Duality between quantum symmetric algebras

Duality for finite Hopf algebras explained by corings

We give a coring version for the duality theorem for actions and coactions of a finitely generated projective Hopf algebra. We also provide a coring analogue for a theorem of H.-J. Schneider, which generalizes and unifies the duality theorem for finite Hopf algebras and its refinements.

A Note on Cyclic Duality and Hopf Algebras

We show that various cyclic and cocyclic modules attached to Hopf algebras and Hopf modules are related to each other via Connes' duality isomorphism for the cyclic category.

Duality Theorems for Infinite Braided Hopf Algebras

Let $H$ be an infinite-dimensional braided Hopf algebra and assume that the braiding is symmetric on $H$ and its quasi-dual $H^d$. We prove the Blattner-Montgomery duality theorem, namely we prove $$ (R # H)# H^{d} \cong R \otimes (H # H^{d}) \hbox {as algebras in braided tensor category} {\cal C}.$$ In particular, we present two duality theorems for infinite braided Hopf algebras in the Yetter-Drinfeld module category.

Symmetric webs, Jones-Wenzl recursions and $q$-Howe duality

We define and study the category of symmetric $\mathfrak{sl}_2$-webs. This category is a combinatorial description of the category of all finite dimensional quantum $\mathfrak{sl}_2$-modules. Explicitly, we show that (the additive closure of) the symmetric $\mathfrak{sl}_2$-spider is (braided monoidally) equivalent to the latter. Our main tool is a quantum version of symmetric Howe duality. As a corollary of our construction, we provide new insight into Jones-Wenzl projectors and the colored Jones polynomials.

A $q$-analogue of derivations on the tensor algebra and the $q$-Schur-Weyl duality

This paper presents a $q$-analogue of an extension of the tensor algebra given by the same author. This new algebra naturally contains the ordinary tensor algebra and the Iwahori-Hecke algebra type $A$ of infinite degree. Namely this algebra can be regarded as a natural mix of these two algebras. Moreover, we can consider natural "derivations" on this algebra. Using these derivations, we can easily prove the $q$-Schur-Weyl duality (the duality between the quantum enveloping algebra of the general linear Lie algebra and the Iwahori-Hecke algebra of type $A$).

Quantum Group Duality and the Cuntz Algebra

The Cuntz algebra carries in a natural way the structure of a module algebra over the quantized universal enveloping algebra $U_q(g)$, and the structure of a co-module algebra over the quantum group $G_q$ associated with $U_q(g)$. These two algebraic structures are dual to each other via the duality between $G_q$ and $U_q(g)$.

正交Howe对偶与动态（分裂）对称对

Orthogonal Howe duality and dynamical (split) symmetric pairs

受李代数相关动态结构启发，为分裂对称对引入动态融合、K-矩阵及Weyl群版本，研究(𝔰𝔬₂ₙ,Oₘ)-对偶性并证明𝔰𝔬₂ₙ侧标准算子与Oₘ侧对称对类似物可交换。

BMW algebra, quantized coordinate algebra and type C Schur--Weyl duality

We prove an integral version of the Schur--Weyl duality between the specialized Birman--Murakami--Wenzl algebra $B_n(-q^{2m+1},q)$ and the quantum algebra associated to the symplectic Lie algebra sp_{2m}. In particular, we deduce that this Schur--Weyl duality holds over arbitrary (commutative) ground rings, which answers a question of Lehrer and Zhang [Strongly multiplicity free modules for Lie algebras and quantum groups, J. Algebra (1) 306 (2006), 138--174] in the symplectic case. As a byproduct, we show that, as $Z[q,q^{-1}]$-algebra, the quantized coordinate algebra defined by Kashiwara is…

Morita Bicategories of Algebras and Duality Involutions

The notion of a weak duality involution on a bicategory was recently introduced by Shulman in [arXiv:1606.05058]. We construct a weak duality involution on the fully dualisable part of $\text{Alg}$, the Morita bicategory of finite-dimensional k-algebras. The 2-category $\text{KV}$ of Kapranov-Voevodsky k-vector spaces may be equipped with a canonical strict duality involution. We show that the pseudofunctor $\text{Rep}: \text{Alg}^{fd} \to \text{KV}$ sending an algebra to its category of finite-dimensional modules may be canonically equipped with the structure of a duality pseudofunctor. Thus …

The duality between $κ$-Poincaré algebra and $κ$-Poincaré group

The full duality between the $κ$-Poincaré algebra and $κ$-Poincaré group is proved.

同调对偶性与 Hopf 代数的正合序列

Homological duality and exact sequences of Hopf algebras

本文研究 Hopf 代数在扩张下同调对偶性质的稳定性，提出并证明了相关正合序列中的保持条件。

Schur-Weyl duality, Verma modules, and row quotients of Ariki-Koike algebras

We prove a Schur-Weyl duality between the quantum enveloping algebra of $\mathfrak{gl}_m$ and certain quotient algebras of Ariki-Koike algebras, which we give explicitly. The duality involves several algebraically independent parameters and is realized through the tensor product of a parabolic universal Verma module and a tensor power of the natural representation of $\mathfrak{gl}_m$. We also give a new presentation by generators and relations of the generalized blob algebras of Martin and Woodcock as well as an interpretation in terms of Schur-Weyl duality by showing they occur as a particul…

Geometric Schur Duality of Classical Type, II

We establish algebraically and geometrically a duality between the Iwahori-Hecke algebra of type D and two new quantum algebras arising from the geometry of N-step isotropic flag varieties of type D. This duality is a type D counterpart of the Schur-Jimbo duality of type A and the Schur-like duality of type B/C discovered by Bao-Wang. The new algebras play a role in the type D duality similar to the modified quantum gl(N) in type A, and the modified coideal subalgebras of quantum gl(N) in type B/C. We construct canonical bases for these two algebras.

Noncommmutative theorems: Gelfand Duality, Spectral, Invariant Subspace, and Pontryagin Duality

We extend the Gelfand-Naimark duality of commutative C*-algebras, "A COMMUTATIVE C*-ALGEBRA -- A LOCALLY COMPACT HAUSDORFF SPACE" to "A C*-ALGEBRA--A QUOTIENT OF A LOCALLY COMPACT HAUSDORFF SPACE". Thus, a C*-algebra is isomorphic to the convolution algebra of continuous regular Borel measures on the topological equivalence relation given by the above mentioned quotient. In commutative case this reduces to Gelfand-Naimark theorem. Applications: 1) A simultaneous extension, to arbitrary Hilbert space operators, of Jordan Canonical Form and Spectral Theorem of normal operators 2) A functional ca…

Duality and products in algebraic (co)homology theories

The origin and interplay of products and dualities in algebraic (co)homology theories is ascribed to a $\times_A$-Hopf algebra structure on the relevant universal enveloping algebra. This provides a unified treatment for example of results by Van den Bergh about Hochschild (co)homology and by Huebschmann about Lie-Rinehart (co)homology.

Koszul duality for locally constant factorization algebras

Generalising Jacob Lurie's idea on the relation between the Verdier duality and the iterated loop space theory, we study the Koszul duality for locally constant factorisation algebras. We formulate an analogue of Lurie's "nonabelian Poincare duality" theorem (which is closely related to earlier results of Graeme Segal, of Dusa McDuff, and of Paolo Salvatore) in a symmetric monoidal stable infinity category carefully, using John Francis' notion of excision. Its proof depends on our earlier study of the Koszul duality for E_n-algebras. As a consequence, we obtain a Verdier type equivalence for f…

Duality for actions of weak Kac algebras and crossed product inclusions of II_1 factors

We show that indecomposable weak Kac algebras are free over their Cartan subalgebras and prove a duality theorem for their actions. Using this result, for any biconnected weak Kac algebra we construct a minimal action on the hyperfinite II_1 factor. The corresponding crossed product inclusion of II_1 factors has depth 2 and an integer index. Its first relative commutant is, in general, non-trivial, so we derive some arithmetic properties of weak Kac algebras from considering reduced subfactors.

Semiinfinite cohomology of associative algebras and bar duality

We describe semiinfinite cohomology of associative algebras in terms of Koszul (or bar) duality. Consider an associative algebra $A$ and two its subalgebras $B$ and $N$ such that $A=B\otimes N$ as a vector space. We prove that the endomorphism algebra of the semiregular $A$-module appears naturally in semiinfinite cohomology theory as a ``two times Koszul dual'' to the algebra $A$. We compare semiinfinite cohomology of universal enveloping algebras with the well-known Lie algebra semiinfinite cohomology. A new description of the critical 2-cocycle is provided. As a consequence we obtain anothe…

Quadratic Duality for Chiral Algebras

We introduce a notion of quadratic duality for chiral algebras. This can be viewed as a chiral version of the usual quadratic duality for quadratic associative algebras. We study the relationship between this duality notion and the Maurer-Cartan equations for chiral algebras, which turns out to be parallel to the associative algebra case. We also present some explicit examples.

Skew Howe duality for Types $\mathbf{BD}$ via $q$-Clifford algebras

We extend a quantized skew Howe duality result for Type $\mathbf{A}$ algebras to orthogonal types via a seesaw. We develop an operator commutant version of the First Fundamental Theorem of invariant theory for $U_q(\mathfrak{so}_n)$ using a double centralizer property inside a quantized Clifford algebra. We obtain a multiplicity-free decomposition of tensor powers of the $U_q(\mathfrak{so}_{2n})$ spin representation by explicitly computing joint highest weights with respect to an action of $U_q(\mathfrak{so}_{2n}) \otimes U_q'(\mathfrak{so}_m)$. Clifford algebras are an essential feature of ou…

From Koszul duality to Poincaré duality

We discuss the notion of Poincaré duality for graded algebras and its connections with the Koszul duality for quadratic Koszul algebras. The relevance of the Poincaré duality is pointed out for the existence of twisted potentials associated to Koszul algebras as well as for the extraction of a good generalization of Lie algebras among the quadratic-linear algebras.

Poincaré duality for Koszul algebras

We discuss the consequences of the Poincaré duality, versus AS- Gorenstein property, for Koszul algebras (homogeneous and non homogeneous). For homogeneous Koszul algebras, the Poincaré duality property implies the existence of twisted potentials which characterize the corresponding algebras while in the case of quadratic linear Koszul algebras, the Poincaré duality is needed to get a good generalization of universal enveloping algebras of Lie algebras. In the latter case we describe and discuss the corresponding generalization of Lie algebras. We also give a short review of the notion of Kosz…

Quantum Langlands duality of representations of W-algebras

We prove duality isomorphisms of certain representations of W-algebras which play an essential role in the quantum geometric Langlands Program and some related results.

T-duality of current algebras and their quantization

In this paper we show that the T-duality transform of Bouwknegt, Evslin and Mathai applies to determine isomorphisms of certain current algebras and their associated vertex algebras on topologically distinct T-dual spacetimes compactified to circle bundles with $H$-flux.

量子群作用于C*-代数的部分Pontryagin对偶

Partial Pontryagin duality for actions of quantum groups on C*-algebras

本文通过建立双交叉积与双重交叉积量子群作用之间的部分Pontryagin对偶，推广了Baaj-Skandalis对偶，并探讨了其对等变Kasparov理论及量子Baum-Connes猜想的意义。

Khovanov弧代数的箭图表示与Schur–Weyl对偶

Quiver presentations and Schur--Weyl duality for Khovanov arc algebras

本文通过给出Khovanov弧代数的Ext-箭图与关系表示，证明了Kleshchev–Martin猜想在该框架下的精确类比，揭示了其与Schur–Weyl对偶的联系。

Langlands duality for finite-dimensional representations of quantum affine algebras

We describe a correspondence (or duality) between the q-characters of finite-dimensional representations of a quantum affine algebra and its Langlands dual in the spirit of q-alg/9708006 and 0809.4453. We prove this duality for the Kirillov-Reshetikhin modules and their irreducible tensor products. In the course of the proof we introduce and construct "interpolating (q,t)-characters" depending on two parameters which interpolate between the q-characters of a quantum affine algebra and its Langlands dual.

Quon language: surface algebras and Fourier duality

Quon language is a 3D picture language that we can apply to simulate mathematical concepts. We introduce the surface algebras as an extension of the notion of planar algebras to higher genus surface. We prove that there is a unique one-parameter extension. The 2D defects on the surfaces are quons, and surface tangles are transformations. We use quon language to simulate graphic states that appear in quantum information, and to simulate interesting quantities in modular tensor categories. This simulation relates the pictorial Fourier duality of surface tangles and the algebraic Fourier duality …