scholarly journals The Serpentine Representation of the Infinite Symmetric Group and the Basic Representation of the Affine Lie Algebra $${\widehat{\mathfrak{sl}_2}}$$ sl 2 ^

2014 ◽  
Vol 105 (1) ◽  
pp. 11-25 ◽  
Author(s):  
Natalia V. Tsilevich ◽  
Anatoly M. Vershik
Keyword(s):  
Symmetric Group ◽  
Lie Algebra ◽  
Infinite Symmetric Group ◽  
Basic Representation ◽  
Affine Lie Algebra

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.

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.

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