Semi-direct products of Lie algebras and covariants
The coadjoint representation of a connected algebraic group with Lie algebra is a thrilling and fascinating object. Symmetric invariants of (= -invariants in the symmetric algebra ) can be considered as a first approximation to the understanding of the coadjoint action and coadjoint orbits. In this article, we study a class of non-reductive Lie algebras, where the description of the symmetric invariants is possible and the coadjoint representation has a number of nice invariant-theoretic properties. If is a semisimple group with Lie algebra and is -module, then we define to be the semi-direct product of and . Then we are interested in the case, where the generic isotropy group for the -action on is reductive and commutative. It turns out that in this case symmetric invariants of can be constructed via certain -equivariant maps from to (”covariants”).
Key words and phrases:index of Lie algebra, coadjoint representation, symmetric invariants
2010 Mathematics Subject Classification:14L30, 17B08, 17B20, 22E46
May 6, 2017
The coadjoint representation of an algebraic group is a thrilling and fascinating object. It encodes information about many other representations of and . Yet, it is a very difficult object to study. Symmetric invariants of can be considered as a first approximation to the understanding of the coadjoint action and coadjoint orbits. The goal of this article is to describe and study a class of non-reductive Lie algebras, where the description of the symmetric invariants is possible and the coadjoint representation has a number of nice invariant-theoretic properties. The ground field is algebraically closed and of characteristic .
Let be a (finite-dimensional rational) representation of a connected algebraic group with
. We form a new Lie algebra as the semi-direct product , where
is an abelian ideal. Then can be regarded as a connected algebraic group with
, where is a commutative unipotent normal subgroup. Here
and the algebra of symmetric invariants contains as a
subalgebra. But finding the other invariants is a difficult and non-trivial problem. Nevertheless,
one can use certain -equivariant morphisms for constructing -invariants in
. Our observation is that if a generic stabiliser for is toral, then this is usually sufficient for obtaining a generating set for .
For -modules and , let denote the graded -module of polynomial morphisms . There is the natural map such that for . If , then one obtains a -invariant polynomial by letting (Lemma 3.1). Furthermore, if is also -equivariant, then . Likewise, if denotes the -module of -equivariant morphisms (covariants), then there is the map , which is the restriction of . Suppose that is reductive and is a generic isotropy group for , with . It is known that [jac], and we prove that whenever the action is stable (Theorem 2.1). Hence if and only if the adjoint representation of is trivial; in particular, must be toral. The main hope behind our considerations is that if is generated by -equivariant morphisms, then and the polynomials with together generate the whole ring . Actually, we prove this under certain additional constraints, see below. For our general theorems, we also need the codimension-2 condition (= CC) on the set of -regular elements in . This means that does not contain divisors.
Our results concern the case in which is semisimple and CC holds for . Suppose that there are linearly independent homogeneous morphisms such that and , where is the minus degree of the Poincaré series of . Then we prove that is a free -module with basis and is a polynomial ring (Theorem 3.3). Under certain additional assumptions (namely, and is not contained in a proper normal subgroup of ), we then prove that such are necessarily -equivariant and hence is a free -module and . Furthermore, if is a polynomial ring, then the Kostant (regularity) criterion holds for (Theorem 3.6). In case , our results are stronger and more precise, see Theorem 3.11.
Using Elashvili’s classification [alela1, alela2], one can write down the arbitrary representations of simple groups and irreducible representations of arbitrary semisimple groups with toral generic stabilisers. We then demonstrate that for most of these representations, the assumptions of our general theorems are satisfied. In each example, an emphasise is made on an explicit construction of morphisms and verification that they belong to . In some cases, the construction is rather intricate and involved, cf. Examples 5.1 and LABEL:ex:sl-2-slag.
The structure of the paper is as follows. In Section 1, we gather some standard well-known facts on semi-direct products, regular elements, and generic stabilisers. In Section 2, we consider the -module of polynomial morphisms and the associated exact sequence . We also compute the rank of the -module . Section 3 is the heart of the article. Here we present our main results on semi-direct products related to the case in which the CC holds for , a generic stabiliser for is toral, and there are linearly independent morphisms such that and . In Section 4, we explain how to verify that the CC holds for a -module . Examples of representations with toral generic stabilisers are presented in Sections 5 and LABEL:sect:primery2. For each example, we explicitly construct the morphisms such that the assumptions of our theorems from Section 3 are satisfied. Our results are summarised in Appendix LABEL:sect:tables, where we provide tables of the representations with toral generic stabilisers.
This is a part of a general project initiated by the second author [Y]: to classify all semi-direct products with semisimple such that the ring is polynomial.
If an algebraic group acts on an irreducible affine variety , then
is the algebra of -invariant regular functions on and
is the field of -invariant rational functions. If
is finitely generated, then , and
the quotient morphism is induced by
the inclusion . If is a -module, then is the null-cone in .
Whenever the ring is graded polynomial,
the elements of any set of algebraically independent homogeneous generators
will be referred to as basic invariants.
For a -module and , is the stabiliser of
in and is the isotropy group of in .
• See also an explanation of the multiplicative (highest weight) notation for representations of semisimple groups in 4.5.
Let be a connected affine algebraic group with Lie algebra . The symmetric algebra is identified with the algebra of polynomial functions on and we also write for it. The algebra has the natural Poisson structure such that for . A subalgebra is said to be Poisson-commutative, if it is a subalgebra in the usual (associative-commutative) sense and also for all . The algebra of invariants is the centraliser of w.r.t , therefore it is the Poisson-centre of .
The index of , denoted , is , where is the stabiliser of with respect to the coadjoint representation of .
Set . If is reductive, then and equals the dimension of a Borel subalgebra. If is Poisson-commutative, then
It is also known that this upper bound is always attained.
Let be a (finite-dimensional rational) -module. The set of -regular elements of is defined to be
As is well-known, is a dense open subset of [VP]. In particular, is the set of -regular elements w.r.t. the coadjoint representation of .
We say that the codimension- condition (= CC ) holds for the action , if .
Suppose that . Then . For any , let denote the differential of at . We say that satisfies the Kostant (regularity) criterion if the following properties hold for and :
is a graded polynomial ring (with basic invariants );
if and only if are linearly independent.
A very useful fact is that if CC holds for , , and there are algebraically independent such that , then freely generate and the Kostant criterion holds for , see [coadj, Theorem 1.2].
Example. If is reductive and nonabelian, then . Hence the (co)adjoint representation of a reductive Lie algebra satisfies the CC .
For a -module , the vector space has a natural structure of Lie algebra, the semi-direct product of and . Explicitly, if and , then
This Lie algebra is denoted by , and is an abelian ideal of . The corresponding connected algebraic group is the semi-direct product of and the commutative unipotent group . The group can be identified with , the product being given by
In particular, . Then can be identified with . If is reductive, then the subgroup is the unipotent radical of , also denoted by .
Let be the moment map, i.e., , where and is the pairing of and . The restriction of the coadjoint representation of to is explicitly described as follows. If and , then
Since if and only if , the maximal dimension of the -orbits in equals .
For . There is a dense open subset such that for any
(i) By [rais], there is a dense open subset such that for any . This yields the desired formula for .
(ii) By Rosenlicht’s theorem [VP, 2.3],
It follows from this lemma that and the equality holds if and only if , i.e., is abelian for generic elements of . By [Y16], if there is a dense open subset of such that is abelian for all , then is Poisson-commutative. Having in mind the general upper bound (11), we conclude that in such a case is a Poisson-commutative subalgebra of of maximal dimension. Moreover, since is the centraliser of in , it is also a maximal Poisson-commutative subalgebra, cf. [BSM14, Theorem 3.3].
We say that the action has a generic stabiliser, if there exists a dense open subset such that all stabilisers , , are -conjugate. Then any subalgebra , , is called a generic stabiliser (= g.s.). Likewise, one defines a generic isotropy group (= ), which is a subgroup of . By [Ri, § 4], the linear action has a generic stabiliser if and only if it has a generic isotropy group. It is also known that always exists if is reductive. A systematic treatment of generic stabilisers in the context of reductive group actions can be found in [VP, §7].
2. On the rank of certain modules of covariants
For finite-dimensional -vector spaces and , let denote the set of polynomial morphisms . Clearly, and it is a free graded -module of rank . Here , if for any and .
If both and are -modules, then acts on by . Therefore, for all if and only if is -equivariant. Write for the set of -equivariant polynomial morphisms . It is also called the module of covariants of type . We have . In the rest of the section, we assume that is reductive. Then is a finitely generated -module, see e.g. [VP, 3.12].
Given a -module , consider the exact sequence of -modules
where for and . Therefore,
Here [jac, Prop. 1.7] and hence
. Recall that if is a domain and is a finitely generated
-module, then the rank of is .
We also consider the “equivariant sequence” that comprises -modules:
Here is the restriction of to . We are interested in conditions under which the -module is generated by -equivariant morphisms. In other words, when is it true that ?
If is a generic isotropy group for and , then we write and for this. Then and hence
Recall that the -action on is said to be stable, if the union of closed -orbits is dense in , see [VP, § 7]. Then is a reductive (not necessarily connected) group. By a general result of Vust [vust, Chap. III], if the action is stable, then
For the reader’s convenience, we outline a proof:
• If is -equivariant, then for any . Applying this to the open set of -generic elements in , we obtain that .
• On the other hand, the ”evaluation” map , , is onto whenever , see [indag02, Theorem 1]. Hence if generic -orbits in are closed (and isomorphic to ), then the upper bound is attained.
Our goal is to compute the rank of the -module .
If the action is stable and , then .
The reductive group acts on . By the Luna-Richardson theorem [lr79], the restriction homomorphism induces an isomorphism of rings of invariants . This common ring will be denoted by . Consider the commutative diagram of -modules
where the vertical arrows denote the restriction of -equivariant morphisms to . Note that the -module is not the Lie algebra of . However, the -module homomorphism is being defined similarly to . By construction, the action is again stable and has trivial generic isotropy groups. Therefore, using Eq. (22), we conclude that
Since is a generic isotropy group, . It follows that both vertical arrows are injective homomorphisms of -modules of equal ranks. Therefore, they give rise to isomorphisms over the field of fractions of and hence . Here
The second equality follows from the fact that for a generic and hence for any . Since , Eq. (22) implies that . ∎
If the action is stable and the -module is generated by -equivariant morphisms, then (i.e., the adjoint representation of is trivial). In particular, is a toral subalgebra of .
There are several cases in which this condition on is also sufficient.
If is the isotropy representation of a symmetric variety, then the condition that is toral does imply that is a free -module generated by -equivariant morphisms, see [coadj, Theorem 5.8].
If is finite, then is a trivial -module.
Next, we provide one more good case. For , let denote the set of zeros of . If , then is a polynomial function on and is a divisor.
Suppose that is semisimple and is a one-dimensional torus. Then is a free -module of rank 1 generated by a -equivariant morphism.
Since is semisimple and is reductive, the action is stable [VP, Theorem 7.15]. Hence in view of Theorem 2.1. Then we can pick a nonzero homogeneous primitive element , i.e., cannot be written as , where and with . Then is also primitive as element of . Indeed, assume that , where , and . Because is a -equivariant morphism, is -stable. Since and is a divisor, is necessarily a -stable divisor in . Because is semisimple, . It follows that . The relation shows that for any . Hence for any , and this contradicts the primitivity of in .
Let be an arbitrary homogeneous element. Since , there are coprime homogenous such that . If , then and, as in the previous paragraph, this leads to a contradiction. Thus, is invertible, and we are done. ∎
3. Semi-direct products with good invariant-theoretic properties
In this section, we describe a class of representations such that is generated by
-equivariant morphisms, satisfies the
Kostant criterion, and has nice invariant-theoretic properties.
For and , we define by , where denote the pairing of dual spaces.
We have if and only if for all , i.e., .
By (12), the invariance with respect to means that
for any . Hence , and we are done. ∎
Thus, any gives rise to . Moreover, it is clear that if is -equivariant, then . It follows from Eq. (12) that if is regarded as a linear function on , then is -invariant. Hence
• both and belong to ;
• both and belong to
We provide below certain conditions that guarantee us that and are generated by the respective subsets.
Recall some properties to the symmetric invariants of semi-direct products:
The decomposition yields a bi-grading of [coadj, Theorem 2.3(i)]. The same argument proves that the algebra is also bi-graded.
The algebra is contained in . Moreover, a minimal generating system for is a part of a minimal generating system of [coadj, Sect. 2 (A)]. In particular, if is a polynomial ring, then so is .
Note that associated with has degree w.r.t. . Conversely, it can be shown that if has degree 1 w.r.t. , then for some , see [Y16, Lemma 2.1]. In other words, there is a natural bijection . It is also true that .
If is reductive, then is finitely generated and stands for the minus degree of the Poincaré series of the graded algebra . More precisely, and its Poincaré series is
Here is a rational function and, by definition, . In particular, if is a polynomial ring, then equals the sum of degrees of the basic invariants. By [kn86, Korollar 5], if is semisimple, then . The arbitrary representations of simple algebraic groups and the irreducible representations of semisimple groups such that are classified in [kl87].
Recall some properties of the linear actions of semisimple groups. If is semisimple, then
is the quotient field of , hence [VP];
is stable if and only if is reductive [VP, Theorem 7.15].
Let be semisimple and . Suppose that and there are linearly independent (over ) homogeneous morphisms such that
are linearly independent for all and is -equivariant;
is a free -module of rank , with basis ;
, that is, ;
The -linear span of (resp. ) is a -stable subspace of (resp. ).
(i) Since a generic isotropy group is -dimensional,
. By [kn86, Satz 1 & Korollar 4], there is a
-equivariant map such that
and if , then . On the other hand,
the map has the same degree and also
for almost all . In other words,
and are proportional for almost all . Consequently, there are coprime
homogeneous such that . Since ,
as well. If , then there is such that and . Then , a contradiction! Hence ,
is -equivariant, and
are linearly independent for all .
(ii) As , the last property also implies that is a basis for the -module . Indeed, recall that . If , then there are such that . Again, if , then there is such that and for all (some) . This contradicts the linear independence of for all . Hence , and we are done.
(iii) Recall that now , is the moment mapping, and the -orbits in are
Hence and . Therefore . Let , , be a basis of (We regard the ’s as linear functions on .) Then are algebraically independent and belong to . Consider the map given by
By the Igusa Lemma [VP, Theorem 4.12], in order to prove that is the quotient morphism by , it suffices to verify the following two conditions:
The closure of does not contain divisors;
There is a dense open subset such that contains a dense -orbit for all .
For : If , then are linearly independent in view of (i). Therefore, the system of linear equations , , has a solution for any . Therefore,