Dual Canonical Bases for the Quantum Special Linear Group and Invariant Subalgebras

Zhang, Hechun; Zhang, R. B.

doi:10.1007/s11005-005-0015-9

Cited by 5 publications

(9 citation statements)

References 20 publications

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“…A brief review of early works on the theory and applications of quantum supergroups can be found in [63]. Partially successful constructions of crystal and canonical bases for quantum supergroups were given in [1,12,37,53,55,54,72,73].…”

Section: Introductionmentioning

confidence: 99%

The First Fundamental Theorem of Invariant Theory for the Orthosymplectic Supergroup

Lehrer

Zhang

2016

Commun. Math. Phys.

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Abstract. Let U q (g) be the quantum supergroup of gl m|n or the modified quantum supergroup of osp m|2n over the field of rational functions in q, and let V q be the natural module for U q (g). There exists a unique tensor functor, associated with V q , from the category of ribbon graphs to the category of finite dimensional representations of U q (g), which preserves ribbon category structures. We show that this functor is full in the cases g = gl m|n or osp 2ℓ+1|2n . For g = osp 2ℓ|2n , we show that the space Hom Uq(g) (V ⊗r q , V ⊗s q ) is spanned by images of ribbon graphs if r + s < 2ℓ(2n + 1). The proofs involve an equivalence of module categories for two versions of the quantisation of U(g).

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Section: Introductionmentioning

confidence: 99%

The First Fundamental Theorem of Invariant Theory for the Orthosymplectic Supergroup

Lehrer

Zhang

2016

Commun. Math. Phys.

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“…Hence, a similar argument as that in [14] shows that the subalgebra O q (G/K) is spanned by the elements of the basis B * m|n which are invariant under L Uq(k) . Let U q (k) be the subalgebra U q (n + ).…”

Section: Zhang Hechunmentioning

confidence: 58%

“…The action of the Kashiwara operators for those even Chevalley generators can be described in the same way as in [14]. A similar argument as that in [14] also shows that for any matrix A ∈ M and i = 1, 2, .…”

Section: Zhang Hechunmentioning

confidence: 63%

“…We need to take care of the signs when switching the odd elements which is elementary. The remaining of the proof is the same as the proof of Proposition 3.2 in [14]. Clearly, all x ij for i = m, m+ 1 are annihilated by E m and F m , so we only need to write down the action of the Kashiwara operators for the following situation.…”

Section: Canonical Bases For Irreducible Tensor Modulesmentioning

confidence: 93%

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The quantum general linear supergroup, canonical bases and Kazhdan-Lusztig polynomials

Zhang

2009

Sci. China Ser. A-Math.

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Canonical bases of the tensor powers of the natural Uq(gl m|n )-module V are constructed by adapting the work of Frenkel, Khovanov and Kirrilov to the quantum supergroup setting. This result is generalized in several directions. We first construct the canonical bases of the Z2-graded symmetric algebra of V and tensor powers of this superalgebra; then construct canonical bases for the superalgebra Oq(M m|n ) of a quantum (m, n) × (m, n)-supermatrix; and finally deduce from the latter result the canonical basis of every irreducible tensor module for Uq(gl m|n ) by applying a quantum analogue of the Borel-Weil construction.

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“…Now, the same argument as in [21] shows that the subalgebra of invariants is spanned by a part of the dual canonical basis. 2…”

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confidence: 80%

Dual canonical bases for the quantum general linear supergroup

Zhang¹,

Zhang²

2006

Journal of Algebra

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Dual canonical bases of the quantum general linear supergroup are constructed which are invariant under the multiplication of the quantum Berezinian. By setting the quantum Berezinian to identity, we obtain dual canonical bases of the quantum special linear supergroup O q (SL m|n ). We apply the dual canonical bases to study invariant subalgebras of the quantum supergroups under left and right translations. In the case n = 1, it is shown that each invariant subalgebra is spanned by a part of the dual canonical bases. This in turn leads to dual canonical bases for any Kac module constructed by using an analogue of Borel-Weil theorem.

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Dual Canonical Bases for the Quantum Special Linear Group and Invariant Subalgebras

Cited by 5 publications

References 20 publications

The First Fundamental Theorem of Invariant Theory for the Orthosymplectic Supergroup

The First Fundamental Theorem of Invariant Theory for the Orthosymplectic Supergroup

The quantum general linear supergroup, canonical bases and Kazhdan-Lusztig polynomials

Dual canonical bases for the quantum general linear supergroup

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