Takiff algebras with polynomial rings of symmetric invariants
aa r X i v : . [ m a t h . R T ] O c t October 9, 2017
TAKIFF ALGEBRAS WITH POLYNOMIAL RINGS OF SYMMETRICINVARIANTS
DMITRI I. PANYUSHEV AND OKSANA S. YAKIMOVA I NTRODUCTION
The ground field k is algebraically closed and of characteristic . Let Q be a connectedalgebraic group with q = Lie Q and S ( q ) = k [ q ∗ ] the symmetric algebra of q . The subalge-bra of Q -invariants in k [ q ∗ ] is denoted by k [ q ∗ ] Q or k [ q ∗ ] q . The elements of k [ q ∗ ] Q are called symmetric invariants of q . Interesting classes of non-reductive groups Q such that k [ q ∗ ] Q isa polynomial ring have recently been found, see e.g. [J07, P07’, PPY, PY12, PY13, CM16,Y17’]. A quest for this type of groups continues. Let q h m i := q ⊗ k [ T ] / ( T m +1 ) be the m -thTakiff algebra (= a truncated current algebra ) of q . Since q h i ≃ q , we may assume that m > .Our main result is that under a mild restriction, the passage from q to q h m i preserves thepolynomiality of symmetric invariants. We also (1) discover a new phenomenon that acertain ideal of q h m i has a polynomial ring of invariants in k [ q h m i ∗ ] , and (2) show that theproperty of q that k [ q ∗ ] is a free k [ q ∗ ] Q -module does not always extend to q h i .The story began in 1971, when Takiff proved that if g is semisimple, then g h i = g ⋉ g ab has a polynomial ring of symmetric invariants whose Krull dimension equals · rk g [Ta71]. Then Ra¨ıs and Tauvel proved a similar result for g h m i with arbitrary m ∈ N [RT92]. This is the classical analogue of the description of the Feigin-Frenkelcentre z ( b g ) ⊂ U ( t − g [ t − ]) , see [FF92]. Recently, Macedo and Savage came up with amulti-parameter generalisation of the Ra¨ıs-Tauvel result. Namely, let(0 · ˆ g = g ⊗ k [ T , . . . , T r ] / ( T m +11 , . . . , T m r +1 r ) =: g h m , . . . , m r i be a truncated multi-current algebra of a semisimple g . Then k [ˆ g ∗ ] ˆ g is a polynomial ring ofKrull dimension ( m + 1) . . . ( m r + 1) · rk g , see [MS16]. The proofs heavily use the fact that g is semisimple, when many structure results are available. For instance, both [RT92] and[MS16] exploit Kostant’s section for the set of the regular elements of g . On the other hand,if g is simple and q = g e is the centraliser of a nilpotent element e ∈ g such that g e has the“codim– property” and e admits a “good generating system” in k [ g ] G , then k [ g e h m i ∗ ] g e h m i is a polynomial ring for all m ∈ N , see [AP17, Theorem 3.1]. In all these cases, the free Mathematics Subject Classification.
Key words and phrases. index of Lie algebra, coadjoint representation, symmetric invariants.The research of the first author was carried out at the IITP RAS at the expense of the Russian Foundationfor Sciences (project } generators of the ring of symmetric invariants of ˆ g or g e h m i are explicitly described viathose of g or g e , respectively. This goes back to a general construction of [RT92].Our main theorem provides a substantial generalisation of all these partial results. Tostate it, we need some notation. The index of q , ind q , is the minimal codimension of the Q -orbits in q ∗ , hence ind q = rk q if q is reductive. Let d f be the differential of f ∈ k [ q ∗ ] .We regard d f as a polynomial mapping from q ∗ to q and write ( d f ) ξ for its value at ξ ∈ q ∗ .If f ∈ k [ q ∗ ] Q , then d f is Q -equivariant. The image of q ⊗ T + · · · + q ⊗ T m in q h m i is anideal of codimension dim q , which is denoted by q h m i u . An open subset of an irreduciblevariety is called big , if its complement does not contain divisors. Then a brief version ofour result is Theorem 0.1.
Let q be an algebraic Lie algebra such that k [ q ∗ ] q = k [ f , . . . , f l ] is a graded poly-nomial ring, where l = ind q . Set Ω q ∗ = { ξ ∈ q ∗ | ( d f ) ξ ∧ · · · ∧ ( d f l ) ξ = 0 } , and assume that Ω q ∗ is big (in q ∗ ). For any m > , we then have (i) k [ q h m i ∗ ] q h m i u is a graded polynomial ring of Krull dimension dim q + ml . (ii) the Takiff algebra q h m i has the same properties as q , i.e., k [ q h m i ∗ ] q h m i is a graded poly-nomial ring of Krull dimension ( m + 1) l = ind q h m i and the similarly defined subset Ω q h m i ∗ ⊂ q h m i ∗ is also big. (See also Theorem 2.2 for a description of free generators and Ω q h m i ∗ .) As is well-known,a semisimple Lie algebra g satisfies the assumptions of Theorem 0.1. (This goes back toChevalley and Kostant.) Therefore, Theorem 0.1 yields another proof and a generalisationof [MS16, Theorem 5.4], see Corollary 2.6. A notable difference between our Theorem 0.1and results of [AP17] is that we do not impose a constraint on P i deg f i , which is a partof the definition of a “good generating system”, and do not require the codim– propertyfor q (see Section 1 for the definition). A weaker assumption that Ω q ∗ is big appears to besufficient. That is, our result applies to a larger supply of non-reductive Lie algebras, seeexamples in Sections 3 and 4. For instance, the canonical truncation , ˜ q , of a Frobenius Liealgebra q satisfies the hypotheses of Theorem 0.1, see Section 3.2.If g is semisimple , then k [ g h m i ∗ ] is a free k [ g h m i ∗ ] g h m i -module for any m [M01, Ap-pendix]. In Section 5, we prove that this property does not generalise to the truncatedmulti-current algebras of g or the truncated current algebras q h m i for arbitrary q such that k [ q h m i ∗ ] is a free k [ q h m i ∗ ] q h m i -module. Namely, k [ g h , , i ] is not a free k [ g h , , i ] g h , , i -module (Theorem 5.5). This can also be interpreted as follows. Since the passage g ❀ g h i preserve freeness of the module [G94, M01], in the chain of Takiff extensions g ❀ g h i ❀ g h ih i ≃ g h , i ❀ g h , ih i ≃ g h , , i , AKIFF ALGEBRAS WITH POLYNOMIAL RINGS OF SYMMETRIC INVARIANTS 3 we loose the freeness of the module at the second or third step (conjecturally, at the thirdstep!). This also implies that, for g h , , , . . . , i =: g h r i and every r > , k [ g h r i ] is not afree module over the ring of symmetric invariants.Notation. Let Q act on an irreducible affine variety X . Then k [ X ] Q is the algebra of Q -invariant regular functions on X and k ( X ) Q is the field of Q -invariant rational functions.If k [ X ] Q is finitely generated, then X//Q := Spec k [ X ] Q , and the quotient morphism π Q : X → X//Q is induced by the inclusion k [ X ] Q ֒ → k [ X ] . If k [ X ] Q is a graded polynomialring, then the elements of any set of algebraically independent homogeneous generatorsare called basic invariants . If V is a Q -module and v ∈ V , then q v = { ζ ∈ q | ζ · v = 0 } isthe stabiliser of v in q and Q v = { s ∈ Q | s · v = v } is the isotropy group of v in Q ; H o is theidentity component of an algebraic group H .1. P RELIMINARIES ON THE COADJOINT REPRESENTATION
Let Q be a connected affine algebraic group with Lie algebra q . The symmetric algebra S ( q ) over k is identified with the graded algebra of polynomial functions on q ∗ and wealso write k [ q ∗ ] for it.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 Rosenlicht’s theorem [VP89, 2.3], one also has ind q = tr . deg k ( q ∗ ) Q . The“magic number” associated with q is b ( q ) = (dim q + ind q ) / . Since the coadjoint orbitsare even-dimensional, the magic number is an integer. If q is reductive, then ind q = rk q and b ( q ) equals the dimension of a Borel subalgebra. The Poisson bracket { , } in k [ q ∗ ] isdefined on the elements of degree (i.e., on q ) by { x, y } := [ x, y ] . The centre of the Poissonalgebra S ( q ) is S ( q ) q = { H ∈ S ( q ) | { H, x } = 0 ∀ x ∈ q } . Since Q is connected, we alsohave S ( q ) q = S ( q ) Q = k [ q ∗ ] Q .The set of Q - regular elements of q ∗ is q ∗ reg = { η ∈ q ∗ | dim Q · η > dim Q · η ′ for all η ′ ∈ q ∗ } .We say that q has the codim– n property if codim ( q ∗ \ q ∗ reg ) > n . The following useful resultappears in [P07’, Theorem 1.2]: Theorem 1.1.
Suppose that q has the codim – property and there are homogeneous algebraicallyindependent f , . . . , f l ∈ k [ q ∗ ] Q such that l = ind q and P li =1 deg f i = b ( q ) . Then (i) k [ q ∗ ] Q = k [ f , . . . , f l ] and (ii) ( d f ) ξ , . . . , ( d f l ) ξ are linearly independent if and only if ξ ∈ q ∗ reg . Furthermore, if q has the codim– property, then for any collection of algebraically inde-pendent homogeneous f , . . . , f l ∈ k [ q ] Q with l = ind q , one has P li =1 deg f i > b ( q ) . Definition 1 (cf. [P08]) . An algebraic Lie algebra q is said to be n - wonderful , if (i) q has the codim– n property. D. PANYUSHEV AND O. YAKIMOVA (ii) k [ q ∗ ] Q is a polynomial algebra of Krull dimension l = ind q ; (iii) If f , . . . , f l are basic invariants in k [ q ∗ ] Q , then P li =1 deg f i = b ( q ) .For instance, any semisimple Lie algebra is -wonderful.It follows from Theorem 1.1 that if q is 2-wonderful, then Ω q ∗ = q ∗ reg is big. Therefore,Theorem 0.1 applies to all 2-wonderful Lie algebras. (A more precise statement is givenin Corollary 2.5 below.) For instance, it applies to all centralisers of nilpotent elements intypes A n or C n , see [PPY, Theorems 4.2 & 4.4] and [AP17, Section 3].Suppose that k [ q ∗ ] Q is a polynomial ring, but nothing is known about the codim– property. Theorem 0.1 suggests that one needs some tools to decide whether Ω q ∗ is big. Inmany cases, the following assertion is helpful. Proposition 1.2 (see [JS10, Prop. 5.2]) . If k [ q ∗ ] Q is a polynomial ring and Q has no propersemi-invariants in k [ q ∗ ] , then Ω q ∗ is big.Remark . Using some ideas of Knop (see [Kn86, Satz 2]), we can prove a more generalassertion, which we do not need here. Namely,
Let an algebraic group Q act on an irreducible affine factorial variety X . Suppose that X//Q exists (i.e., k [ X ] Q is finitely generated) and k [ X ] contains no proper Q -semi-invariants. Let X sm denote the smooth locus of X and π Q : X → Y := X//Q the quotient morphism. Set Ω X = { x ∈ X sm | π Q ( x ) ∈ Y sm & ( d π Q ) x is onto } . Then Ω X is big.
2. T
AKIFF ALGEBRAS AND THEIR SYMMETRIC INVARIANTS
By definition, the m -th Takiff algebra of q is q h m i := q ⊗ k [ T ] / ( T m +1 ) . In particular, q h i = q ⋉ q ab is the semi-direct product, where the second factor is an abelian ideal. For j m ,the image of q ⊗ T j in q h m i is denoted by q [ j ] . A typical element of q h m i can be writtenas x = ( x , x , . . . , x m ) , where x j ∈ q [ j ] . Likewise, we have q h m i ∗ ≃ L mj =0 ( q ∗ [ j ] ) as vectorspace, and ξ = ( ξ , ξ , . . . , ξ m ) is an element of q h m i ∗ , where ξ j ∈ q ∗ [ j ] . Then the pairingof q h m i and q h m i ∗ is given by < x , ξ > q h m i = P mi =0
Then q h m i u = L mj =1 q [ j ] is an ad -nilpotent ideal of q h m i and the corresponding connectedalgebraic group is Q h m i ≃ Q ⋉ exp( q h m i u ) = Q ⋉ Q h m i u . (If Q is reductive, then Q h m i u is the unipotent radical of Q h m i .) For a non-Abelian Q , theunipotent group Q h m i u is commutative if and only if m = 1 .By [RT92, 2.8], one has ind q h m i = ( m + 1) · ind q . Hence also b ( q h m i ) = ( m + 1) · b ( q ) .Moreover,(2 · ξ ∈ q h m i ∗ reg if and only if ξ m ∈ q ∗ reg . Therefore, the presence of codim– n property for q implies that for q h m i .A general method for constructing symmetric invariants of q h m i is presented in [RT92].Suppose that f ∈ k [ q ∗ ] is homogeneous. Recall that d f ∈ Mor( q ∗ , q ) is the differential of f . Consider ξ ǫ as an element of q ∗ ⊗ k [ ǫ ] with ǫ m +1 = 0 , and expand f ( ξ ǫ ) as a polynomialin ǫ : f ( ξ m + ǫξ m − + · · · + ǫ m − ξ + ǫ m ξ ) = m X j =0 F j ( ξ ) ǫ j . It is readily seen that F ( ξ ) = f ( ξ m ) and F ( ξ ) = < ( d f ) ξ m , ξ m − > q . More generally, thefollowing assertion is true. Proposition 2.1 (see [RT92, Section III]) . For any j ∈ { , , . . . , m } , we have (i) F j ( ξ ) = < ( d f ) ξ m , ξ m − j > q + H j ( ξ m , . . . , ξ m − j +1 ) for some H j ∈ k [ q h m i ∗ ] ; (ii) If f ∈ k [ q ∗ ] Q , then every F j is a symmetric invariant of q h m i , i.e., F j ∈ k [ q h m i ∗ ] Q h m i . Let f , . . . , f l be a set of basic invariants in k [ q ∗ ] Q , where l = ind q . Using the aboveconstruction of [RT92], we associate to each f i the set of Q h m i -invariants F i , . . . , F mi . Now,we are ready to state precisely our main result. Theorem 2.2.
Let Q be a connected algebraic group such that k [ q ∗ ] Q = k [ f , . . . , f l ] is a gradedpolynomial ring, where l = ind q . Set Ω q ∗ = { ξ ∈ q ∗ | ( d f ) ξ ∧ · · · ∧ ( d f l ) ξ = 0 } , and assumethat Ω q ∗ is big. For any m > , we then have (i) k [ q h m i ∗ ] Q h m i u is a graded polynomial ring of Krull dimension dim q + ml , which is freelygenerated by the coordinate functions on q ∗ [ m ] and the { F ji } ’s with i = 1 , . . . , l and j =1 , . . . , m . (ii) the Takiff algebra q h m i has the same properties as q , i.e., – k [ q h m i ∗ ] Q h m i is a graded polynomial ring of Krull dimension ind q h m i = ( m + 1) l .(It is freely generated by the { F ji } ’s with i = 1 , . . . , l and j = 0 , . . . , m .) – Ω q h m i ∗ = L m − j =0 q ∗ [ j ] × Ω q ∗ is big, where Ω q ∗ ⊂ q ∗ [ m ] ≃ q ∗ . D. PANYUSHEV AND O. YAKIMOVA
Proof. (i)
Recall that q h m i ≃ q ⋉ q h m i u , where q = q [0] and q h m i u = L mj =1 q [ j ] , and Q h m i = Q ⋉ Q h m i u . Here Q h m i u is a unipotent normal subgroup of Q h m i .Note that the subspace q [ m ] ⊂ q h m i regarded as a subset of k [ q h m i ∗ ] belongs to thesubalgebra of Q h m i u -invariants, and F i = f i ∈ S [ q [ m ] ] . Let A denote the subalgebra of k [ q h m i ∗ ] Q h m i u generated by q [ m ] and { F ji } with j = 1 , . . . , m and i = 1 , . . . , l . (Note that wedo not include F , . . . , F l in the generating set for A !)For x = ( x , . . . , x m ) with x i ∈ q [ i ] , we say that x j = 0 is the lowest component of x , if x = · · · = x j − = 0 . Now, ( d F ji ) ξ ∈ q h m i and using Proposition 2.1(i), one readily verifiesthat its lowest component is (cid:0) ( d F ji ) ξ (cid:1) m − j = ( d f i ) ξ m ∈ q [ m − j ] , where j = 0 , , . . . , m − .Clearly, these lowest components are linearly independent if and only if ξ m ∈ Ω q ∗ . If v , . . . , v dim q is a basis for q [ m ] , then ( d v i ) ξ = v i ∈ q [ m ] . Since all these differentials havea block-triangular form w.r.t. the decomposition q h m i = L mi =1 q [ i ] (cf. Table 1), it followsthat the differentials per se are linearly independent at ξ if and only if ξ m ∈ Ω q ∗ . Therefore,the polynomials v , . . . , v dim q , and { F ji } with j = 1 , . . . , m, i = 1 , . . . , l are algebraically independent and generate A . As the differentials of this family are lin-early independent on the big open subset L m − j =0 q ∗ [ j ] × Ω q ∗ of q h m i ∗ , Theorem 1.1 in [PPY]guarantee us that A is an algebraically closed subalgebra in k [ q h m i ∗ ] , of Krull dimension dim q + ml .On the other hand, if ξ = (0 , . . . , , ξ m ) and ξ m ∈ q ∗ reg , then dim Q h m i u · ξ = m (dim q − l ) .Hence tr . deg k [ q h m i ∗ ] Q h m i u dim q h m i − dim Q h m i u · ξ = dim q + ml . Therefore A ⊂ k [ q h m i ∗ ] Q h m i u is an algebraic extension, which implies that A = k [ q h m i ∗ ] Q h m i u . In otherwords, k [ q h m i ∗ ] Q h m i u = k [ q ∗ [ m ] ][ F ji , i = 1 , . . . , l ; j = 1 , . . . , m ] . (ii) Since Q h m i ≃ Q ⋉ Q h m i u and the F ji ’s are already Q h m i -invariant (Prop. 2.1(ii)), itfollows from part (i) that k [ q h m i ∗ ] Q h m i = ( k [ q h m i ∗ ] Q h m i u ) Q = k [ q ∗ [ m ] ] Q [ { F ji } , i l ; 1 j m ]= k [ { F ji } , i l ; 0 j m ] . Furthermore, the differentials of the total set of generators { F ji } , with the value j = 0 in-cluded, are also linearly independent if and only if ξ m ∈ Ω q ∗ ⊂ q ∗ [ m ] , see [RT92, Lemma 3.3]and Table 1. Therefore, Ω q h m i ∗ = L m − j =0 q ∗ [ j ] × Ω q ∗ is big. (cid:3) For future use, we record a by-product of the proof:
Corollary 2.3. ξ ∈ Ω q h m i ∗ ⇐⇒ ξ m ∈ Ω q ∗ .Remark . It appears that Theorem 2.2 is fully analogous to [P07, Theorem 11.1], wherethe polynomiality of invariants for the adjoint representation of q h m i is studied. AKIFF ALGEBRAS WITH POLYNOMIAL RINGS OF SYMMETRIC INVARIANTS 7 T ABLE
1. Components of the differentials of basic invariants q [ m ] q [ m − q [ m − . . . . . . . . . q [0] ( d F ) ξ ( d f ) ξ m ( d F l ) ξ ( d f l ) ξ m ( d F ) ξ ∗ ( d f ) ξ m ( d F l ) ξ ∗ ( d f l ) ξ m ( d F ) ξ ∗ ∗ ( d f ) ξ m . . . . . . 0... ... ... ... . . . . . . 0 ( d F l ) ξ ∗ ∗ ( d f l ) ξ m . . . . . . 0. . . . . . . . . . . . . . . . . . . . . Corollary 2.5. If q is an n -wonderful algebra for n > , then so is q h m i for any m ∈ N .Proof. Let us check that the properties of Definition 1 carry over from q to q h m i . • As noted above, the presence of codim– n property for q implies that for q h m i . Wealso have dim q h m i = ( m + 1) · dim q and ind q h m i = ( m + 1) · ind q . • If q is 2-wonderful, then ( d f ) ξ , . . . , ( d f l ) ξ are linearly independent if and only if ξ ∈ q ∗ reg (Theorem 1.1). Hence Ω q ∗ = q ∗ reg and its complement does not contain divisors.Therefore, k [ q h m i ∗ ] Q h m i is polynomial ring of Krull dimension ( m + 1) l = ( m + 1) · ind q ,freely generated by the F ji ’s. • Clearly, deg F ji = deg f i for all i and j . Therefore l X i =1 m X j =0 deg F ji = ( m + 1) l X i =1 deg f i = ( m + 1) b ( q ) = b ( q h m i ) . (cid:3) Corollary 2.6 (cf. [MS16, Thm. 5.4]) . For any r -tuple m , . . . , m r , the truncated multi-currentalgebra q h m , . . . , m r i has a polynomial ring of symmetric invariants.Proof. A truncated multi-current algebra of any q is obtained as an iteration of variousTakiff algebras. That is,(2 · ˆ q := q h m , . . . , m r i ≃ (cid:16) . . . (cid:0) ( q h m i ) h m i (cid:1) . . . (cid:17) h m r i . Therefore, if q satisfies the hypotheses of Theorem 2.2, then so is ˆ q . In particular, k [ˆ q ∗ ] ˆ Q isa polynomial ring. (cid:3) D. PANYUSHEV AND O. YAKIMOVA
Note that if q = g is semisimple, then one can use results of [RT92] only for the firstiteration g ❀ g h m i , because afterwards the algebra in question becomes non-reductive. Remark . An essential point in our proof of Theorem 2.2 is the use of Theorem 1.1in [PPY]. This ensures that the subalgebra A is algebraically closed in k [ q h m i ∗ ] and hence A = k [ q h m i ∗ ] Q h m i u for the dimension reason. However, one can use instead an invariant-theoretic (geometric) argument related to Igusa’s lemma (see e.g. [VP89, Theorem 4.12])or [P07, Lemma 6.1]). Namely, consider the morphism τ : q h m i ∗ → q ∗ [ m ] × A ml =: Y given by τ ( ξ ) = ( ξ m , F ( ξ ) , . . . , F l ( ξ ) , . . . , F m ( ξ ) , . . . , F ml ( ξ )) . From the assumption on Ω q ∗ and a ”triangular” form of { F ji } (see Prop. 2.1(i)), one derives that(1) Im τ ⊃ Ω q ∗ × A ml , where the RHS is a big open subset of Y ;(2) for any y ∈ Ω q ∗ × A ml , the fibre τ − ( y ) is a sole Q h m i u -orbit.Then Igusa’s lemma asserts that k [ Y ] ≃ k [ q h m i ∗ ] Q h m i u , i.e., Y ≃ q h m i ∗ //Q h m i u and τ = π Q h m i u . (Cf. the similar use of Igusa’s lemma in [P07, Theorems 6.2 & 11.1] and [P07’,Theorem 5.2].)3. P REHOMOGENEOUS VECTOR SPACES AND RINGS OF SEMI - INVARIANTS
Here we show that some old results of Sato–Kimura [SK77] on prehomogeneous vectorspaces allow us to construct Lie algebras satisfying the hypotheses of Theorem 2.2.3.1.
Prehomogeneous vector spaces.
Let H ⊂ GL ( V ) be a representation of a con-nected group H having an open orbit in V , i.e., V is a prehomogeneous vector space w.r.t. H . By [SK77, § H -semi-invariants in k [ V ] , denoted k [ V ] h H i , is poly-nomial. More precisely, let O ⊂ V be the open H -orbit and D , . . . , D l all simple di-visors in V \ O (we do not need the irreducible components of codimension > in V ). If D i = { f i = 0 } , then f i ∈ k [ V ] h H i , f , . . . , f l are algebraically independent, and k [ V ] h H i = k [ f , . . . , f l ] . Moreover, let λ i : H → k × be the H -character corresponding to f i , i.e., h · f i = λ i ( h ) f i for all h ∈ H . Then the differentials of λ i ’s are linearly independentand ˜ H := { h ∈ H | λ i ( h ) = 1 ∀ i } o is of codimension l in H . Then [ H, H ] ⊂ ˜ H ⊂ H and k [ V ] [ H,H ] = k [ V ] ˜ H = k [ V ] h H i is a polynomial ring.3.2. Frobenius Lie algebras.
Suppose that ind h = 0 , i.e., h is Frobenius . Then H hasan open orbit in h ∗ and the above results apply to V = h ∗ . Then k [ h ∗ ] h H i = k [ h ∗ ] ˜ H is apolynomial ring of Krull dimension dim H − dim ˜ H = ind ˜ h . Note that k [˜ h ∗ ] = S (˜ h ) ⊂ S ( h ) = k [ h ∗ ] , and an important additional feature of the “coadjoint” situation is that k [ h ∗ ] ˜ H ⊂ k [˜ h ∗ ] ,see [BGR, Kap. II, § k [ h ∗ ] ˜ H = k [˜ h ∗ ] ˜ H , i.e., ˜ h has a polynomial ring of symmetric AKIFF ALGEBRAS WITH POLYNOMIAL RINGS OF SYMMETRIC INVARIANTS 9 invariants whose Krull dimension equals ind ˜ h . By the very construction, ˜ H has no propersemi-invariants in k [ h ∗ ] and hence in k [˜ h ∗ ] . It then follows from Proposition 1.2 that Ω ˜ h ∗ isbig. Thus, Theorem 2.2 applies to ˜ h , and hence ˜ h h m i has a polynomial ring of symmetricinvariants for any m > . Remark . More generally, for any
Lie algebra h , the ring of symmetric semi-invariants k [ h ∗ ] h H i (i.e., the Poisson semi-centre of S ( h ) = k [ h ∗ ] ) is isomorphic to the ring of symmetricinvariants of a canonically defined subalgebra ˜ h ⊂ h [BGR, Kap. II, § ˜ h is called the canonical truncation of h . It has the property that dim h − dim ˜ h = ind ˜ h − ind h [OVdB, Lemma 3.7], hence b ( h ) = b (˜ h ) . Furthermore, since ˜ H has no proper semi-invariants in k [˜ h ∗ ] , k (˜ h ∗ ) ˜ H is the field of fractions of k [˜ h ∗ ] ˜ H and theKrull dimension of k [˜ h ∗ ] ˜ H equals ind ˜ h . Therefore, if k [ h ∗ ] h H i = k [˜ h ∗ ] ˜ H is a polynomial ring,then Proposition 1.2 and Theorem 2.2 apply to ˜ h , and hence ˜ h h m i has a polynomial ringof symmetric invariants for all m > . In the special case, where h is Frobenius, this isalready explained in the previous paragraph.Let us illustrate this theory in both Frobenius and non-Frobenius cases. Example 3.2.
Let G be a simple algebraic group with Lie ( G ) = g , b a Borel subalgebraof g , and [ b , b ] = u . The corresponding connected subgroups of G are B and U . Herewe are interested in the symmetric invariants of b , u , and the canonical truncation of b .Most of these results are due to Kostant [K12] and Joseph [J77]. (Actually, many Kostant’sresults are rather old and had been cited in [J77].) Our idea is to demonstrate utility of theSato–Kimura theory in this context. ( ♦ ) If ind b = 0 , then ind u = rk g and ˜ b = [ b , b ] = u . Hence S ( b ) U = S ( u ) U is apolynomial ring of Krull dimension rk g . As explained above, Theorem 2.2 applies to u = ˜ b . It is well known that ind b = 0 ifand onlyif g ∈ { B n , C n , D n , E , E , F , G } .Let f , . . . , f rk g be the basic invariants in S ( u ) U . Their weights and degrees are pointedout in [J77, Tables I,II], with some corrections in [FJ05, Annexe A]. It follows from thosedata that P rk g i =1 deg f i < b ( u ) = dim b unless g = C n . This means that, for all but one case,the codim– property does not hold for u (use Theorem 1.1 !). ( ♦ ) If ind b > , then ind u < rk g and S ( u ) U is a proper subalgebra of S ( b ) U . (Actually,one always has ind u + ind b = rk g .) There are two possibilities to construct a suitablesubalgebra of b : one is related to the Sato–Kimura approach, and the other exploits thecanonical truncation. h - 1 - i Since B has a dense orbit in u ∗ [K12], one applies Sato–Kimura results to V = u ∗ , H = B , and U = [ B, B ] . This shows that S ( u ) U is still a polynomial ring. Moreover, Ω u ∗ is big for the same reason as above. For all these cases (i.e., g ∈ { A n , D n +1 , E } ), we have P i deg f i < b ( u ) . Hence there is no codim– property for u , but Theorem 2.2 applies to u . h - 2 - i Now, the canonical truncation of b is a subalgebra that properly contains u .Namely, the toral part of ˜ b has dimension ind b . If b = t ⊕ u and ∆ + is the set of positiveroots (= roots of u ), then one canonically constructs the cascade K of strongly orthogonalroots in ∆ + ( Kostant’s cascade ), see [J77, Section 2]. If K = { γ , . . . , γ t } , then ind b = dim t − t and ˜ b = ˜ t ⊕ u , where ˜ t = { γ , . . . , γ t } ⊥ . Thus, we obtain that k [ b ∗ ] U = k [ b ∗ ] h B i = k [˜ b ∗ ] ˜ B . By [J77, 4.16], S ( b ) U is a polynomial ring of Krull dimension rk g . Hence Theorem 2.2applies to ˜ b .The output of this example is that, for any simple Lie algebra g , our main theoremapplies to both ˜ b (the canonical truncation of b ) and u = [ b , b ] . These two subalgebras of b coincide ifand only if b isFrobenius.4. M ORE EXAMPLES
We provide other applications of Theorem 2.2 to Lie algebras with or without the codim– property. Example 4.1.
Let G ⊂ SL ( V ) be a representation of a connected semisimple algebraicgroup. Consider the semi-direct product q = g ⋉ V ab . The corresponding connectedgroup Q = G ⋉ exp( V ) has no non-trivial characters, hence k [ q ∗ ] does not contain proper Q -semi-invariants. Therefore, if (we know that) k [ q ∗ ] Q is a polynomial ring, then Ω q ∗ isbig (use Proposition 1.2) and Theorem 2.2 applies to q . The classification of representa-tion ( G : V ) of simple algebraic groups G such that k [ q ∗ ] Q is a polynomial ring is thesubject of an ongoing project initiated by the second author. First non-trivial results for G = SL n are found in [Y17’], and the representations of the exceptional groups are consid-ered in [PY17]. The representations of SO n and Sp n will be handled in our forthcomingpublication. (However, it is not always easy to decide whether the codim– propertyholds for such q .)Consider a concrete elementary example, where everything can be verified by hand.For an n -dimensional vector space V with n > , take the semi-direct product q = sl ( V ) ⋉ n V = sl n ⋉ n k n . The elements of n V (resp. n V ∗ ) are regarded as n × n matrices, where sl n acts via left (resp. right) multiplications. Since k [ n V ∗ ] SL ( V ) = k [det] and generic stabilisersfor the action ( SL ( V ) : n V ) are trivial, we have k [ q ∗ ] Q = k [ n V ∗ ] SL ( V ) = k [det] . (The first equality here stems from [P07, Theorem 6.4].) Hence ind q = 1 and b ( q ) = n .For an n × n matrix η , one has d (det) η = 0 ⇔ rk η < n − . Therefore, d (det) vanishes onthe determinantal variety of matrices of rank n − , which is of codimension in n V ∗ .Thus, Ω q ∗ is big. AKIFF ALGEBRAS WITH POLYNOMIAL RINGS OF SYMMETRIC INVARIANTS 11
On the other hand, q ∗ reg = sl ∗ n × ( n V ) ∗ det is a principal open subset, i.e., q ∗ \ q ∗ reg = sl ∗ n ×{ det = 0 } is a divisor. Hence the codim– property does not hold here. This also followsfrom the fact that n = deg(det) < b ( q ) = n . Example 4.2.
Let
Hei n be the Heisenberg Lie algebra of dimension n + 1 . It has a basis x , . . . , x n , y , . . . , y n , z such that the only nonzero brackets are [ x i , y i ] = z , i = 1 , . . . , n .Then ind ( Hei n ) = 1 and k [ Hei ∗ n ] Hei n = k [ z ] . Therefore, Ω Hei n = Hei ∗ n and Theorem 2.2applies here. It is easily seen that the hyperplane { ξ ∈ Hei ∗ n | <ξ, z> = 0 } consists of thefixed points of the Heisenberg group. Hence, Hei n does not have the codim– property.This has the following application to centralisers of nilpotent elements:Let G be a simple group of type G . If G · e ⊂ g is the subregular nilpotent orbit, then dim g e = 4 and g e ≃ Hei ⊕ k e . Example 4.3.
Let e ∈ g be nilpotent. Methods of [CM16] provide the polynomiality of k [ g ∗ e ] g e for some nilpotent orbits that are not treated in [PPY]. In particular, Tables 2 and 3in [CM16] list such orbits for G of type E and F . For those of them, where the reductivepart of g e is semisimple, we know for sure that k [ g ∗ e ] has no proper G oe -semi-invariants,and hence Theorem 2.2 applies. Specifically, the four suitable E -orbits have the Dynkin-Bala-Carter labels E ( a ) , A , D , A + A , whereas all six F -orbits are suitable for us. Example 4.4.
Associated with any parabolic subalgebra p of g , there is an interesting con-traction of g , which is called a parabolic contraction , see [PY13]. If p = b , then such a con-traction has much better properties [PY12]. Let b − be an opposite Borel and u − = [ b − , b − ] .Then g = b ⊕ u − is a vector space sum. The contraction in question is q := b ⋉ ( u − ) ab ,where ( u − ) ab is an abelian ideal of q and ( u − ) ab is regarded as b -module via isomorphism g / b ≃ u − . Note that q is solvable.By [PY12, Section 3], we have (1) ind q = rk g , (2) k [ q ∗ ] Q is a polynomial ring, and (3) the degrees of basic invariants are the same as those for g . In particular, b ( q ) = b ( g ) and if f , . . . , f l are the basic invariants in k [ q ∗ ] Q , then P li =1 deg f i = b ( q ) .However, q does not have the codim– property unless g is of type A l [PY12, Theo-rem 4.2]. Furthermore, Ω q ∗ is not big, if g = A l [Y14, Remark 5.3]. Therefore, Theorem 2.2does not apply to b ⋉ ( u − ) ab , if g = A l . But one can look at the canonical truncation of q ,where the situation improves considerably. Following [Y14, Sect. 5], consider ˜ q = u ⋉ ( u − ) ab ⊂ b ⋉ ( u − ) ab = q . Here one has ˜ q = [ q , q ] , ind ˜ q = ind q +(dim q − dim ˜ q ) = 2rk g , and hence b (˜ q ) = b ( q ) = b ( g ) .By [Y14, Theorem 5.9], S (˜ q ) ˜ q is a polynomial ring of Krull dimension g . The situationwith the codim– property for ˜ q remains the ”same” as for q , but Ω ˜ q ∗ is already a big opensubset of ˜ q ∗ (see the proof of Theorem 5.9 in [Y14]). Thus, Theorem 2.2 applies to ˜ q for all simple g . Example 4.5.
Let g = g ⊕ g be a Z -grading of a simple Lie algebra g and q = g ⋉ g ab the related Z -contraction. Then ind q = rk g and the codim– property is always satisfiedhere (see [P07’]). Here g is reductive but not necessarily semisimple, and k [ q ∗ ] Q is apolynomial ring (in rk g variables) if and only if the restriction homomorphism k [ g ] G → k [ g ] G is onto [Y17, Sect. 6]. This excludes only four Z -gradinds related to the algebrasof type E n . Example 4.6.
Let p and p ′ be two parabolic subalgebras of g such that p + p ′ = g . Then s = p ∩ p ′ is called a seaweed (or biparabolic ) subalgebra of g [P01]. By work of Joseph andhis collaborators, it is known in many cases that k [ s ∗ ] h S i is a polynomial ring. In particular,this is true for any s , if g is of type A n or C n [J07]. (See also a summary of known resultsand other good cases in [FP].) Therefore, in all such good cases, the canonical truncationof s (= truncated biparabolic in Joseph’s terminology) is a good example for Theorem 2.2.5. O N THE EQUIDIMENSIONALITY
Whenever a connected algebraic group Q has the property that k [ q ∗ ] Q is a polynomialring, it is natural to inquire whether it is true that k [ q ∗ ] is a free k [ q ∗ ] Q -module. Thelatter is equivalent to that the enveloping algebra U ( q ) is a free module over its centre Z ( q ) ≃ k [ q ∗ ] Q . Assuming that k [ q ∗ ] Q is a polynomial ring, i.e., q ∗ //Q is an affine space, thewell-known geometric answer to this inquiry is that k [ q ∗ ] is a free k [ q ∗ ] Q -module if and only if π Q : q ∗ → q ∗ //Q is equidimensional,i.e., equivalently, the zero-fibre of π Q , π − Q ( π Q (0)) , has the ‘right’ dimension dim q − dim q ∗ //Q . In the setting of Takiff algebras, one can raise the following: Question 1.
Suppose that the hypotheses of Theorem 2.2 hold for q and π Q is equidimensional. Isit true that π Q h m i : q h m i ∗ → q h m i ∗ //Q h m i is equidimensional, too? As we shall see below, the general answer to this question is “no”. The celebrated positiveresult is that if g is semisimple, then the zero-fibre of π G h m i is irreducible and π G h m i isequidimensional for any m ∈ N [M01, Appendix]. The reason is that the usual nilpotentcone N ⊂ g ≃ g ∗ is an irreducible complete intersection, and it has rational singularities.Here N h m i := π − G h m i ( π G h m i (0)) is a jet scheme of N .For m = 1 , these results are obtained in [G94] via a case-by-case argument. (See alsoanother approach and a generalisation in [P07, Theorem 10.2].)In this section, we prove that the equidimensionality does not carry over to the multi-current setting, even for semisimple g . Let ˆ q = q h m , . . . , m r i be a truncated multi-currentalgebra of q , cf. (0 · ˆ q = L i ,...,i r q [ i ,...,i r ] and likewise for ˆ q ∗ ,where i j m j , j = 1 , . . . , r . It then follows from (2 ·
1) and the iteration process (2 · AKIFF ALGEBRAS WITH POLYNOMIAL RINGS OF SYMMETRIC INVARIANTS 13 that ξ = ( ξ [ i ,...,i r ] ) ∈ ˆ q ∗ reg ⇐⇒ ξ [ m ,...,m r ] ∈ q ∗ reg (see also Prop. 4.1(b) in [MS16]). Assume that q satisfies all the assumptions of Theo-rem 2.2 and set N = π − Q ( π Q (0)) ⊂ q ∗ . Then N h m , . . . , m r i ⊂ ˆ q ∗ stands for the zero-fibreof π ˆ Q : ˆ q ∗ → ˆ q ∗ // ˆ Q . We work below with the case in which all m i = 1 . Then ˆ q is obtainedas iteration of semi-direct products, the first step being q ❀ q ⋉ q ab = q h i . Let us inves-tigate the relation between N and N h i . This will also apply below to the passage from N h i to N h , i .Recall that ξ = ( ξ , ξ ) is an element of q h i ∗ . If k [ q ∗ ] Q = k [ f , . . . , f l ] with l = ind q , then k [ q h i ∗ ] Q h i is freely generated by F , . . . , F l , F , . . . , F l , where F i depends only on ξ and F i ( ξ , ξ ) = < ( d f i ) ξ , ξ > q . Therefore(5 · N h i = { ( ξ , ξ ) | ξ ∈ N & < ( d f i ) ξ , ξ > q = 0 ∀ i } . Since d f i is a Q -equivariant morphism from q ∗ to q , we have ( d f i ) ξ ∈ q ξ . Moreover, if ξ ∈ q ∗ reg ∩ Ω q ∗ , then { ( d f ) ξ , . . . , ( d f l ) ξ } is a basis for q ξ . Consider the stratification of N determined by the basic invariants f , . . . , f l . Set X i, N = { ξ ∈ N | dim span (cid:0) { ( d f ) ξ , . . . , ( d f l ) ξ } (cid:1) i } . Then { } = X , N ⊂ X , N ⊂ · · · ⊂ X l, N = N . If N = S j N j is the irreducible decompo-sition, then X i, N j is similarly defined for any j . Set X oi, N j = X i, N j \ X i − , N j for i > and X o , N j = { } . Clearly, each X oi, N j is irreducible and open in X i, N j . However, X oi, N j can beempty for some i, j . It follows from (5 ·
1) that p : N h i → N , ( ξ , ξ ) ξ , is a surjectiveprojection and dim p − ( X oi, N j ) = dim X oi, N j + dim q − i. Since q h i has a polynomial ring of symmetric invariants, with l basic invariants F . . . , F l , F , . . . , F l , one can consider the corresponding stratification of N h i : { } = X , N h i ⊂ X , N h i ⊂ · · · ⊂ X l, N h i = N h i . Lemma 5.1.
We have p − ( X oi, N ) ⊂ l + i [ j =2 i X oj, N h i .Proof. By definition, dim span (cid:0) { ( d f ) ξ , . . . , ( d f l ) ξ } (cid:1) = i for ξ ∈ X oi, N . This clearly im-plies that, for ξ = ( ξ , ξ ) ∈ p − ( ξ ) , we have dim span (cid:0) { ( d F ) ξ , . . . , ( d F l ) ξ } (cid:1) = i and dim span (cid:0) { ( d F ) ξ , . . . , ( d F l ) ξ } (cid:1) > i (cf. Table 1 with m = 1 ). Furthermore, the lowestcomponents of ( d F j ) ξ and ( d F j ) ξ belong to different graded pieces of q h i . (cid:3) By Lemma 5.1, the closures of p − ( X ol, N j ) for all j are the only subvarieties of N h i thatmeet Ω q h i ∗ . Therefore, if X ol, N j = ∅ , then p − ( X ol, N j ) is an irreducible component of N h i of dimension dim N j + dim q − l . Since dim N j > dim q − l for all j , one readily obtains Proposition 5.2. If q satisfies all the assumptions of Theorem 2.2, then the following two condi-tions are equivalent: (1) π Q h i is equidimensional, i.e., dim N h i = dim q h i − ind q h i = 2(dim q − l ) ; (2) (i) π Q is equidimensional, i.e., dim N j = dim q − l for all j ; (ii) X ol, N j = ∅ for all j (i.e., N j ∩ Ω q ∗ = ∅ ); (iii) codim N j ( X oi, N j ) > l − i for i < l . This yields a sufficient condition for the absence of equidimensionality of π Q h i : Corollary 5.3.
If there is an irreducible component N j of N such that N j ∩ Ω q ∗ = ∅ , then dim p − ( N j ) > q − l ) . Hence π Q h i is not equidimensional. We say that such N j is a bad irreducible component of N . Remark . If N is irreducible and dim N = dim q − l , then a similar analysis shows that N h i is irreducible if and only if conditions (i), (ii), and (iii)’ hold, where (i), (ii) are asabove, with N in place of N j , and the last one is a bit stronger than (iii) : (iii)’ codim N ( X oi, N ) > l − i for i < l .For, the closure of p − ( o X l, N ) is always an irreducible component of N h i of the ‘right’dimension q − l ) , and we need the condition that p − ( X oi, N ) does not yield anothercomponent, i.e., dim p − ( X oi, N ) < q − l ) for i < l .From now on, we assume that q = g is a simple Lie algebra of rank l . Let us recall someproperties of the nilpotent cone N ⊂ g ∗ ≃ g : • N is irreducible and contains finitely many G -orbits; • X ol, N is the principal (or, regular ) nilpotent orbit; • X l − , N is irreducible of dimension dim N − and X ol − , N = ∅ (it contains the sub-regular nilpotent orbit as a dense open subset). Moreover, if deg f . . . deg f l ,then deg f l − < deg f l and ( d f l ) ξ = 0 for all ξ ∈ X l − , N .Then N h i is also irreducible, and for the projection p : N h i → N , we have: – p − ( X ol, N ) is the open dense G h i -orbit in N h i , of dimension g − l ) ; – dim p − ( X ol − , N ) = 2(dim g − l ) − . Hence the closure of p − ( X ol − , N ) is a simpledivisor, say D , in N h i . By Lemma 5.1, p − ( X ol − , N ) ⊂ X o l − , N h i ∪ X o l − , N h i .The next iteration replaces g h i with g h , i ≃ g h i ⋉ g h i ab and provides the surjectiveprojection p : N h , i ≃ N h ih i → N h i . Here we are interested in p − ( D ) . There is adichotomy: either (1) D ∩ X o l − , N h i = ∅ or (2) D ⊂ X l − , N h i . • In the first case, dim p − ( D ) = 4(dim g − l ) and it is an irreducible component of N h , i that is different from the closure of p − ( X o l, N h i ) . In other words, p − ( D ) is a badirreducible component of N h , i . Hence π g h , , i is not equidimensional by Corollary 5.3. AKIFF ALGEBRAS WITH POLYNOMIAL RINGS OF SYMMETRIC INVARIANTS 15 • In the second case, dim p − ( D ) = 4(dim g − l ) + 1 . Hence π g h , i is already not equidi-mensional and then π g h , , i is not equidimensional, too.Thus, we have proved Theorem 5.5.
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OSCOW
USSIA
E-mail address : [email protected] (O. Yakimova) U NIVERSIT ¨ AT ZU
K ¨
OLN , M
ATHEMATISCHES I NSTITUT , W
EYERTAL
OLN ,D EUTSCHLAND
E-mail address ::