PRESENTATION OF QUANTUM GENERALIZED SCHUR ALGEBRAS

2005 ◽  
Vol 04 (05) ◽  
pp. 567-575
Author(s):  
LIBIN LI
Keyword(s):  
Tensor Product ◽  
Quantum Group ◽  
Simple Module ◽  
Schur Algebras ◽  
Schur Algebra ◽  
Root Of Unity ◽  
Annihilator Ideal ◽  
Finite Dimensional ◽  
Weight Property

We obtain the explicit generators of the annihilator ideal for the tensor product of any finite dimensional simple module over quantum group [Formula: see text], by using the weight property of ideals in [Formula: see text] when q is not a root of unity. As an application, we give a presentation of quantum generalized Schur algebra.

scholarly journals On semi-infinite cohomology of finite-dimensional graded algebras

2010 ◽  
Vol 146 (2) ◽  
pp. 480-496 ◽  
Author(s):  
Roman Bezrukavnikov ◽  
Leonid Positselski
Keyword(s):  
Quantum Group ◽  
General Setting ◽  
Derived Categories ◽  
Graded Algebras ◽  
Root Of Unity ◽  
Finite Dimensional ◽  
Small Quantum ◽  
Definition Of

AbstractWe describe a general setting for the definition of semi-infinite cohomology of finite-dimensional graded algebras, and provide an interpretation of such cohomology in terms of derived categories. We apply this interpretation to compute semi-infinite cohomology of some modules over the small quantum group at a root of unity, generalizing an earlier result of Arkhipov (posed as a conjecture by B. Feigin).

scholarly journals DIFFERENTIAL CALCULUS AND CONNECTIONS ON A QUANTUM PLANE AT A CUBIC ROOT OF UNITY

2000 ◽  
Vol 12 (02) ◽  
pp. 227-285 ◽  
Author(s):  
R. COQUEREAUX ◽  
A. O. GARCÍA ◽  
R. TRINCHERO
Keyword(s):  
Representation Theory ◽  
Quantum Group ◽  
Lorentz Group ◽  
Differential Calculus ◽  
Differential Algebra ◽  
Space Time ◽  
Quantum Plane ◽  
Root Of Unity ◽  
Finite Dimensional

We consider the algebra of N×N matrices as a reduced quantum plane on which a finite-dimensional quantum group ℋ acts. This quantum group is a quotient of [Formula: see text], q being an Nth root of unity. Most of the time we shall take N=3; in that case dim(ℋ)=27. We recall the properties of this action and introduce a differential calculus for this algebra: it is a quotient of the Wess–Zumino complex. The quantum group ℋ also acts on the corresponding differential algebra and we study its decomposition in terms of the representation theory of ℋ. We also investigate the properties of connections, in the sense of non commutative geometry, that are taken as 1-forms belonging to this differential algebra. By tensoring this differential calculus with usual forms over space-time, one can construct generalized connections with covariance properties with respect to the usual Lorentz group and with respect to a finite-dimensional quantum group.

scholarly journals Existence and dimensionality of simple weight modules for quantum enveloping algebras

1993 ◽  
Vol 48 (1) ◽  
pp. 35-40
Author(s):  
Zhiyong Shi
Keyword(s):  
Necessary Conditions ◽  
Simple Module ◽  
Weight Space ◽  
Semisimple Lie Algebra ◽  
Simple Modules ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Root Of Unity ◽  
Finite Dimensional ◽  
Sufficient And Necessary Conditions

We give sufficient and necessary conditions for simple modules of the quantum group or the quantum enveloping algebra Uq(g) to have weight space decompositions, where g is a semisimple Lie algebra and q is a nonzero complex number. We show that(i) if q is a root of unity, any simple module of Uq(g) is finite dimensional, and hence is a weight module;(ii) if q is generic, that is, not a root of unity, then there are simple modules of Uq(g) which do not have weight space decompositions.Also the group of units of Uq(g) is found.

Global crystal bases and q-Schur algebras

2015 ◽  
Vol 14 (08) ◽  
pp. 1550117
Author(s):  
Anna Stokke
Keyword(s):  
Invertible Matrix ◽  
Crystal Bases ◽  
Crystal Basis ◽  
Schur Algebras ◽  
Schur Algebra ◽  
Finite Dimensional ◽  
Basis Vectors

We prove that the quantized Carter–Lusztig basis for a finite-dimensional irreducible Uq(𝔤𝔩n(ℂ))-module V(λ) is related to the global crystal basis for V(λ) by an upper triangular invertible matrix. We express the global crystal basis in terms of the q-Schur algebra and provide an algorithm for obtaining global crystal basis vectors for V(λ) using the q-Schur algebra.

scholarly journals Affine quantum Schur algebras at roots of unity

2017 ◽  
Vol 28 (07) ◽  
pp. 1750056
Author(s):  
Qiang Fu
Keyword(s):  
London Mathematical Society ◽  
Roots Of Unity ◽  
Hall Algebra ◽  
Schur Algebras ◽  
Schur Algebra ◽  
Finite Dimensional ◽  
Irreducible Modules ◽  
Algebra Approach ◽  
Weyl Theory ◽  
Littelmann Paths

Finite dimensional irreducible modules for the affine quantum Schur algebra [Formula: see text] were classified in [B. Deng, J. Du and Q. Fu, A Double Hall Algebra Approach to Affine Quantum Schur–Weyl Theory, London Mathematical Society Lecture Note Series, Vol. 401 (Cambridge University Press, Cambridge, 2012), Chapt. 4] when [Formula: see text] is not a root of unity. We will classify finite-dimensional irreducible modules for affine quantum Schur algebras at roots of unity and generalize [J. A. Green, Polynomial Representations of [Formula: see text] , 2nd edn., with an appendix on Schensted correspondence and Littelmann paths by K. Erdmann, J. A. Green and M. Schocker, Lecture Notes in Mathematics, Vol. 830 (Springer-Verlag, Berlin, 2007), (6.5f) and (6.5g)] to the affine case in this paper.

scholarly journals Modified Turaev-Viro invariants from quantum 𝔰𝔩(2|1)

2020 ◽  
Vol 29 (04) ◽  
pp. 2050018
Author(s):  
Cristina Ana-Maria Anghel ◽  
Nathan Geer
Keyword(s):  
Quantum Group ◽  
Root Of Unity ◽  
Finite Dimensional ◽  
Modular Category ◽  
Dimension 2 ◽  
Quantum Superalgebra ◽  
Construction Works

The category of finite dimensional modules over the quantum superalgebra [Formula: see text] is not semi-simple and the quantum dimension of a generic [Formula: see text]-module vanishes. This vanishing happens for any value of [Formula: see text] (even when [Formula: see text] is not a root of unity). These properties make it difficult to create a fusion or modular category. Loosely speaking, the standard way to obtain such a category from a quantum group is: (1) specialize [Formula: see text] to a root of unity; this forces some modules to have zero quantum dimension, (2) quotient by morphisms of modules with zero quantum dimension, (3) show the resulting category is finite and semi-simple. In this paper, we show an analogous construction works in the context of [Formula: see text] by replacing the vanishing quantum dimension with a modified quantum dimension. In particular, we specialize [Formula: see text] to a root of unity, quotient by morphisms of modules with zero modified quantum dimension and show the resulting category is generically finite semi-simple. Moreover, we show the categories of this paper are relative [Formula: see text]-spherical categories. As a consequence, we obtain invariants of 3-manifold with additional structures.

scholarly journals Nuclear subalgebras of UHF C*-algebras

1986 ◽  
Vol 29 (1) ◽  
pp. 97-100 ◽  
Author(s):  
R. J. Archbold ◽  
Alexander Kumjian
Keyword(s):  
Tensor Product ◽  
Inductive Limit ◽  
Inductive Limits ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional ◽  
The Family ◽  
Algebraic Tensor Product

A C*-algebra A is said to be approximately finite dimensional (AF) if it is the inductive limit of a sequence of finite dimensional C*-algebras(see [2], [5]). It is said to be nuclear if, for each C*-algebra B, there is a unique C*-norm on the *-algebraic tensor product A ⊗B [11]. Since finite dimensional C*-algebras are nuclear, and inductive limits of nuclear C*-algebras are nuclear [16];,every AF C*-algebra is nuclear. The family of nuclear C*-algebras is a large and well-behaved class (see [12]). The AF C*-algebras for a particularly tractable sub-class which has been completely classified in terms of the invariant K0 [7], [5].

Invariants of 3-manifolds derived from universal invariants of framed links

1995 ◽  
Vol 117 (2) ◽  
pp. 259-273 ◽  
Author(s):  
Tomotada Ohtsuki
Keyword(s):  
Hopf Algebra ◽  
Quantum Group ◽  
Independent Method ◽  
Topological Invariant ◽  
Finite Dimensional

Reshetikhin and Turaev [10] gave a method to construct a topological invariant of compact oriented 3-manifolds from a ribbon Hopf algebra (e.g. a quantum group Uq(sl2)) using finite-dimensional representations of it. In this paper we give another independent method to construct a topological invariant of compact oriented 3-manifolds from a ribbon Hopf algebra via universal invariants of framed links without using representations of the algebra. For Uq(sl2) these two methods give different invariants of 3-manifolds.

scholarly journals Quantum relativistic Toda chain at root of unity: isospectrality, modifiedQ-operator, and functional Bethe ansatz

2002 ◽  
Vol 31 (9) ◽  
pp. 513-553 ◽  
Author(s):  
Stanislav Pakuliak ◽  
Sergei Sergeev
Keyword(s):  
Bethe Ansatz ◽  
Weyl Algebra ◽  
Separation Of Variables ◽  
Toda Chain ◽  
Special Choice ◽  
Finite Dimensional Representation ◽  
Classical Counterpart ◽  
Discrete Integrable System ◽  
Root Of Unity ◽  
Finite Dimensional

We investigate anN-state spin model called quantum relativistic Toda chain and based on the unitary finite-dimensional representations of the Weyl algebra withqbeingNth primitive root of unity. Parameters of the finite-dimensional representation of the local Weyl algebra form the classical discrete integrable system. Nontrivial dynamics of the classical counterpart corresponds to isospectral transformations of the spin system. Similarity operators are constructed with the help of modified Baxter'sQ-operators. The classical counterpart of the modifiedQ-operator for the initial homogeneous spin chain is a Bäcklund transformation. This transformation creates an extra Hirota-type soliton in a parameterization of the chain structure. Special choice of values of solitonic amplitudes yields a degeneration of spin eigenstates, leading to the quantum separation of variables, or the functional Bethe ansatz. A projector to the separated eigenstates is constructed explicitly as a product of modifiedQ-operators.

scholarly journals Nested Polyhedra and Indices of Orbits of Coxeter Groups of Non-Crystallographic Type

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1737
Author(s):  
Mariia Myronova ◽  
Jiří Patera ◽  
Marzena Szajewska
Keyword(s):  
Tensor Product ◽  
Lie Algebras ◽  
Coxeter Groups ◽  
Reflection Groups ◽  
Geometrical Structures ◽  
Simple Lie Algebras ◽  
Finite Dimensional ◽  
Branching Rules ◽  
Definition Of ◽  
Representation Orbit

The invariants of finite-dimensional representations of simple Lie algebras, such as even-degree indices and anomaly numbers, are considered in the context of the non-crystallographic finite reflection groups H2, H3 and H4. Using a representation-orbit replacement, the definitions and properties of the indices are formulated for individual orbits of the examined groups. The indices of orders two and four of the tensor product of k orbits are determined. Using the branching rules for the non-crystallographic Coxeter groups, the embedding index is defined similarly to the Dynkin index of a representation. Moreover, since the definition of the indices can be applied to any orbit of non-crystallographic type, the algorithm allowing to search for the orbits of smaller radii contained within any considered one is presented for the Coxeter groups H2 and H3. The geometrical structures of nested polytopes are exemplified.

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