Symmetric groups, Schur–Weyl duality and positive self-adjoint Hopf algebras

2018 ◽  
pp. 105-147
Author(s):  
Caroline Gruson ◽  
Vera Serganova
Keyword(s):  
Hopf Algebras ◽  
Symmetric Groups ◽  
Weyl Duality

scholarly journals Dominant and global dimension of blocks of quantised Schur algebras

2021 ◽  
Author(s):  
Ming Fang ◽  
Wei Hu ◽  
Steffen Koenig
Keyword(s):  
Hecke Algebras ◽  
Symmetric Groups ◽  
Global Dimension ◽  
Group Algebras ◽  
Dominant Dimension ◽  
Schur Algebras ◽  
Homological Dimensions ◽  
Weyl Duality

AbstractGroup algebras of symmetric groups and their Hecke algebras are in Schur-Weyl duality with classical and quantised Schur algebras, respectively. Two homological dimensions, the dominant dimension and the global dimension, of the indecomposable summands (blocks) of these Schur algebras S(n, r) and $$S_q(n,r)$$ S q ( n , r ) with $$n \geqslant r$$ n ⩾ r are determined explicitly, using a result on derived invariance in Fang, Hu and Koenig (J Reine Angew Math 770:59–85, 2021).

scholarly journals On pointed Hopf algebras associated with the symmetric groups

2008 ◽  
Vol 128 (3) ◽  
pp. 359-371 ◽  
Author(s):  
Nicolás Andruskiewitsch ◽  
Fernando Fantino ◽  
Shouchuan Zhang
Keyword(s):  
Hopf Algebras ◽  
Symmetric Groups ◽  
Pointed Hopf Algebras

scholarly journals Blocks of symmetric groups, semicuspidal KLR algebras and zigzag Schur-Weyl duality

2018 ◽  
Vol 188 (2) ◽  
pp. 453-512 ◽  
Author(s):  
Anton Evseev ◽  
Alexander Kleshchev
Keyword(s):  
Symmetric Groups ◽  
Klr Algebras ◽  
Weyl Duality

A REALIZATION OF THE ENVELOPING SUPERALGEBRA

2021 ◽  
pp. 1-36
Author(s):  
JIE DU ◽  
QIANG FU ◽  
YANAN LIN
Keyword(s):  
Symmetric Groups ◽  
Loop Algebra ◽  
Natural Representation ◽  
Schur Algebras ◽  
Tensor Spaces ◽  
Weyl Duality ◽  
Geometric Setting

Abstract In [2], Beilinson–Lusztig–MacPherson (BLM) gave a beautiful realization for quantum $\mathfrak {gl}_n$ via a geometric setting of quantum Schur algebras. We introduce the notion of affine Schur superalgebras and use them as a bridge to link the structure and representations of the universal enveloping superalgebra ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ of the loop algebra $\widehat {\mathfrak {gl}}_{m|n}$ of ${\mathfrak {gl}}_{m|n}$ with those of affine symmetric groups ${\widehat {{\mathfrak S}}_{r}}$ . Then, we give a BLM type realization of ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ via affine Schur superalgebras. The first application of the realization of ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ is to determine the action of ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ on tensor spaces of the natural representation of $\widehat {\mathfrak {gl}}_{m|n}$ . These results in epimorphisms from $\;{\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ to affine Schur superalgebras so that the bridging relation between representations of ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ and ${\widehat {{\mathfrak S}}_{r}}$ is established. As a second application, we construct a Kostant type $\mathbb Z$ -form for ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ whose images under the epimorphisms above are exactly the integral affine Schur superalgebras. In this way, we obtain essentially the super affine Schur–Weyl duality in arbitrary characteristics.

scholarly journals QUANTUM ISOMETRY GROUPS OF SYMMETRIC GROUPS

2012 ◽  
Vol 23 (07) ◽  
pp. 1250074 ◽  
Author(s):  
JAN LISZKA-DALECKI ◽  
PIOTR M. SOŁTAN
Keyword(s):  
Hopf Algebras ◽  
Isometry Group ◽  
Nearest Neighbor ◽  
Symmetric Groups ◽  
Geometric Interpretation ◽  
Group Algebras ◽  
Isometry Groups ◽  
Spectral Triples ◽  
Multiplier Hopf Algebras ◽  
Quantum Isometry Groups

We identify the quantum isometry groups of spectral triples built on the symmetric groups with length functions arising from the nearest-neighbor transpositions as generators. It turns out that they are isomorphic to certain "doubling" of the group algebras of the respective symmetric groups. We discuss the doubling procedure in the context of regular multiplier Hopf algebras. In the last section we study the dependence of the isometry group of Sn on the choice of generators in the case n = 3. We show that two different choices of generators lead to nonisomorphic quantum isometry groups which exhaust the list of noncommutative noncocommutative semisimple Hopf algebras of dimension 12. This provides noncommutative geometric interpretation of these Hopf algebras.

scholarly journals Projective limits of quantum symmetry groups and the doubling construction for Hopf algebras

2014 ◽  
Vol 17 (02) ◽  
pp. 1450012 ◽  
Author(s):  
Adam Skalski ◽  
Piotr M. Sołtan
Keyword(s):  
Hopf Algebras ◽  
Discrete Group ◽  
Inductive Limit ◽  
Symmetric Groups ◽  
Projective Limit ◽  
Symmetry Groups ◽  
C Algebras ◽  
Quantum Symmetry ◽  
Doubling Construction ◽  
Projective Limits

The quantum symmetry group of the inductive limit of C*-algebras equipped with orthogonal filtrations is shown to be the projective limit of the quantum symmetry groups of the C*-algebras appearing in the sequence. Some explicit examples of such projective limits are studied, including the case of quantum symmetry groups of the duals of finite symmetric groups, which do not fit directly into the framework of the main theorem and require further specific study. The investigations reveal a deep connection between quantum symmetry groups of discrete group duals and the doubling construction for Hopf algebras.

Left counital Hopf algebras on bi-decorated planar rooted forests and Rota-Baxter systems

2020 ◽  
Vol 27 (2) ◽  
pp. 219-243 ◽  
Author(s):  
Xiao-Song Peng ◽  
Yi Zhang ◽  
Xing Gao ◽  
Yan-Feng Luo
Keyword(s):  
Hopf Algebras

scholarly journals Monadic cointegrals and applications to quasi-Hopf algebras

2021 ◽  
Vol 225 (10) ◽  
pp. 106678
Author(s):  
Johannes Berger ◽  
Azat M. Gainutdinov ◽  
Ingo Runkel
Keyword(s):  
Hopf Algebras
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