξ ) . If ξ ∈ U srr , then d ξ Z ⊂ L ( ξ ) . Thereby d ξ Z × ⊂ d ξ Z ⊂ L ( ξ ) , whichleads to the equality d ξ Z × = d ξ Z . (cid:3) Corollary 3.8.
For a generic ξ ∈ U srr , we have dim L ( ξ ) = tr . deg Z × = tr . deg Z . In particular, Z × ⊂ Z is an algebraic extension. Although three dimensions in Corollary 3.8 are equal, we do not know yet their exactvalue. For a given ξ ∈ U sr , dim L ( ξ ) depends on the number of singular lines in P ( ξ ) . Proposition 3.9 ([PY20, Proposition 1.5]) . If ξ ∈ U sr , then (i) dim L ( ξ ) = b ( g ) if and only if all nonzero -forms in P ( ξ ) are regular; (ii) if P ( ξ ) contains a singular line, say k π t ′ ( ξ ) , then dim L ( ξ ) l + rk π t ′ ( ξ ) ; (iii) furthermore, dim L ( ξ ) = l + rk π t ′ ( ξ ) if and only if – k π t ′ ( ξ ) is the only singular line in P ( ξ ) , and ERIODIC AUTOMORPHISMS AND COMPATIBLE BRACKETS 13 – rk ( π ( ξ ) | V ) = dim V − l for V = ker π t ′ ( ξ ) . By Corollary 3.8, Z is algebraic over Z × , and it follows from (2 ·
2) that Z × ⊂ S ( g ) g . Since,being an algebra of invariants, S ( g ) g is an algebraically closed subalgebra of S ( g ) , weconclude that(3 · Z ⊂ S ( g ) g . By [MY19, Prop. 1.1], if A is a Poisson-commutative subalgebra of S ( g ) g , then(3 · tr . deg A b ( g ) − b ( g ) + rk g =: b ( g , ϑ ) . Note that b ( g , ϑ ) b ( g ) and the equality occurs if and only if g is abelian. The mostinteresting case is that in which the upper bound in (3 ·
7) is attained for A = Z . Theorem 3.10. If ϑ has the property that ind g (0) = ind g , then tr . deg Z = b ( g , ϑ ) .Proof. Suppose that ξ ∈ U srr . Then rk π t ( ξ ) = dim g − l if and only if t ∈ P reg . Since tr . deg Z b ( g , ϑ ) , it suffices to show that dim L ( ξ ) = b ( g , ϑ ) whenever ξ ∈ U srr is generic.Suppose first that ∞ ∈ P reg . Then rk π t ( ξ ) = dim g − l for any t ∈ P and, by Proposi-tion 3.9 (i) , we obtain dim L ( ξ ) = b ( g ) . And the equality ind g = ind g ( ∞ ) means that g isabelian, hence b ( g ) = b ( g , ϑ ) .Suppose that ∞ 6∈ P reg . Then π ∞ ( ξ ) spans the unique singular line in the pencil cπ t ( ξ ) .By Lemma 3.5, there is a dense open subset U ⊂ g ∗ such that, for any η ∈ U and V =ker π ∞ ( η ) , we have rk ( π ( η ) | V ) = dim V − l . If ξ ∈ U srr ∩ U , then conditions (ii) and (iii) ofProposition 3.9 are satisfied for π t ( ξ ) . Thus, in this case, dim L ( ξ ) = l + 12 rk π ∞ = l + 12 (dim g − dim g − l + rk g ) = b ( g ) − b ( g ) + rk g = b ( g , ϑ ) , since rk π ∞ ( ξ ) = rk π ∞ = dim g − ind g ( ∞ ) and ind g ( ∞ ) is computed in Theorem 3.2. (cid:3) Properties of g (0) . Study of Lie algebras g (0) associated with arbitrary periodic auto-morphisms of g have been initiated in [P09], where they are called cyclic contractions or Z k - contractions . In [P09], these algebras are denoted by g h k i , where k = ord ( ϑ ) , becauseof their interpretation as the fixed-point subalgebras of an extension of ϑ to an automor-phism of k -th Takiff algebra modelled on g . (We discuss Takiff algebras in Section 6.)The case in which ord ( ϑ ) > appears to be more difficult than that of involutions. If ord ( ϑ ) = 2 , then ind g (0) = ind g and g (0) has the codim– property [P07]. Whereas, for ord ( ϑ ) > , it can happen that g (0) does not have the codim– property, and the equality ind g (0) = ind g is only known under certain constraints.To state some sufficient conditions, we first recall some results of E.B. Vinberg [Vi76].Associated with Z m -grading g = L i ∈ Z m g i , one has the linear action of G on g . Then • the algebra of invariants k [ g ] G is a polynomial (free) algebra; • the morphism π : g → g //G = Spec ( k [ g ] G ) is flat and surjective; • π − π (0) = N ∩ g and each fibre of π contains finitely many G -orbits.(It is worth mentioning that this is only a tiny fraction of fundamental results obtained inthat great paper.) The fibre π − π (0) is customary called the null-cone (w.r.t. the G -actionon g ) and we denote it by N . Definition 3.11.
Following [P09], we say that(1) ϑ is S - regular , if g contains a regular semisimple element of g ;(2) ϑ is N - regular , if g (i.e., N ) contains a regular nilpotent element of g ;(3) ϑ is very N -regular , if each irreducible component of N contains a regular nilpotentelement of g .If ord ( ϑ ) = 2 , then it follows from [KR71] that properties (1)–(3) are equivalent. But thisis not always the case if ord ( ϑ ) > . It can happen that ϑ is S -regular, but not N -regular;and vice versa. It can also happen that N is reducible and some irreducible componentsof N are not reduced (in the scheme-theoretic sense). Examples of automorphisms oforder > such that good properties of Definition 3.11 hold are given in Examples 5.9and 5.10 in [P09]. The following assertion provides sufficient conditions for some goodproperties of g (0) to hold. Theorem 3.12 ([P09, Sect. 5]) . Let ϑ ∈ Aut ( g ) be of finite order. • If g ∩ g reg = ∅ , then ind g (0) = ind g ; • If ϑ is both S -regular and very N -regular, then – g (0) has the codim – -property; – Z is is freely generated by H • , . . . , H • l , l = rk g , where { H , . . . , H l } is any set ofhomogeneous generators of S ( g ) g = Z that consists of ϑ -eigenvectors. The assumptions of Theorem 3.12 are not always satisfied, and the property of being“very N -regular” is difficult to check directly. Some methods for handling these proper-ties and related examples can be found in Section 5 in [P09].There is a nice special case, where all properties of Definition 3.11 hold and Theo-rem 3.12 applies. Namely, let g be the direct sum of m copies of a semisimple Lie algebra h and ϑ a cyclic permutation of the summands; hence ord ( ϑ ) = m . Here we obtain acomplete description of the Poisson-commutative subalgebra Z , see Section 6.4. O N ALGEBRAICALLY INDEPENDENT GENERATORS
If it is known that tr . deg Z = b ( g , ϑ ) , then it becomes a meaningful task to compute theminimal number of generators of Z or Z × .Let { H , . . . , H l } be a set of homogeneous algebraically independent generators of S ( g ) g and d i = deg H i . Then P li =1 d i = b ( g ) . Recall from Sections 1.2 and 2.1 that associated with ϑ , we have the polynomial homomorphism ϕ : k ⋆ → GL( g ) and its extension to invertible ERIODIC AUTOMORPHISMS AND COMPATIBLE BRACKETS 15 linear transformations of S j ( g ) for all j . Therefore, each H j decomposes as H j = P i > H j,i ,where ϕ s ( H j ) = P i > s i H j,i . The polynomials H j,i are called bi-homogeneous componentsof H j . By definition, the ϕ - degree of H j,i is i , also denoted by deg ϕ H j,i .Let H • j be the nonzero bi-homogeneous component of H j with maximal ϕ -degree. Then deg ϕ H j = deg ϕ H • j and we set d • j = deg ϕ H • j . By Proposition 2.1 (i) , Z t is generated by ϕ − s ( H ) , . . . , ϕ − s ( H l ) , where t = s m ∈ k ⋆ . By the standard argument with the Vander-monde determinant, we then conclude that Z × is generated by all bi-homogeneous com-ponents of H , . . . , H l , i.e.,(4 · Z × = alg h H j,i | j l, i d • j i . Definition 4.1.
Let us say that H , . . . , H l is a good generating system in S ( g ) g ( g.g.s. for short )for ϑ , if H • , . . . , H • l are algebraically independent. Then we also say that ϑ admits a g.g.s.The property of being ‘good’ really depends on a generating system. For instance, for g = sl n and m = 2 , the coefficients of the characteristic polynomial of A ∈ sl n yield a g.g.s. , while the polynomials tr( A i ) , i = 2 , . . . , n do not provide a g.g.s. , see [P07].The importance of g.g.s. is manifestly seen in the following fundamental result. Theorem 4.2 ([Y14, Theorem 3.8]) . Let H , . . . , H l be an arbitrary set of homogeneous alge-braically independent generators of S ( g ) g . Then (i) P lj =1 deg ϕ H j > P m − j =1 j dim g j =: D ϑ ; (ii) H , . . . , H l is a g.g.s. if and only if P lj =1 deg ϕ H j = D ϑ ; (iii) if g (0) has the codim– property, ind g (0) = l , and H , . . . , H l is a g.g.s. , then Z = S ( g (0) ) g (0) is a polynomial algebra freely generated by H • , . . . , H • l . Recall that dim g j = dim g m − j for any < j m − . Hence D ϑ = m − X j =1 j dim g j = 12 m − X j =1 m dim g j = m g − dim g ) . Not every i ∈ { , , . . . , d • j } provides a nonzero bi-homogeneous component H j,i . To makethis precise, we first consider the case of inner automorphisms, which is technically easier. Theorem 4.3.
Suppose that ϑ ∈ Aut ( g ) is inner and admits a g.g.s. Let H , . . . , H l be such a g.g.s. in S ( g ) g and d • j = deg ϕ H j . Then (i) each d • j ∈ m Z and H j,i = 0 only if i ∈ m Z and i d • j ; (ii) if ind g (0) = l , then H j,i = 0 for every i ∈ m Z with i d • j and the polynomials { H j,i | j = 1 , . . . , l ; & i = 0 , m, . . . , d • j } are algebraically independent (and generate Z × ).Proof. (i) Since ϑ is inner, rk g = rk g and ϑ ( H j ) = H j for all j . On the other hand, ϑ ( H j,i ) = ζ i H j,i . This proves everything. (ii) It follows from (i) that the number of nonzero bi-homogeneous components of H j is at most ( d • j /m ) + 1 . Hence the total number of nonzero bi-homogeneous componentsof all H j is at most l X j =1 (cid:18) d • j m + 1 (cid:19) = D ϑ m + l = 12 (dim g − dim g ) + l = b ( g ) − b ( g ) + rk g = b ( g , ϑ ) , where the equality rk g = rk g is used. On the other hand, the hypothesis ind g (0) = l guarantee us that tr . deg Z × = tr . deg Z × = b ( g , ϑ ) , cf. Theorem 3.10 and Corollary 3.8. Asthe bi-homogeneous components { H j,i } generate Z × (4 · H j,i with i/m ∈ Z must benonzero and algebraically independent. (cid:3) With extra technicalities, the same idea works for the arbitrary automorphisms as well.Let ϑ ∈ Aut ( g ) be an arbitrary automorphism of order m . Since ϑ acts on S ( g ) g , thereis a set of homogeneous generators { H j } ⊂ S ( g ) g such that each H j is an eigenvector of ϑ , i.e., ϑ ( H j ) = ζ r j H j for some r j ∈ Z . However, we need a set of free generators thatsimultaneously is a g.g.s . and consists of ϑ -eigenvectors. For m = 2 , this is proved in [PY,Lemma 3.4]. The following is an adaptation of that argument to any m > . Lemma 4.4. If ϑ admits a g.g.s. , then there is also a g.g.s. that consists of ϑ -eigenvectors.Proof. Let H , . . . , H l be a g.g.s. , hence P lj =1 deg ϕ H j = D ϑ in view of Theorem 4.2.Let A + be the ideal in S ( g ) g generated by all homogeneous invariants of positive degree.Then A := A + / A is a finite-dimensional k -vector space. If H ∈ A + , then ¯ H := H + A ∈ A . As is well-known, F , . . . , F m is a generating system for S ( g ) g if and only if the k -linearspan of ¯ F , . . . , ¯ F m is the whole of A . In our situation, dim k A = l and A = h ¯ H , . . . , ¯ H l i k .If H i is not a ϑ -eigenvector, i.e., ϑ ( H i ) k H i , then we consider the (non-minimal) gen-erating set { H , . . . , H i − , H i +1 , . . . , H l } [ ( H [ k ] i = m − X j =0 ζ jk ϑ j ( H i ) | k < m ) for S ( g ) g that includes l + m − polynomials. Since ¯ H , . . . , ¯ H i − , ¯ H i +1 , . . . , ¯ H l are linearlyindependent in A , one can pick up a suitable H [ k ] i such that ¯ H , . . . , ¯ H i − , H [ k ] i , ¯ H i +1 , . . . , ¯ H l is again a minimal generating set. Let us demonstrate that there is actually a uniquesuitable choice of H [ k ] i , and this yields a g.g.s. as well. Recall that d • j = deg ϕ H • j = deg ϕ H j .Clearly deg ϕ H [ k ] i d • i for each k . Since P m − j =0 ζ jk ′ = 0 if k ′ m − , we see that deg ϕ H [ k ] i = d • i if ( k + d • i ) ∈ m Z and deg ϕ H [ k ] i < d • i for all other k . If deg ϕ H [ k ] i < d • i , then thesum of ϕ -degrees for H , . . . , H i − , H [ k ] i , H i +1 , . . . , H l is less than D ϑ . By Theorem 4.2, thismeans that such choice of H [ k ] i in place of H i provides a set of algebraically dependentpolynomials. Thus, the only right choice is to take k such that k + d • i ≡ m ) , whenTheorem 4.2 also guarantee us that we obtain a g.g.s. ERIODIC AUTOMORPHISMS AND COMPATIBLE BRACKETS 17
The procedure reduces the number of generators that are not ϑ -eigenvectors, and weeventually obtain a g.g.s. that consists of ϑ -eigenvectors. (cid:3) Let { H , . . . , H l } ⊂ S ( g ) g be a generating set consisting of ϑ -eigenvectors. Then ϑ ( H i ) = ζ r i H i with r i < m . The integers { r i } depend only on the connected component of Aut ( g ) that contains ϑ , and if a is the order of ϑ in Aut ( g ) / Int ( g ) , then ζ ar i = 1 . Therefore,if g is simple, then ζ r i = ± for all types but D . Lemma 4.5.
For any ϑ ∈ Aut ( g ) of order m , we have (1) ϑ ( H j ) = H j if and only if d • j ∈ m Z ; (2) P lj =1 r j = m (rk g − rk g ) ; (3) rk g = { j | ϑ ( H j ) = H j } .Proof. (1) The proof is similar to that of Theorem 4.3(i). (2)
Recall that g always contains regular semisimple elements of g [Ka83, §8.8]. There-fore, if t is a Cartan subalgebra of g , then t = z g ( t ) is a ϑ -stable Cartan subalgebra of g .Since κ | t is non-degenerate, dim( t ∩ g j ) = dim( t ∩ g m − j ) for all j < m .Let us apply Kostant’s regularity criterion (cf. (3 · x ∈ t ∩ g reg . According tothis criterion, h d x H j | j l i k = t . Here d x H j ∈ t ∩ g i if and only if r j = i . Therefore l X j =1 r j = m − X i =1 i dim( t ∩ g i ) = 12 m (dim t − dim t ) . (3) To prove this, it suffices to notice that d x H j ∈ g if and only if ϑ ( H j ) = H j . (cid:3) Now, we can prove our main result on Z × . Theorem 4.6.
Suppose that ϑ ∈ Aut ( g ) admits a g.g.s. and ind g (0) = rk g . Then (i) Z × is a polynomial Poisson-commutative subalgebra of S ( g ) g having the maximal transcen-dence degree. (ii) More precisely, if H , . . . , H l is a g.g.s. that consists of ϑ -eigenvectors, then Z × is freelygenerated by the nonzero bi-homogeneous components of all { H j } .Proof. We already know that Z × is Poisson-commutative and tr . deg Z × = b ( g , ϑ ) is maxi-mal possible (Theorem 3.10). Hence we have to prove the polynomiality and (ii).Recall that { H j } is a g.g.s. if and only if P lj =1 d • j = D ϑ = m (dim g − dim g ) / . If ϑ ( H j ) = H j , i.e., d • j ∈ m Z , then H j has at most ( d • j /m ) + 1 nonzero bi-homogeneouscomponents, as in the proof of Theorem 4.3. In general, if ϑ ( H j ) = ζ r j H j , then H j,i can benonzero only if i ≡ r j (mod m ) . Therefore, d • j ≡ r j (mod m ) and H j has at most d • j − r j m nonzero bi-homogeneous components. Using Lemma 4.5, we see that the total number of all nonzero bi-homogeneous components is at most l X j = l (cid:18) d • j − r j m + 1 (cid:19) = l X j =1 d • j m ! − l X j =1 r j m ! + l = D ϑ m − l − rk g l = b ( g ) − b ( g ) + rk g = b ( g , ϑ ) . On the other hand, it follows from Eq. (4 ·
1) that the total number of bi-homogeneouscomponents of H , . . . , H l is at least b ( g , ϑ ) . Therefore, all admissible bi-homogeneouscomponents must be nonzero and algebraically independent. (cid:3) A precise relationship between Z and Z × depends on further properties of ϑ . Twocomplementary assertion are given below. Corollary 4.7.
In addition to the hypotheses of
Theorem 4.6 , suppose that g (0) has the codim – property and g = g ϑ is not abelian. Then Z = Z × is the polynomial algebra freely generated byall nonzero bi-homogeneous components H j,i .Proof. In this case, ∈ P reg and it follows from Theorem 4.2 (iii) that Z = k [ H • , . . . , H • l ] ,which is contained in Z × . On the other hand, ∞ ∈ P sing (Corollary 3.3), hence Z ∞ is notrequired for Z . Thus, Z = Z × . (cid:3) Corollary 4.8.
In addition to the hypotheses of
Theorem 4.6 , suppose that g (0) has the codim – property, ϑ is inner, and g = g ϑ is abelian. Then Z ∞ = S ( g ) and Z = alg h Z × , g i is apolynomial algebra.Proof. Recall that the subspace g lies in the centre of g ( ∞ ) , hence S ( g ) ⊂ Z ∞ . If ϑ is inner,then ind g ( ∞ ) = dim g = rk g (Theorem 3.2). Hence S ( g ) ⊂ Z ∞ is an algebraic extension.Since S ( g ) is an algebraically closed subalgebra of S ( g ) , it coincides with Z ∞ . As in theprevious corollary, we have Z ⊂ Z × . Therefore, Z = alg h Z × , Z ∞ i = alg h Z × , g i . Among the algebraically independent generators of Z × , one has l nonzero functions H j, ∈ S ( g ) , j = 1 , . . . , l . (Note that d • j ∈ m Z for each j , since ϑ is inner.) Hence the passage from Z × to Z merely means that we have to replace { H j, } with a basis for g . (cid:3) Remark . If ϑ does not admit a g.g.s. , then H • , . . . , H • l are algebraically dependent. Since { H • i } are certain bi-homogeneous components, this means that the number of all nonzerobi-homogeneous components of { H j } is larger than tr . deg Z . Moreover, the case of invo-lutions ( m = 2 ) shows that then the algebra Z = S ( g (0) ) g (0) , which is contained in Z , isnot polynomial, see [Y17, Sect. 5]. Therefore, it would be unwise to expect really goodproperties of Z × or Z without presence of g.g.s. ERIODIC AUTOMORPHISMS AND COMPATIBLE BRACKETS 19
5. O
N THE MAXIMALITY PROBLEM
Since Z ⊂ S ( g ) g by (3 · ˜ Z := alg h Z , S ( g ) g i is still Poisson-commutative.It was proved in [PY] that if ϑ is an involution admitting a g.g.s. , then ˜ Z is a maximal Poisson-commutative subalgebra of S ( g ) g . For m > , the same problem becomes moredifficult, and we obtain only partial results in this section. Our line of argument employsproperties of graded polynomial algebras.Let F , . . . , F N ∈ k [ { x i } ] = k [ A n ] be algebraically independent homogeneous polyno-mials. Each differential d F i is a regular -form on A n with polynomial coefficients. Then(5 ·
1) d F ∧ . . . ∧ d F N = f R, where f ∈ k [ A n ] and R is a regular differential N -form that is nonzero on a big opensubset. Note that f is defined uniquely up to multiplication with scalars. Let F = k [ { F j } ] be the subalgebra of k [ A n ] generated by the polynomials F j . Theorem 5.1 ([PPY, Theorem 1.1]) . If f = 1 , then F is an algebraically closed subalgebra of k [ A n ] , i.e., if H ∈ k [ A n ] is algebraic over the field k ( F , . . . , F N ) , then H ∈ F . Consider the following conditions on the Lie algebra g (0) : ( ♦ ) ind g (0) = rk g = l , i.e., ∈ P reg ; ( ♦ ) Z is a polynomial ring generated by H • i with i l ; ( ♦ ) dim g ∗ (0) , sing dim g − , i.e., g (0) has the codim– property; ( ♦ ) either g is non-abelian or g is abelian and ϑ is inner.These conditions imply that Z is a polynomial algebra, see Corollary 4.7 and 4.8. More-over, the following is true. Proposition 5.2.
If conditions ( ♦ ) – ( ♦ ) are satisfied, then ˜ Z is a polynomial algebra, too.Proof. By Lemma 4.5, we have H j, = 0 if and only if ϑ ( H j ) = H j , and the number of such j ’s equals rk g . Then H j, ∈ S ( g ) g . In the passage from Z to ˜ Z , these nonzero generators H j, are replaced with the basic symmetric invariants of g . (cid:3) Lemma 5.3.
Assume that ( ♦ ) and ( ♦ ) hold. Suppose that there is a divisor D ⊂ g ∗ such that dim d η Z × < b ( g , ϑ ) for any η ∈ D . Then D = Y × g ∗ > , where Y ⊂ g ∗ ∩ g ∗ sing is a G -stableconical divisor in g ∗ .Proof. By Proposition 3.1, there is a big open subset U sr ⊂ g ∗ such that U sr ⊂ T t =0 , ∞ g ∗ ( t ) , reg .By ( ♦ ) , g ∗ (0) , reg is also a big open subset of g ∗ . Hence ˜ U := D ∩ U sr ∩ g ∗ (0) , reg is open anddense in D . Take any η ∈ ˜ U . If we write η = η + η ′ with η ∈ g ∗ and η ′ ∈ g ∗ > , then wemay also assume that η ∈ ( g ) ∗ reg . Assume that η ∈ g ∗ reg . Then η ∈ g ∗∞ , reg by Lemma 3.6.Thus, in that case, η ∈ U srr . Moreover, such η is generic in the sense of Lemma 3.5 and the conclusion of that lemma holds for it. Arguing as in the proof of Theorem 3.10 and using(3 · dim L ( η ) = b ( g , ϑ ) , a contradiction!Therefore we must have η ∈ g ∗ sing for a generic η ∈ D and hence for any η ∈ D . Since g ∗ ∩ g ∗ sing is a proper subset of g ∗ , the divisor D is indeed of the form Y × g ∗ > , where Y ⊂ g ∗ ∩ g ∗ sing .The algebra Z × consists of G -invariants and the group G is connected. Thereby eachirreducible component of the subset { γ ∈ g ∗ | dim d γ Z × < b ( g , ϑ ) } is G -stable. In par-ticular, D , and hence Y as well, is G -stable. Since Z × is a homogeneous subalgebra, thedivisor Y ⊂ g ∗ is conical. (cid:3) Theorem 5.4.
Suppose that ( ♦ ) , ( ♦ ) , and ( ♦ ) hold. If g ∩ g ∗ sing does not contain divisors, then S ( g ) g ⊂ Z × and Z × = Z = ˜ Z is a maximal Poisson-commutative subalgebra of S ( g ) g .Proof. We know that Z × is a polynomial algebra (Theorem 4.6) and tr . deg Z × = b ( g , ϑ ) (Theorem 3.10). Suppose that Z × ⊂ A ⊂ S ( g ) g and A is Poisson-commutative. Then wehave tr . deg A b ( g , ϑ ) by (3 · F i of the alge-braically independent generators F i ∈ Z × are linearly independent on a big open subset.Then by Theorem 5.1, Z × is an algebraically closed subalgebra of S ( g ) . In particular, wemust have A = Z × . This applies to Z and ˜ Z as well. (cid:3) Remark . Conditions ( ♦ ) – ( ♦ ) are satisfied for involutions ϑ that have g.g.s. . Example . Theorem 5.4 applies to several outer automorphisms of semisimple Lie al-gebras, for instance, to ϑ ∈ Aut ( g ) of order m in case g = ( sl n ) m and g is a diagonallyembedded sp n , cf. [P09, (4.2)]. Automorphisms of this form are considered in Section 8.Theorem 3.12 provides a bunch of automorphisms ϑ with ord ( ϑ ) > such that ˜ Z is apolynomial algebra, see Examples 5.9 and 5.10 in [P09]. In all these cases, Z = ˜ Z and weconjecture that ˜ Z is a maximal Poisson-commutative subalgebra of S ( g ) g .Summarising our previous considerations, we can say that in order to guarantee somegood properties of the Poisson-commutative subalgebras Z × , Z , and ˜ Z , the following prop-erties of ϑ ∈ Aut ( g ) and thereby of g (0) are needed: (a) ind g (0) = ind g ; (b) g (0) has the codim– property; (c) ϑ admits a g.g.s. If ord ( ϑ ) = 2 , then (a) and (b) are always satisfied [P07], and a complete description ofinvolutions admitting a g.g.s. is available [Y17]. In a forthcoming paper, we are going toundertake a thorough substantial investigation of these properties for arbitrary ϑ . ERIODIC AUTOMORPHISMS AND COMPATIBLE BRACKETS 21
6. T
HE CASE OF A CYCLIC PERMUTATION
Let h be a simple non-abelian Lie algebra and g = h m the direct sum of m > copies of h . Then l = rk g = m · rk h . Let ϑ ∈ Aut ( g ) be a cyclic permutation of the summands of g .That is, ϑ ( x , . . . , x m ) = ( x m , x , . . . , x m − ) , x i ∈ h . Then, for i = 0 , , . . . , m − and ζ = m √ , we have(6 · g i = { ( x, ζ − i x, . . . , ζ (1 − m ) i x ) | x ∈ h } . In particular, g = ∆ h ≃ h is the diagonal and each g i is isomorphic to h as vector spaceand as g -module. Here the Lie algebra g (0) is isomorphic to the truncated current algebra h h m i := h ⊗ k [ t ] / ( t m ) = h [ t ] / ( t m ) , see [P09, Corollary 3.6]. The isomorphism g (0) ≃ h h m i takes g i to the image of h t i ⊂ h ⊗ k [ t ] in h h m i . The Lie algebra h h m i is also known as a (generalised) Takiff algebra modelled on h . By [RT92, Theorem 2.8], we have ind g (0) = m · rk h = rk g , i.e., ∈ P reg . It then followsfrom Theorem 3.10 that in this case tr . deg Z = b ( h m , ϑ ) = 12 (cid:0) ( m −
1) dim h + ( m + 1)rk h (cid:1) = ( m − b ( h ) + rk h . On the other hand, g is not abelian, hence P sing = {∞} and Z ∞ is not required for Z . Since h h m i is N -graded and the zero part is semisimple, the nilpotent radical h h m i u is equalto L m − i =1 h t i . Comparing this with Proposition 2.3 on the graded structure of q ( ∞ ) , weconclude that here g ( ∞ ) ≃ h h m +1 i u . Since ind g ( ∞ ) is computed for any ϑ in Theorem 3.2,we obtain a new observation that(6 · ind h h m i u = ( m − h + dim h . Upon the identification g ≃ h , see Eq. (6 · x ∈ h is nilpotent (resp. semisim-ple, regular) in h if and only if ( x, ζ − i x, . . . , ζ (1 − m ) i x ) ∈ g is nilpotent (resp. semisimple,regular) in g . This also implies that the null-cone N is isomorphic to the null-cone of h .Hence N is irreducible. Thus, ϑ is both S -regular and very N -regular.Let { H , . . . , H l } be s set of homogeneous generators of S ( g ) g consisting of ϑ -eigenvectors.Since ϑ is S -regular and very N -regular, it follows from Theorem 3.12 that g (0) has thecodim– -property and Z = k [ H • , . . . , H • l ] , see also [RT92]. The last relation also meansthat { H j } is a g.g.s. for ϑ . Theorem 6.1. If g = h m and ϑ is a cyclic permutation, then the algebra Z = Z ( h m , ϑ ) is freelygenerated by the nonzero bi-homogeneous components H j,i , j l = m · rk h . Moreover, Z is a maximal Poisson-commutative subalgebra of S ( g ) g . Proof.
The above discussion shows that conditions ( ♦ ) – ( ♦ ) hold for ϑ . Hence Z is freelygenerated by the nonzero bi-homogeneous components H j,i by Corollary 4.7.A point ξ ∈ g ∗ is regular in g ∗ if and only if ξ is regular in g ∗ ≃ h ∗ . Thereby dim( g ∗ ∩ g ∗ sing ) = dim h − and this intersection does not contain divisors of g ∗ . Therefore Z is a maximal Poisson-commutative subalgebra of S ( g ) g by Theorem 5.4. (cid:3) It is not hard to produce a generating set { H j } ⊂ S ( g ) g consisting of ϑ -eigenvectors.Suppose that rk h = r and that { F , . . . , F r } is a generating set of S ( h ) h ⊂ S ( g ) for the firstcopy of h in g . For k < m , set(6 · F [ k ] i = 1 m (cid:0) F i + ζ k ϑ ( F i ) + ζ k ϑ ( F i ) + . . . + ζ k ( m − ϑ m − ( F i ) (cid:1) . Then ϑ ( F [ k ] i ) = ζ − k F [ k ] i . Thus, we can take { H j | j l } = { F [ k ] i | i r, k < m } .In Section 7, we will need an explicit description of the bi-homogeneous components H j,i , which exploits the polarisation construction of [RT92]. Suppose that each F i is ho-mogeneous and b i = deg F i .For ξ ∈ h ∗ , let ξ ( k ) ∈ g ∗ be the corresponding element sitting in the k -th copy of h ∗ in g ∗ .Then the induced action of ϑ in g ∗ is given by the formula: ϑ (cid:0) ξ (1)1 + . . . + ξ ( m − m − + ξ ( m ) m (cid:1) = ξ (1)2 + . . . + ξ ( m − m + ξ ( m )1 . Accordingly, g ∗ j = { ξ (1) + ζ j ξ (2) + . . . + ζ ( m − j ξ ( m ) | ξ ∈ h ∗ } and we fix the correspondingisomorphisms ν j : h ∗ ∼ → g ∗ j , where(6 · ξ ν j (cid:0) ξ (1) + ζ j ξ (2) + . . . + ζ ( m − j ξ ( m ) (cid:1) ∈ g ∗ j . For ξ , . . . , ξ m − with ξ j ∈ g ∗ j , let ¯ ξ = ( ξ , . . . , ξ m − ) be the corresponding element of g ∗ and ξ + . . . + ξ m − be an element of h ∗ . For s ∈ k , set φ s ( ¯ ξ ) = ( s m − ξ , s m − ξ , . . ., ξ m − ) anddefine the φ -polarisations F [ k ] of F ∈ S d ( h ) in the following way(6 · F ( φ s ( ¯ ξ )) := F ( s m − ξ + s m − ξ + . . . + ξ m − ) =: F [0] ( ¯ ξ ) + sF [1] ( ¯ ξ ) + . . . + s d ( m − F [ d ( m − ( ¯ ξ ) . It is convenient to assume that F [ k ] = 0 for k > d ( m − . Example . If x ∈ h and k < m , then x [ k ] = m ( x, ζ − i x, . . . , ζ (1 − m ) i x ) ∈ g i with i = m − − k .According to [RT92], Z is freely generated by ( F i ) [ k ] with i r , k < m , where S ( h ) h = k [ F , . . . , F r ] . Proposition 6.3.
We have { H j,i =0 | j l, i } = { ( F i ) [ k ] | i r, k b i ( m − } ,where ( F i ) [ k ] are the φ -polarisations of F i defined by (6 · . ERIODIC AUTOMORPHISMS AND COMPATIBLE BRACKETS 23
Proof.
Now we need to distinguish the first copy h (1) ⊂ g of h from an abstract h . Supposethat F (1) ∈ S d ( h (1) ) is the image of F ∈ S d ( h ) under the tautological isomorphism S ( h ) ≃ S ( h (1) ) . The combination of (6 ·
4) and the definition of ϕ s , see Section 1.2, leads to ϕ s ( F (1) ) = F [( m − d ] + sF [( m − d − + . . . + s ( m − d − F [1] + s ( m − d F [0] for s ∈ k ⋆ . In this notation, ϑ ( F [ k ] ) = ζ − d − k F [ k ] . Next we plug the formula for ϕ s ( F (1) ) into (6 ·
3) and conclude that each bi-homogeneous component of each F [ k ] i is a φ -polarisationof F i ∈ S b i ( h ) . By Theorem 6.1, the total number of the nonzero bi-homogeneous compo-nents ( F [ k ] i ) i ′ is tr . deg Z = b ( g , ϑ ) = ( m − · b ( h ) + r . Since h is semisimple, we have b ( h ) = P ri =1 b i . Finally, the total number of φ -polarisations ( F i ) [ k ] equals r + ( m − r X i =1 b i = r + ( m − · b ( h ) = b ( g , ϑ ) . Hence the two sets in question coincide. (cid:3)
7. G
AUDIN SUBALGEBRAS
Let h be the same as in the previous section. The enveloping algebra U ( t − h [ t − ]) con-tains a remarkable commutative subalgebra z ( b h ) , which is known as the Feigin–Frenkelcentre [FF92] (see also [F07]). This subalgebra is defined as the centre of the universalaffine vertex algebra associated with the affine Kac–Moody algebra b h at the critical level.In particular, each element of z ( b h ) is annihilated by the adjoint action of h . The elements of z ( b h ) give rise to higher Hamiltonians of the Gaudin model, which describes a completelyintegrable quantum spin chain [FFR].A Gaudin model consists of n copies of h and the Hamiltonians H k = X j = k P dim h i =1 x ( k ) i x ( j ) i z k − z j , k n, where z , . . . , z n ∈ k are pairwise distinct and { x ( k ) i | i dim h } is a basis for the k -thcopy of h that is orthonormal w.r.t. κ . Letting g = h n , these Gaudin Hamiltonians {H k } can be regarded as elements of either U ( h ) ⊗ n ≃ U ( g ) or S ( g ) . They commute (and hencePoisson-commute) with each other. Then higher Gaudin Hamiltonians are elements of U ( h ) ⊗ n that commute with all H k .Recall the construction of [FFR] that provides a Gaudin subalgebra G of U ( h ) ⊗ n . Set b h − = t − h [ t − ] and let ∆ U ( b h − ) ≃ U ( b h − ) be the diagonal of U ( b h − ) ⊗ n . Then any vector ~z = ( z , . . . , z n ) ∈ ( k ⋆ ) n defines a natural homomorphism ρ ~z : ∆ U ( b h − ) → U ( h ) ⊗ n , where ρ ~z ( xt k ) = z k x (1) + z k x (2) + . . . + z kn x ( n ) for x ∈ h . By definition, G = G ( ~z ) is the image of z ( b h ) under ρ ~z , and one can prove that G contains H k for all k . Hence [ G ( ~z ) , H k ] = 0 for each k . One also has G ⊂ U ( g ) ∆ h by the construction. Remark . Gaudin subalgebras have attracted a great deal of attention, see e.g. [CFR] andreferences therein. It is standard to work with complex Lie algebras in this framework.Gaudin algebras are closely related to quantum MF subalgebras [R06] and share some oftheir properties. In particular, for a generic ~z ∈ ( R ) n , the action of G ( ~z ) on an irreduciblefinite-dimensional g -module V ( λ ) ⊗ . . . ⊗ V ( λ n ) is diagonalisable and has a simple spec-trum on the subspace of highest weight vectors w.r.t. the diagonal h ≃ ∆ h ⊂ h ⊕ n [R18].The cyclic permutation ϑ is an automorphism of g of order n . Let now ζ be a primitive n -th root of unity. Then the ϑ -eigenspace g j corresponding to ζ j is(7 · g j = { x (1) + ζ − j x (2) + ζ − j x (3) + . . . + ζ j (1 − n ) x ( n ) | x ∈ h } ≃ h . Let ¯ t denote the image of t − n in k [ t − ] / ( t − n − . Then the quotient h [ t − ] / ( t − n − hasthe canonical Z n -grading h ⊕ h ¯ t ⊕ h ¯ t ⊕ . . . ⊕ h ¯ t n − . In particular, h [ t − ] / ( t − n − ≃ g asa Z n -graded Lie algebra. Fix an isomorphism g j → ¯ t j h as the projection pr : g → h onthe first summand of g combined with the multiplication by ¯ t j . Now we regard g as thequotient of h [ t − ] .The above identifications related to ϑ provide a simpler approach to constructing cer-tain Gaudin subalgebras. Proposition 7.2.
Take z k = ζ − k and consider the corresponding Gaudin subalgebra G = G ( ~z ) in U ( h ) ⊗ n . Then G coincides with the image of z ( b h ) in the quotient U ( b h − ) / ( t − n −
1) = U ( g ) .Proof. Take a ∈ Z > and write a = nq + j with j n . Then for xt − a ∈ h [ t − ] , we have ρ ~z ( xt − a ) = x (1) + ζ j x (2) + ζ j x (3) + . . . + ζ j ( n − x ( n ) ∈ g n − j . Therefore ρ ~z ( xt − a ) identifies with the image x ¯ t n − j of xt − a in h [ t − ] / ( t − n − . (cid:3) By a theorem of Feigin and Frenkel [FF92], z ( b h ) is a polynomial ring in infinitely manyvariables, with a distinguished set of generators. Set τ = − ∂ t . There are algebraicallyindependent elements S , . . . , S r ∈ z ( b h ) such that z ( b h ) = k [ τ k ( S i ) | i r, k > . The symbols gr( S i ) are homogeneous elements of S ( h t − ) and if F i = gr( S i ) | t =1 , then k [ F , . . . , F r ] = S ( h ) h . The set { S i } is called a complete set of Segal–Sugawara vectors . Keepthe notation b i = deg F i . Lemma 7.3.
For k > , let F i,k be the symbol of ρ ~z ( τ k ( S i )) . In the case z k = ζ − k for k =1 , . . . , n , we have then h F i,k | k ( n − b i i k = (cid:10) ( F i ) [ k ] | k ( n − b i (cid:11) k for the φ -polarisations ( F i ) [ k ] related to g ∗ = L n − j =0 g ∗ j , cf. (6 · . ERIODIC AUTOMORPHISMS AND COMPATIBLE BRACKETS 25
Proof.
Since gr( S i ) ∈ S ( t − h ) , we have F i, ∈ S ( g n − ) and F i, = n b i ( F i ) [0] . Assuming n > ,we can state that F i, = n b i ( F i ) [1] . If k < n , then clearly k ! F i,k = n b i ( F i ) [ k ] . More generally,as long as k ( n − b i , we have k ! n b i F i,k = ( F i ) [ k ] + X Let h be the same as before. Let now ϑ ∈ Aut ( h ) be an automorphism of order m and ζ = m √ be primitive. To any such ϑ , one associates a ϑ -twisted loop algebra h [ t, t − ] ϑ =( h [ t, t − ]) ϑ , where ϑ ( t ) = ζ − t and ϑ ( t − ) = ζ t − . If ϑ is an outer automorphism of h , then h [ t, t − ] ϑ is a subalgebra of a twisted Kac–Moody algebra, see [Ka83, Chap. 8] for details.Similarly to the case of b h − , we identify b h ϑ − := ( b h − ) ϑ with ( h [ t, t − ] / h [ t ]) ϑ . This leads toan action of ( h [ t ]) ϑ on b h ϑ − and correspondingly on S ( b h ϑ − ) . In this section, we consider theinvariants of ( h [ t ]) ϑ in S ( b h ϑ − ) , i.e., the subalgebra Z ( b h − , ϑ ) = S ( b h ϑ − ) ( h [ t ]) ϑ , which can be thought of as a ϑ -twisted analogue of gr( z ( b h )) , cf. (0 · g = h n . Let ˜ ϑ ∈ Aut ( g ) be the composition of ϑ applied to one copy of h only and acyclic permutation of the summands. Formally speaking, ˜ ϑ ( y (1)1 + y (2)2 + . . . + y ( n ) n ) = y (1) n + ϑ ( y ) (2) + y (3)2 + . . . + y ( n ) n − for any y , . . . , y n ∈ h . The order of ˜ ϑ is N = nm . Similarly to the case of b h − , there areisomorphisms g ≃ h [ t ] ϑ / ( t N − and g (0) ≃ h [ t ] ϑ / ( t N ) .Set q = h (0) transferring the notation of Section 3 to h . Next we want to understandthe connection between the Takiff algebra q h n i = q [ t ] / ( t n ) , modelled on q , and g (0) . Let ˜ ζ = N √ be such that ζ = ˜ ζ n . Set also ω = ˜ ζ m . Lemma 8.1 (cf. [P09, Sect. 3]) . (i) The Takiff algebra q h n i is a contraction of g (0) . (ii) If ind q = rk h , then ind g (0) = rk g . (iii) If there is a g.g.s. for ϑ , then there is a g.g.s. for ˜ ϑ . (iv) If q has the codim – property, then so does g (0) .Proof. (i) Let g = L j 1) = nD ϑ + N n − 12 dim h = D ˜ ϑ . Now the claim follows from Theorem 4.2. (iv) Suppose that q has the codim– property, then so does the Takiff algebra q h n i [RT92,PY20’]. Furthermore, if the index does not change under a polynomial contraction, thenthe dimension of the singular subset can only increase [Y17, (4.1)]. Applying this to g (0) and its contraction q h n i , we conclude that g (0) has the codim– property. (cid:3) Theorem 8.2. (i) If ind h (0) = rk h , then Z ( b h − , ϑ ) is a Poisson-commutative subalgebra. (ii) The algebra Z ( b h − , ϑ ) is a polynomial ring (in infinitely many variables) if and only if S ( g (0) ) g (0) is a polynomial ring for each n > and g = h n with ˜ ϑ ∈ Aut ( g ) as above.Proof. For N = nm , set W N := ( h t − N ⊕ . . . ⊕ h t − ) ϑ ⊂ b h ϑ − . Then h [ t ] ϑ acts on W N via its quotient g (0) and W N ≃ g (0) as a g (0) -module. Therefore thereis an isomorphism of commutative algebras S ( W N ) h [ t ] ϑ ≃ S ( g (0) ) g (0) .If f , f ∈ S ( b h ϑ − ) , then there is N ′ = n ′ m such that f , f ∈ W N ′ . Set n = 2 n ′ . Then { f , f } = 0 if and only if the images ¯ f , ¯ f of f , f in S ( g ) ≃ S ( b h θ ) / ( t − N − Poisson-commute. The Z N -grading of g inherited from b h − coincides with the Z N -grading definedby ˜ ϑ . If f , f ∈ Z ( b h − , ϑ ) , then ¯ f , ¯ f ∈ S ( g (0) ) g (0) by the construction. The assumption ind h (0) = rk h implies that ind g (0) = rk g , see Lemma 8.1. Thus S ( g (0) ) g (0) is contained inthe Poisson-commutative subalgebra Z ( g , ˜ ϑ ) ⊂ S ( g ) and hence { ¯ f , ¯ f } = 0 . (ii) The algebra Z ( b h − , ϑ ) = lim −→ S ( W nm ) h [ t ] ϑ has a direct limit structure. If Z ( b h − , ϑ ) is apolynomial ring, then S ( W N ) h [ t ] ϑ ≃ S ( g (0) ) g (0) has to be a polynomial ring for each n .In order to prove the opposite implication, suppose that S ( W N ) h [ t ] ϑ is a polynomial ringfor each n . By a standard argument on graded algebras, see e.g. the proof of Lemma 4.4,any algebraically independent set of generators of S ( W N ) h [ t ] ϑ extends to an algebraicallyindependent set of generators of S ( W N + m ) h [ t ] ϑ . In the direct limit, one obtains a set ofalgebraically independent generators of Z ( b h − , ϑ ) . (cid:3) If ind h (0) = rk h , there is a g.g.s. for ϑ , and h (0) has the codim– property, then S ( g (0) ) g (0) is a polynomial ring, see Lemma 8.1 and Theorem 4.2. Set b i, • = b i ( m − − b • i . Proposition 8.3. If ind h (0) = rk h , there is a g.g.s. F , . . . , F r ∈ S ( h ) h for ϑ , and h (0) has the codim – property, then Z ( b h − , ϑ ) is freely generated by the ( t − ) -polarisations ( F i ) [ b i, • + km ] with k > related to the subspaces (( h t − j ) ϑ ) ∗ . Proof. First we enlarge on t − -polarisations. For k ∈ Z , let ¯ k ∈ { , . . . , m − } be the residueof k modulo m . Then b h ϑ − = L k − h ¯ k t k . Let φ s : ( b h ϑ − ) ∗ → ( b h ϑ − ) ∗ with s ∈ k ⋆ be the linearmap multiplying elements of ( h ¯ k t k ) ∗ with s − k − . Next we canonically identify ( h ¯ k t k ) ∗ with h ∗ ¯ k . Then an element of ( b h ϑ − ) ∗ is a finite sequence ξ = ( ξ − , ξ − , . . . , ξ − L ) with ξ k ∈ h ∗ ¯ k . Forsuch a ξ , set | ξ | = P Li =1 ξ − i ∈ h ∗ . If F ∈ S ( h ) , then F ( | φ s ( ξ ) | ) = P k > s k F [ k ] ( ξ ) , where thesum is actually finite. If n = 1 , then W m ≃ h and ( F i ) [ b i, • ] identifies with F • i ∈ S ( h (0) ) h (0) .Recall that Z ( b h − , ϑ ) = lim −→ S ( W nm ) h [ t ] ϑ and that S ( W N ) h [ t ] ϑ ≃ S ( g (0) ) g (0) for N = nm . Nowwe identify the vector spaces W N and g . For k < n , let F [ k ] i ∈ S ( g ) be defined by(8 · S ( g (0) ) g (0) is freely generated by the highest components ( F [ k ] i ) • with i r , k < n . The discussion in the proof of Lemma 8.1 (iii) shows that if B • i − ℓ i + mk ∈ N Z ,then ( F [ k ] i ) • = n bi ( F i ) [ b i, • ] ∈ S ( W m ) . Analogously to the case of ϑ = id h , cf. Proposition 6.3,one can see that { n b i ( F [ k ] i ) • | k < n } = { ( F i ) [ b i, • + km ] | k < n } . This completes the proof. (cid:3) Remark . Theorem 8.2 emphasises the importance of the equality ind h (0) = rk h . Let usrecall that it holds for the involutions by [P07]. 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