The Second Fundamental Theorem of Invariant Theory for the Orthosymplectic Supergroup

Lehrer, G. I.; Zhang, R. B.

doi:10.1017/nmj.2019.25

Cited by 12 publications

(28 citation statements)

References 27 publications

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“…Remark 8.1.2. (i) Theorem 8.1.1(i) was recently proved by Zhang in [Zh,Theorem 5.12], using a different approach, and was conjectured in [LZ4]. (ii) In [HX,LZ1,Zh], Theorem 8.1.1(ii) is proved for the special cases Sp(2n), O(m) and OSp(1|2n).…”

Section: The Second Fundamental Theorem Of Invariant Theorymentioning

confidence: 97%

“…That in all other cases the ideals are still generated by one element is somewhat unexpected, see e.g. [LZ4,Remark 5.9]. (iii) That the generating element for O(m) and Sp(2n) can be chosen to be an idempotent as in Theorem 8.1.1(iii) is known by [HX,LZ1].…”

Section: The Second Fundamental Theorem Of Invariant Theorymentioning

confidence: 99%

“…which contains a cup. Furthermore, for all k + l ≥ r c , the kernel of φ k,l is generated as a two-sided ideal in B k,l (δ) by the element (ii) The special case l = 0 of Theorem 8.2.1(ii) is implied by [LZ4,Theorem 2.3]. The case l > 0 seems to be new.…”

Section: The General Linear Casementioning

confidence: 99%

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Tensor ideals, Deligne categories and invariant theory

Coulembier

2018

Sel. Math. New Ser.

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We derive some tools for classifying tensor ideals in monoidal categories. We use these results to classify tensor ideals in Deligne's universal categories RepO δ , RepGL δ and RepP . These results are then used to obtain new insight into the second fundamental theorem of invariant theory for the algebraic supergroups of types A, B, C, D, P .We also find new short proofs for the classification of tensor ideals in RepSt and in the category of tilting modules for SL2(k) with char(k) > 0 and for Uq(sl2) with q a root of unity. In general, for a simple Lie algebra g of type ADE, we show that the lattice of such tensor ideals for Uq(g) corresponds to the lattice of submodules in a parabolic Verma module for the corresponding affine Kac-Moody algebra.2010 Mathematics Subject Classification. 18D10, 17B45, 17B10, 15A72. Key words and phrases. Monoidal (super)category, tensor ideal, thick tensor ideal, Deligne category, algebraic (super)group, second fundamental theorem of invariant theory, tilting modules, quantum groups. 1 3.2. Proofs. By definition, for a submodule M of P C ½ , we have M(X) ⊂ C(½, X), for all X ∈ ObC.Proposition 3.2.1. (i) For a submodule M of P ½ , we define J M as J M (X, Y ) := ι XY M(X ∨ ⊗ Y ) , for all X, Y ∈ ObC.Then J M is a left-tensor ideal in C.

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Section: The Second Fundamental Theorem Of Invariant Theorymentioning

confidence: 97%

Section: The Second Fundamental Theorem Of Invariant Theorymentioning

confidence: 99%

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Tensor ideals, Deligne categories and invariant theory

Coulembier

2018

Sel. Math. New Ser.

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“…One formulation of the invariant theory of gl m|n seeks to describe the subalgebra of gl m|n -invariants of S k|l r|s . The first fundamental theorem (FFT) provides a finite set of generators for the subalgebra of invariants [7,16,29], and the second fundamental theorem (SFT) describes the relations among the generators [17,30].…”

Section: Introductionmentioning

confidence: 99%

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

Zhang

2020

Journal of Pure and Applied Algebra

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We develop the non-commutative polynomial version of the invariant theory for the quantum general linear supergroup U q pgl m|n q. A non-commutative U q pgl m|n q-module superalgebra P k|l r|s is constructed, which is the quantum analogue of the supersymmetric algebra over C k|l b C m|n ' C r|s b pC m|n q˚. We analyse the structure of the subalgebra of U q pgl m|n q-invariants in P k|l r|s by using the quantum super analogue of Howe duality.The subalgebra of U q pgl m|n q-invariants in P k|l r|s 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 U q pgl m|n q.We show that the above mentioned superalgebra homomorphism is an isomorphism if and only if m ě mintk, ru and n ě mintl, su, 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 U q pgl m|n q.We consider the special case with n " 0 in greater detail, obtaining a complete treatment of the non-commutative polynomial version of the invariant theory for U q pgl m q. In particular, the explicit SFT proved here is believed to be new. We also recover the FFT and SFT of the invariant theory for the general linear superalgebra from the classical limit (i.e., q Ñ 1) of our results.

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“…. By the invariant theory for Osp(1|2n) [Ser01], [LZ16], [LZ14] we have a fixpoint Lie subalgebra L −2n+1 :…”

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confidence: 99%

Simplicities of VOAs associated to Jordan algebras of type B and character formulas for simple quotients

Zhao

2018

Journal of Algebra

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Let V be a Z ≥0 graded vertex operator algebra(VOA), with V 0 = C1, V 1 = {0}. Then V 2 has a commutative(but not necessarily associative) algebra structure with operation a • b = a(1)b. This algebra V 2 is called Griess algebra of V . In [Lam96] and [Lam99], Lam constructed vertex algebras whose Griess algebras are simple Jordan algebras of type A, B, C. For type D simple Jordan algebras the construction was given by Ashihara in [Ash11]; In [AM09] Ashihara and Miyamoto constructed a family of vertex algebras V J ,r parametrized by r ∈ C, whose Griess algebras are isomorphic to type B Jordan algebras J . The VOA V J ,r is further studied by Niibori and Sagaki in [NS10].One of the main results in [NS10] claims that if J is not the Jordan algebra of 1 × 1 matrix, then V J ,r is simple if and only if r / ∈ Z. This suggests that r ∈ Z are special and may deserve further study. We show that the simple quotients V J ,r , r ∈ Z =0 can be constructed by a dual-pair type construction. We also apply the construction to compute the character Tr|V J ,r q L(0) , r = −2n, n ≥ 1. The Clebsch-Gordan coefficients appear naturally in the character formula.We give more details about this paper. All vector spaces and Lie groups are assumed to be over C. Let (h, (·, ·)) be a finite dimensional vector space with a non-degenerate symmetric bilinear form (·, ·), dim(h) = d. Then h ⊗ h has an associative algebra structure:which induces a Jordan algebra structure on h ⊗ h:Let J be the Jordan subalgebra of h ⊗ h consists of symmetric tensors:Then J is the type B simple Jordan algebra of rank d [JJ49].1 Throughout this paper we assume that d ≥ 2 unless otherwise stated. Let V J ,r be the VOA constructed in [AM09] andV J ,r be the corresponding simple quotient. In [NS10] it is shown that V J ,r =V J ,r if and only if r / ∈ Z. Our results further show that we can constructV J ,r , r ∈ Z =0 explicitly. We divide our constructions into three cases:Case 1, r = m, m ≥ 1: Let (V m , (·, ·)) be a m-dimensional vector space with a non-degenerate symmetric bilinear form. The tensor product space h ⊗ V m is a dm-dimensional vector space with the non-degenerate symmetric bilinear form:). Let H(h ⊗ V m ) be the Heisenberg VOA associated to the vector space h ⊗ W m [FLM89]. The group O(m) acts on the component V m , therefore it acts as automorphism on H(h ⊗ V m ). We constructV J ,m as: V J ,m def.

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The Second Fundamental Theorem of Invariant Theory for the Orthosymplectic Supergroup

Cited by 12 publications

References 27 publications

Tensor ideals, Deligne categories and invariant theory

Tensor ideals, Deligne categories and invariant theory

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

Simplicities of VOAs associated to Jordan algebras of type B and character formulas for simple quotients

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