The first and second fundamental theorems of invariant theory for the quantum general linear supergroup
Introduction
Let be the natural module for the general linear Lie superalgebra , and let be its dual. Denote by the supersymmetric algebra over , which is isomorphic to , the tensor product of two supersymmetric algebras. One formulation of the invariant theory of seeks to describe the subalgebra of -invariants of . 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 . The invariant theory of in this non-commutative algebraic setting was poorly understood previously. In fact, even the SFT of invariant theory for (the case ) 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 The main issue hindering progress is that commutative algebraic techniques used in classical invariant theory are no longer applicable to the quantum super case, as all algebras involved now are not commutative.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 of , which forms a module superalgebra (see Section 2.3) over the quantum general linear supergroup. We investigate the subalgebra of -invariants of , 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 which has as classical limit ().
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 -graded vector space W, the braided supersymmetric algebra (defined in Section 3.2) is viewed as a q-deformation of the supersymmetric algebra , i.e. it has as classical limit (). Recall that and are both -graded. We say that is a flat deformation of if for all integers . Furthermore, if W is a module over a quantum general linear supergroup and 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 be the natural module for , and similarly introduce and . Our discussion at the beginning suggests that may be defined as a “tensor product” of and . 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, is an associative algebra carrying -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 becomes a module superalgebra, on which the two actions of and graded-commute with each other. The subspace of -invariants in is a subalgebra. Hence the notion of “generators” of -invariant subalgebra of (i.e. FFT) makes sense in this context.
However, the multiplication in at the present stage is quite unwieldy to use, and furthermore, the -action on 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 in terms of generators and relations, and also characterise the module structure on by using quantum Howe duality of type .
We now briefly describe this new construction of . Let be the natural representation. We recall from [37] that the coordinate superalgebra is a bi-superalgebra which is generated by the matrix elements defined by (2.8). By truncation we obtain a subalgebra of with and . This subalgebra is a module superalgebra over . Direct calculation shows that is isomorphic to as a module superalgebra. Similarly, we can introduce and , which are both generated by matrix elements of the dual module . The latter is isomorphic to the braided supersymmetric algebra . Now we can define as a twisted tensor product of these two module superalgebras and ; see Definition 4.1. An explicit presentation of in terms of generators and relations is given in Lemma 4.3.
To prove the flatness of and analyse its module structure, we make extensive use of quantum Howe duality. It was noted in [37] that both and admit multiplicity-free decompositions as -modules by a partial analogue of the quantum Peter-Weyl theorem. We apply truncation to (resp. ), producing a multiplicity-free decomposition of the subalgebra (resp. ) as -module, where and . This is called quantum Howe duality of type (see Theorem 3.4, and also [33, Theorem 2.2]). Using this we obtain the following results:
- (1)
is a flat deformation of ;
- (2)
A PBW basis for is constructed;
- (3)
Quantum Howe duality for implies quantum Schur-Weyl duality between and the Hecke algebra .
Now the following problem arises naturally.
Question 1.2 Describe the invariant subalgebra in terms of generators (FFT) and defining relations (SFT). In particular, determine whether is a quantum polynomial superalgebra or a quotient thereof.
As we have mentioned, the -invariant subalgebra 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 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 , 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 , we give a reformulation of FFT. Some elementary quadratic relations among invariants from are obtained in Lemma 4.5, which leads to an auxiliary quadratic superalgebra with same relevant quadratic relations (defined in Section 4.3). This quadratic superalgebra is shown to be isomorphic to the braided supersymmetric algebra 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 in Theorem 4.21. This implies that the invariant subalgebra is the quotient of the quantum polynomial superalgebra by the two-sided ideal .
The SFT of invariant theory now seeks to describe the kernel of , 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 can be embedded into a larger superalgebra with and . We are led to consider the kernel of , since the restriction exactly coincides with . Using quantum super Howe duality, we show that as a -module admits a multiplicity-free decomposition over all -hook partitions which contain the partition . This multiplicity-free decomposition can be used to characterise as a two-sided ideal of , which we shall explain below.
Our method relies essentially on some favourable properties of matrix elements by relating to the coordinate superalgebra . By a partial analogue of the Peter-Weyl theorem, decomposes into a direct sum of subspaces , which are spanned by matrix elements of the simple tensor modules for . We prove in Theorem 5.4 that the two-sided ideal in generated by the subspace admits a multiplicity-free decomposition into direct sum of over all -hook partitions γ containing λ. A key observation is that this result can be translated to the quadratic superalgebra , though is generally not isomorphic to as a superalgebra. Consequently, using the multiplicity-free decomposition of mentioned before, we can prove that is generated by the subspace with as a two-sided ideal of . Here is an analogue of , which is isomorphic to as a -module.
Our SFT of invariant theory asserts that is generated as a two-sided ideal of by the subspace with . In particular, we show that the invariant subalgebra is a quantum polynomial superalgebra isomorphic to if and only if and . These results are given in Theorem 5.13
We consider two special cases of our FFT and SFT of invariant theory for .
The quantum general linear group is the special case of with . 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 with , which is shown to be spanned by quantum determinants of order . 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 is another special case, where . Using the language of matrix elements, we provide a new approach to the FFT and SFT of invariant theory for , 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 -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 . We also note that the FFT and SFT of invariant theory for 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 and (see Cautis, Kamnitzer and Morrison's spider category [3] and other relevant developments [25], [26]). Here we extend quantum Howe duality of type established in [33] to type , 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 with , 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 , where q is an indeterminate. For any vector superspace , we let be the parity map, that is, if . Tensor products of vector superspaces are again vector superspaces. We define the functorial permutation map such that for homogeneous and , and generalise to inhomogeneous elements linearly. If A is an associative superalgebra, we define the super bracket such that
Quantum Howe duality of type
In this section we present the quantum Howe duality of type , which states that a subalgebra of admits a multiplicity-free decomposition as a -module. In Section 3.1 we describe the superalgebra as an invariant subalgebra of and prove the quantum Howe duality. In Section 3.2 we show that is isomorphic to the braided supersymmetric algebra , which has the supersymmetric algebra as the classical
The FFT of invariant theory for
In this section we give the first fundamental theorem (FFT) of invariant theory for . In Section 4.1 we construct a -module superalgebra , together with its explicit -invariants. The FFT states that these -invariants generate the -invariant subalgebra of ; 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 in terms of a superalgebra homomorphism, which will be
The SFT of invariant theory for
In this section we describe the kernel of the superalgebra epimorphism given in Theorem 4.21 as a two-sided ideal of . This is equivalent to the SFT of invariant theory for , since elements of the kernel give rise to new relations among invariants in apart from the quadratic relations (4.8). Our main result is given in Theorem 5.13.
The main idea is to identify as a subalgebra of with and . 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 and . Basically, the invariant theory of [29], [30] can be obtained in our language of matrix elements by specialising q to 1. Also, the FFT of invariant theory for recovers [19, Theorem 6.10], while the SFT for 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.
References (38)
- et al.
Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras
Adv. Math.
(1987) - et al.
The first fundamental theorem of invariant theory for the orthosymplectic super group
Adv. Math.
(2018) - et al.
Quantum Schur superalgebras and Kazhdan-Lusztig combinatorics
J. Pure Appl. Algebra
(2011) Lie superalgebras
Adv. Math.
(1977)On a theorem of Benson and Curtis
J. Algebra
(1981)- et al.
Dual canonical bases for the quantum general linear supergroup
J. Algebra
(2006) - et al.
Mixed quantum skew Howe duality and link invariants of type A
J. Pure Appl. Algebra
(2019) Classical invariant theory for the quantum symplectic group
Adv. Math.
(1996)- et al.
The general linear supergroup and its Hopf superalgebra of regular functions
J. Algebra
(2002) - et al.
Unitary highest weight representations of quantum general linear superlagebra
J. Algebra
(2009)
Braided symmetric and exterior algebras
Trans. Am. Math. Soc.
Webs and quantum skew Howe duality
Math. Ann.
Howe duality for Lie superalgebras
Compos. Math.
Super duality and Kazhdan-Lusztig polynomials
Trans. Am. Math. Soc.
Young diagrams and determinantal varieties
Invent. Math.
Quantum determinantal ideals
Duke Math. J.
The first fundamental theorem of coinvariant theory for the quantum general linear group
Publ. Res. Inst. Math. Sci.
Realizations of quantum hom-spaces, invariant theory, and quantum determinantal ideals
J. Algebra
Remarks on classical invariant theory
Trans. Am. Math. Soc.
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