The first and second fundamental theorems of invariant theory for the quantum general linear supergroup

Dedicated to Professor Gus Lehrer on the occasion of his 70th birthday
https://doi.org/10.1016/j.jpaa.2020.106411Get rights and content

Abstract

We develop a non-commutative polynomial version of the invariant theory of the quantum general linear supergroup Uq(glm|n). A non-commutative Uq(glm|n)-module superalgebra Pr|sk|l is constructed, which is the quantum analogue of the supersymmetric algebra over Ck|lCm|nCr|s(Cm|n). We analyse the structure of the subalgebra of Uq(glm|n)-invariants in Pr|sk|l by using a quantum super analogue of Howe duality.

The subalgebra of Uq(glm|n)-invariants in Pr|sk|l is shown to be finitely generated. We determine its generators and establish a surjective superalgebra homomorphism from a braided supersymmetric algebra onto it. This establishes the first fundamental theorem of invariant theory for Uq(glm|n).

We show that the above mentioned superalgebra homomorphism is an isomorphism if and only if mmin{k,r} and nmin{l,s}, and obtain a PBW basis for the subalgebra of invariants in this case. When the homomorphism is not injective, we give a representation theoretical description of the generating elements of the kernel. This way we obtain the relations obeyed by the generators of the subalgebra of invariants, producing the second fundamental theorem of invariant theory for Uq(glm|n).

We consider the special case n=0 in greater detail, obtaining a complete treatment of the non-commutative polynomial version of the invariant theory of Uq(glm). In particular, the explicit SFT proved here is believed to be new. We also recover the FFT and SFT of invariant theory for the general linear superalgebra from the classical limit (i.e. q1) of our results.

Introduction

Let Cm|n be the natural module for the general linear Lie superalgebra glm|n, and let (Cm|n) be its dual. Denote by Sr|sk|l the supersymmetric algebra over Ck|lCm|nCr|s(Cm|n), which is isomorphic to S(Ck|lCm|n)S(Cr|s(Cm|n)), the tensor product of two supersymmetric algebras. One formulation of the invariant theory of glm|n seeks to describe the subalgebra of glm|n-invariants of Sr|sk|l. The first fundamental theorem (FFT) provides a finite set of generators for the subalgebra of invariants [7], [17], [29], and the second fundamental theorem (SFT) describes relations among these generators [18], [30].

The aim of this paper is to develop quantum analogues of FFT and SFT of invariant theory for the quantum general linear supergroup Uq(glm|n). The invariant theory of Uq(glm|n) in this non-commutative algebraic setting was poorly understood previously. In fact, even the SFT of invariant theory for Uq(glm) (the case n=0) in this setting was unknown. The main issue hindering progress is that commutative algebraic techniques used in classical invariant theory are no longer applicable to the quantum super case, as all algebras involved now are not commutative.

In this paper, we construct a quantum analogue Pr|sk|l of Sr|sk|l, which forms a module superalgebra (see Section 2.3) over the quantum general linear supergroup. We investigate the subalgebra of Uq(glm|n)-invariants of Pr|sk|l, which is again non-commutative. We construct a finite set of generators for the subalgebra of invariants, and determine all algebraic relations obeyed by these generators. These results amount to an FFT and SFT, which are respectively given in Theorem 4.6 (also see Theorem 4.21) and Theorem 5.13.

The first problem we need to address is the following, which is absent in the classical (i.e. non-quantum) case.

Question 1.1

Define an appropriate quantum polynomial superalgebra Pr|sk|l which has Sr|sk|l as classical limit (q1).

As discussed below, the problem in itself is highly nontrivial. We address it following ideas of reference [2], where braided symmetric algebras were taken as quantum polynomial algebras. We generalise this notion to the super setting.

Given a Z2-graded vector space W, the braided supersymmetric algebra Sq(W) (defined in Section 3.2) is viewed as a q-deformation of the supersymmetric algebra S(W), i.e. it has S(W) as classical limit (q1). Recall that S(W) and Sq(W) are both Z+-graded. We say that Sq(W) is a flat deformation of S(W) if dimS(W)N=dimSq(W)N for all integers N0. Furthermore, if W is a module over a quantum general linear supergroup and Sq(W) is a flat deformation, then W is called a flat module. However, a well known fact is that even in the quantum group case almost all the braided symmetric algebras are not flat deformations (cf. [2], [19]), that is, they are “smaller” than the corresponding polynomial algebras. Little is known about braided supersymmetric algebras in the quantum super case. Here we will construct some flat deformations, which are useful for the study of invariant theory.

Let Vm|n be the natural module for Uq(glm|n), and similarly introduce Vk|l and Vr|s. Our discussion at the beginning suggests that Pr|sk|l may be defined as a “tensor product” of Sq(Vk|lVm|n) and Sq(Vr|s(Vm|n)). Here the “tensor product” needs to be defined properly.

The concept of a module superalgebra in the sense of [24, §4.1] is therefore brought into our picture. For instance, Sq(Vk|lVm|n) is an associative algebra carrying Uq(glk|l)Uq(glm|n)-module structure, whose algebraic structure is preserved by the quantum supergroup action. A useful observation originating from Hopf algebra theory (cf. [28], [19]) is that the tensor product of two module superalgebras is again a module superalgebra (see Proposition 2.3). Then Pr|sk|l becomes a module superalgebra, on which the two actions of Uq(glk|l)Uq(glr|s) and Uq(glm|n) graded-commute with each other. The subspace of Uq(glm|n)-invariants in Pr|sk|l is a subalgebra. Hence the notion of “generators” of Uq(glm|n)-invariant subalgebra of Pr|sk|l (i.e. FFT) makes sense in this context.

However, the multiplication in Pr|sk|l at the present stage is quite unwieldy to use, and furthermore, the Uq(glm|n)-action on Pr|sk|l is non-semisimple. This makes it highly nontrivial to describe the invariants. To overcome these difficulties, we shall use known results on coordinate superalgebras [37] of quantum general linear supergroups following ideas of [19]. This enables us to give a new formulation of Pr|sk|l in terms of generators and relations, and also characterise the module structure on Pr|sk|l by using quantum Howe duality of type (Uq(glk|l),Uq(glr|s)).

We now briefly describe this new construction of Pr|sk|l. Let π:Uq(glm|n)End(Vm|n) be the natural representation. We recall from [37] that the coordinate superalgebra Mm|n is a bi-superalgebra which is generated by the matrix elements tab defined by (2.8). By truncation we obtain a subalgebra Mr|sk|l of Mm|n with k,rm and l,sn. This subalgebra is a module superalgebra over Uq(glk|l)Uq(glr|s). Direct calculation shows that Mr|sk|l is isomorphic to Sq(Vk|lVr|s) as a module superalgebra. Similarly, we can introduce Mm|n and Mr|sk|l, which are both generated by matrix elements of the dual module (Vm|n). The latter is isomorphic to the braided supersymmetric algebra Sq((Vk|l)(Vr|s)). Now we can define Pr|sk|l as a twisted tensor product of these two module superalgebras Mm|nk|l and Mm|nr|s; see Definition 4.1. An explicit presentation of Pr|sk|l in terms of generators and relations is given in Lemma 4.3.

To prove the flatness of Pr|sk|l and analyse its module structure, we make extensive use of quantum Howe duality. It was noted in [37] that both Mm|n and Mm|n admit multiplicity-free decompositions as Uq(glm|n)Uq(glm|n)-modules by a partial analogue of the quantum Peter-Weyl theorem. We apply truncation to Mm|n (resp. Mm|n), producing a multiplicity-free decomposition of the subalgebra Mr|sk|l (resp. Mr|sk|l) as Uq(glk|l)Uq(glr|s)-module, where k,rm and l,sn. This is called quantum Howe duality of type (Uq(glk|l),Uq(glr|s)) (see Theorem 3.4, and also [33, Theorem 2.2]). Using this we obtain the following results:

  • (1)

    Pr|sk|l is a flat deformation of Sr|sk|l;

  • (2)

    A PBW basis for Mr|sk|l is constructed;

  • (3)

    Quantum Howe duality for Mr|0k|l implies quantum Schur-Weyl duality between Uq(glk|l) and the Hecke algebra Hq(r).

In particular, we show that there are three flat Uq(glk|l)Uq(glr|s)-modules: Vk|lVr|s, (Vk|l)(Vr|s) and Vk|l(Vr|s).

Now the following problem arises naturally.

Question 1.2

Describe the invariant subalgebra Xr|sk|l:=(Pr|sk|l)Uq(glm|n) in terms of generators (FFT) and defining relations (SFT). In particular, determine whether Xr|sk|l is a quantum polynomial superalgebra or a quotient thereof.

As we have mentioned, the Uq(glm|n)-invariant subalgebra Xr|sk|l is non-commutative; there exist no results which can be readily applied to show that it is a (quotient of a) quantum polynomial superalgebra in the sense of Question 1.1.

Motivated by the approach in [19], we show that the invariant subalgebra Xr|sk|l is finitely generated, and construct the invariants explicitly using bi-superalgebra structure of coordinate superalgebra. This amounts to the FFT; see Theorem 4.6. Note that, due to the non-commutative nature of the quantum polynomial superalgebra Pr|sk|l, techniques from classical invariant theory based on commutative algebra fail in our case, especially those techniques addressing finite generation.

To fully explore the algebraic structure of Xr|sk|l, we give a reformulation of FFT. Some elementary quadratic relations among invariants from Xr|sk|l are obtained in Lemma 4.5, which leads to an auxiliary quadratic superalgebra M˜r|sk|l with same relevant quadratic relations (defined in Section 4.3). This quadratic superalgebra M˜r|sk|l is shown to be isomorphic to the braided supersymmetric algebra Sq(Vk|l(Vr|s)) as a superalgebra, and similarly admits quantum super Howe duality and hence is a flat deformation. The FFT is then reformulated in terms of the surjective superalgebra homomorphism Ψr|sk|l:M˜r|sk|lXr|sk|l in Theorem 4.21. This implies that the invariant subalgebra Xr|sk|l is the quotient of the quantum polynomial superalgebra M˜r|sk|l by the two-sided ideal KerΨr|sk|l.

The SFT of invariant theory now seeks to describe the kernel of Ψr|sk|l, as the images of nonzero elements in kernel give rise to non-elementary relations among invariants. This can be reduced to the following situation. Note that M˜r|sk|l can be embedded into a larger superalgebra M˜K|L:=M˜K|LK|L with K=max{k,r} and L=max{l,s}. We are led to consider the kernel of ΨK|L:M˜K|LXK|LK|L, since the restriction KerΨK|LM˜r|sk|l exactly coincides with KerΨr|sk|l. Using quantum super Howe duality, we show that KerΨK|L as a Uq(glK|L)Uq(glK|L)-module admits a multiplicity-free decomposition over all (K|L)-hook partitions which contain the partition λc=((n+1)m+1). This multiplicity-free decomposition can be used to characterise KerΨK|L as a two-sided ideal of M˜K|L, which we shall explain below.

Our method relies essentially on some favourable properties of matrix elements by relating M˜K|L to the coordinate superalgebra MK|L. By a partial analogue of the Peter-Weyl theorem, MK|L decomposes into a direct sum of subspaces Tλ, which are spanned by matrix elements of the simple tensor modules LλK|L for Uq(glK|L). We prove in Theorem 5.4 that the two-sided ideal in MK|L generated by the subspace Tλ admits a multiplicity-free decomposition into direct sum of Tγ over all (K,L)-hook partitions γ containing λ. A key observation is that this result can be translated to the quadratic superalgebra M˜K|L, though M˜K|L is generally not isomorphic to MK|L as a superalgebra. Consequently, using the multiplicity-free decomposition of KerΨK|L mentioned before, we can prove that KerΨK|L is generated by the subspace T˜λc with λc=((n+1)m+1) as a two-sided ideal of M˜K|L. Here T˜λM˜K|L is an analogue of Tλ, which is isomorphic to LλK|LLλK|L as a Uq(glK|L)Uq(glK|L)-module.

Our SFT of invariant theory asserts that KerΨr|sk|l is generated as a two-sided ideal of M˜r|sk|l by the subspace T˜λcM˜r|sk|l with λc=((n+1)m+1). In particular, we show that the invariant subalgebra Xr|sk|l is a quantum polynomial superalgebra isomorphic to M˜r|sk|l if and only if mmin{k,r} and nmin{l,s}. These results are given in Theorem 5.13

We consider two special cases of our FFT and SFT of invariant theory for Uq(glm|n).

The quantum general linear group Uq(glm) is the special case of Uq(glm|n) with n=0. We immediately obtain the generators of the subalgebra of invariants, recovering the FFT of invariant theory given in [19, Theorem 6.10]. The kernel of the surjective algebra homomorphism mentioned above is now generated by the subspace of T˜λc with λc=(1m+1), which is shown to be spanned by quantum determinants of order m+1. Therefore, Theorem 6.6 and Theorem 6.8 together give a complete treatment for the non-commutative invariant theory for quantum general linear group, especially the SFT appears to be new.

The universal enveloping algebra U(glm|n) is another special case, where q1. Using the language of matrix elements, we provide a new approach to the FFT and SFT of invariant theory for glm|n, which was originally obtained in [29], [30]. This is interesting in its own right.

There has been some earlier work on the non-commutative polynomial version of invariant theory for quantum groups.

The works [9], [10], [11] investigate coinvariant theory for the quantum general linear group. In the non-super setting, our formalism differs from that in [9], [10], [11] in that our tensor product of coordinate algebras of quantum matrices is both a module and a comodule algebra (compare Section 6.2 with [10, §2]). In addition to treating the more general super case, our formalism also avoids the lengthy computation in [9] with a basis of the coordinate algebra indexed by bitableaux.

In [19], Lehrer, Zhang and Zhang gave a general method to construct the quantum analogues of polynomial rings, by using module algebras and the braiding of quantum group arising from the universal R-matrix. Then they gave a complete treatment of FFT for each quantum group associated with a classical Lie algebra. However, there is no complete treatment of SFTs for quantum groups. One of our main results in this paper is the SFT for Uq(glm). We also note that the FFT and SFT of invariant theory for Uq(sp2n) are obtained in [28], but it is rather difficult to generalise the construction of the underlying non-commutative polynomial algebra therein.

It was shown in [19], [33], [38] that (skew) Howe duality [12], [13] survives quantisation for the quantum general linear (super) group, and the resulting quantum Howe duality was applied to develop the q-deformed Segal-Shale-Weil representations. More recently, quantum skew Howe dualities turned out to be a powerful tool in the categorification of representations of Uq(sln) and Uq(glm|n) (see Cautis, Kamnitzer and Morrison's spider category [3] and other relevant developments [25], [26]). Here we extend quantum Howe duality of type (Uq(glk|l),Uq(glr)) established in [33] to type (Uq(glk|l),Uq(glr|s)), and simplify the original proof in [33].

Another formulation of non-commutative invariant theory provides a description for the endomorphism algebra over quantum (super) groups. The paper [20] establishes a full tensor functor from the category of ribbon graphs to the category of finite dimensional representations of Uq(g) with g=glm|n,osp(m|2n), giving the FFT of invariant theory in this endomorphism algebra setting. However, very little was known previously about the non-commutative polynomial version of invariant theory for quantum supergroups.

Section snippets

Quantum general linear supergroup

We shall work over the filed K:=C(q), where q is an indeterminate. For any vector superspace A=A0¯A1¯, we let []:A0¯A1¯Z2:={0¯,1¯} be the parity map, that is, [a]=i¯ if aAi¯. Tensor products of vector superspaces are again vector superspaces. We define the functorial permutation mapP:ABBA such that ab(1)[a][b]ba for homogeneous aA and bB, and generalise to inhomogeneous elements linearly. If A is an associative superalgebra, we define the super bracket [,]:AAA such that [X,Y]:=XY

Quantum Howe duality of type (Uq(glk|l),Uq(glr|s))

In this section we present the quantum Howe duality of type (Uq(glk|l),Uq(glr|s)), which states that a subalgebra Mr|sk|l of Mm|n admits a multiplicity-free decomposition as a Uq(glk|l)Uq(glr|s)-module. In Section 3.1 we describe the superalgebra Mr|sk|l as an invariant subalgebra of Mm|n and prove the quantum Howe duality. In Section 3.2 we show that Mr|sk|l is isomorphic to the braided supersymmetric algebra Sq(Vk|lVr|s), which has the supersymmetric algebra S(Vk|lVr|s) as the classical

The FFT of invariant theory for Uq(glm|n)

In this section we give the first fundamental theorem (FFT) of invariant theory for Uq(glm|n). In Section 4.1 we construct a Uq(glm|n)-module superalgebra Pr|sk|l, together with its explicit Uq(glm|n)-invariants. The FFT states that these Uq(glm|n)-invariants generate the Uq(glm|n)-invariant subalgebra of Pr|sk|l; see Theorem 4.6. The proof of the FFT is given in Section 4.2. Finally, in Section 4.3 we reformulate the FFT for Uq(glm|n) in terms of a superalgebra homomorphism, which will be

The SFT of invariant theory for Uq(glm|n)

In this section we describe the kernel of the superalgebra epimorphism Ψr|sk|l:M˜r|sk|lXr|sk|l given in Theorem 4.21 as a two-sided ideal of M˜r|sk|l. This is equivalent to the SFT of invariant theory for Uq(glm|n), since elements of the kernel give rise to new relations among invariants in Xr|sk|l apart from the quadratic relations (4.8). Our main result is given in Theorem 5.13.

The main idea is to identify M˜r|sk|l as a subalgebra of M˜K|L with K=max{k,r} and L=max{l,s}. This way we obtain

Examples

In this section, we shall elucidate how our main results can be applied to derive “polynomial” versions of invariant theory for U(glm|n) and Uq(glm). Basically, the invariant theory of glm|n [29], [30] can be obtained in our language of matrix elements by specialising q to 1. Also, the FFT of invariant theory for Uq(glm) recovers [19, Theorem 6.10], while the SFT for Uq(glm) appears to be new.

Acknowledgements

I would like to thank Professor Gus Lehrer and Professor Ruibin Zhang for advices and help during the course of this work. I am also grateful to the referee for reading this article thoroughly and making many helpful suggestions. This work was supported at different stages by student stipends from the China Scholarship Council and the Australian Research Council Discovery Projects DP150104507.

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