scholarly journals Quantised affine algebras and parameter-dependent R-matrices

1995 ◽  
Vol 51 (2) ◽  
pp. 177-194 ◽  
Author(s):  
Anthony J. Bracken ◽  
Mark D. Gould ◽  
Yao-Zhong Zhang
Keyword(s):  
Explicit Formula ◽  
Lie Algebra ◽  
Simple Lie Algebra ◽  
Affine Lie Algebra ◽  
Affine Algebras ◽  
Parameter Dependent ◽  
Concrete Representations

Let Uq(G(1)) be a quantised non-twisted affine Lie algebra with Uq(G) the corresponding quantised simple Lie algebra. Using the previously obtained universal R-matrices for and , explicitly spectral-dependent universal R-matrices for Uq(A1) and Uq(A2) are determined. These spectral-dependent universal R-matrices are evaluated in some concrete representations; well-known results for the fundamental representations are reproduced, and an explicit formula for the spectral-dependent R-matrix associated with the V(3) ⊗ V(6) module is derived, where V(3) and V(6) carry the 3- and 6-dimensional representations of Uq(A2), respectively.

scholarly journals D5(1)-Geometric crystal corresponding to the Dynkin spin node i = 5 and its ultra-discretization

2019 ◽  
Vol 18 (12) ◽  
pp. 1950227 ◽  
Author(s):  
Mana Igarashi ◽  
Kailash C. Misra ◽  
Suchada Pongprasert
Keyword(s):  
Explicit Formula ◽  
Lie Algebra ◽  
Perfect Crystal ◽  
Index Set ◽  
Limit Formula ◽  
Affine Lie Algebra ◽  
Level Zero

Let [Formula: see text] be an affine Lie algebra with index set [Formula: see text] and [Formula: see text] be its Langlands dual. It is conjectured that for each Dynkin node [Formula: see text] the affine Lie algebra [Formula: see text] has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of certain coherent family of perfect crystals for [Formula: see text]. In this paper, we construct a positive geometric crystal [Formula: see text] in the level zero fundamental spin [Formula: see text]-module [Formula: see text]. Then we define explicit [Formula: see text]-action on the level [Formula: see text] known [Formula: see text]-perfect crystal [Formula: see text] and show that [Formula: see text] is a coherent family of perfect crystals with limit [Formula: see text]. Finally, we show that the ultra-discretization of [Formula: see text] is isomorphic to [Formula: see text] as crystals which proves the conjecture in this case.

scholarly journals A combinatorial identity for the derivative of a theta series of a finite type root lattice

2003 ◽  
Vol 172 ◽  
pp. 1-30
Author(s):  
Satoshi Naito
Keyword(s):  
Representation Theory ◽  
Lie Algebra ◽  
Finite Type ◽  
Cartan Subalgebra ◽  
Theta Series ◽  
Combinatorial Identity ◽  
Root Lattice ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Affine Lie Algebra

AbstractLet be a (not necessarily simply laced) finite-dimensional complex simple Lie algebra with the Cartan subalgebra and Q ⊂ * the root lattice. Denote by ΘQ(q) the theta series of the root lattice Q of . We prove a curious “combinatorial” identity for the derivative of ΘQ(q), i.e. for by using the representation theory of an affine Lie algebra.

scholarly journals BGG reciprocity for current algebras

2015 ◽  
Vol 151 (7) ◽  
pp. 1265-1287 ◽  
Author(s):  
Vyjayanthi Chari ◽  
Bogdan Ion
Keyword(s):  
Lie Algebra ◽  
Current Algebra ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Affine Lie Algebra ◽  
A Current ◽  
Current Algebras

In Bennett et al. [BGG reciprocity for current algebras, Adv. Math. 231 (2012), 276–305] it was conjectured that a BGG-type reciprocity holds for the category of graded representations with finite-dimensional graded components for the current algebra associated to a simple Lie algebra. We associate a current algebra to any indecomposable affine Lie algebra and show that, in this generality, the BGG reciprocity is true for the corresponding category of representations.

scholarly journals Lowest degree invariant second-order PDEs over rational homogeneous contact manifolds

2019 ◽  
Vol 21 (01) ◽  
pp. 1750089 ◽  
Author(s):  
Dmitri V. Alekseevsky ◽  
Jan Gutt ◽  
Gianni Manno ◽  
Giovanni Moreno
Keyword(s):  
Explicit Formula ◽  
Lie Algebra ◽  
Contact Manifold ◽  
Second Order ◽  
Contact Manifolds ◽  
Simple Lie Algebra ◽  
Type Formula ◽  
Monge Ampere

For each simple Lie algebra [Formula: see text] (excluding, for trivial reasons, type [Formula: see text]), we find the lowest possible degree of an invariant second-order PDE over the adjoint variety in [Formula: see text], a homogeneous contact manifold. Here a PDE [Formula: see text] has degree [Formula: see text] if [Formula: see text] is a polynomial of degree [Formula: see text] in the minors of [Formula: see text], with coefficient functions of the contact coordinate [Formula: see text], [Formula: see text], [Formula: see text] (e.g., Monge–Ampère equations have degree 1). For [Formula: see text] of type [Formula: see text] or [Formula: see text], we show that this gives all invariant second-order PDEs. For [Formula: see text] of types [Formula: see text] and [Formula: see text], we provide an explicit formula for the lowest-degree invariant second-order PDEs. For [Formula: see text] of types [Formula: see text] and [Formula: see text], we prove uniqueness of the lowest-degree invariant second-order PDE; we also conjecture that uniqueness holds in type [Formula: see text].

scholarly journals Proofs of some partition identities conjectured by Kanade and Russell

2021 ◽  
Author(s):  
Hjalmar Rosengren
Keyword(s):  
Lie Algebra ◽  
Partition Identities ◽  
Partition Identity ◽  
Affine Lie Algebra ◽  
Level 2

AbstractKanade and Russell conjectured several Rogers–Ramanujan-type partition identities, some of which are related to level 2 characters of the affine Lie algebra $$A_9^{(2)}$$ A 9 ( 2 ) . Many of these conjectures have been proved by Bringmann, Jennings-Shaffer and Mahlburg. We give new proofs of five conjectures first proved by those authors, as well as four others that have been open until now. Our proofs for the new cases use quadratic transformations for Askey–Wilson and Rogers polynomials. We also obtain some related results, including a partition identity conjectured by Capparelli and first proved by Andrews.

SPECTRAL FLOW AND FREE FIELD REALIZATIONS OF BOSONIC STRINGS ON NAPPI–WITTEN GROUP MANIFOLD

2011 ◽  
Vol 26 (01) ◽  
pp. 149-160
Author(s):  
GANG CHEN
Keyword(s):  
Lie Algebra ◽  
Bosonic Strings ◽  
Free Field ◽  
Geometric Interpretation ◽  
Closed String ◽  
Space Time ◽  
Spectral Flow ◽  
Affine Lie Algebra ◽  
Group Manifold ◽  
Free Field Realization

In this paper we study some aspects of closed string theories in the Nappi–Witten space–time. The effects of spectral flow on the geodesics are studied in terms of an explicit parametrization of the group manifold. The worldsheets of the closed strings under the spectral flow of the geodesics can be classified into four classes, each with a geometric interpretation. We also obtain a free field realization of the Nappi–Witten affine Lie algebra in the most general conditions using a different but equivalent parametrization of the group manifold.

scholarly journals Combinatorial bases of modules for affine Lie algebra B 2(1)

2013 ◽  
Vol 11 (2) ◽  
Author(s):  
Mirko Primc
Keyword(s):  
Lie Algebra ◽  
Vertex Operator ◽  
Vertex Operator Algebra ◽  
Intertwining Operators ◽  
Type B ◽  
Highest Weight Module ◽  
Highest Weight ◽  
Affine Lie Algebra ◽  
Algebra Theory ◽  
Simple Currents

AbstractWe construct bases of standard (i.e. integrable highest weight) modules L(Λ) for affine Lie algebra of type B 2(1) consisting of semi-infinite monomials. The main technical ingredient is a construction of monomial bases for Feigin-Stoyanovsky type subspaces W(Λ) of L(Λ) by using simple currents and intertwining operators in vertex operator algebra theory. By coincidence W(kΛ0) for B 2(1) and the integrable highest weight module L(kΛ0) for A 1(1) have the same parametrization of combinatorial bases and the same presentation P/I.

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