Quantum Weyl group and multiplicative formula for the R-matrix of a simple Lie algebra

1991 ◽  
Vol 25 (2) ◽  
pp. 143-145 ◽  
Author(s):  
S. Z. Levendorskii ◽  
Ya. S. Soibel'man
Keyword(s):  
Weyl Group ◽  
Lie Algebra ◽  
R Matrix ◽  
Simple Lie Algebra ◽  
Multiplicative Formula

THE -SCHUR ALGEBRAS AND -SCHUR DUALITIES OF FINITE TYPE

2020 ◽  
pp. 1-32 ◽  
Author(s):  
Li Luo ◽  
Weiqiang Wang
Keyword(s):  
Weyl Group ◽  
Lie Algebra ◽  
Finite Type ◽  
Canonical Basis ◽  
Hecke Algebra ◽  
Schur Algebras ◽  
Schur Algebra ◽  
Simple Lie Algebra ◽  
Finite Set

We formulate a $q$ -Schur algebra associated with an arbitrary $W$ -invariant finite set $X_{\text{f}}$ of integral weights for a complex simple Lie algebra with Weyl group $W$ . We establish a $q$ -Schur duality between the $q$ -Schur algebra and Hecke algebra associated with $W$ . We then realize geometrically the $q$ -Schur algebra and duality and construct a canonical basis for the $q$ -Schur algebra with positivity. With suitable choices of $X_{\text{f}}$ in classical types, we recover the $q$ -Schur algebras in the literature. Our $q$ -Schur algebras are closely related to the category ${\mathcal{O}}$ , where the type $G_{2}$ is studied in detail.

Icosahedral symmetry breaking: C60to C78, C96and to related nanotubes

2014 ◽  
Vol 70 (6) ◽  
pp. 650-655 ◽  
Author(s):  
Mark Bodner ◽  
Emmanuel Bourret ◽  
Jiri Patera ◽  
Marzena Szajewska
Keyword(s):  
Symmetry Breaking ◽  
Weyl Group ◽  
Lie Algebra ◽  
Icosahedral Symmetry ◽  
Simple Lie Algebra ◽  
Icosahedral Group ◽  
Symmetric Subgroup ◽  
Surface Shell

Exact icosahedral symmetry of C60is viewed as the union of 12 orbits of the symmetric subgroup of order 6 of the icosahedral group of order 120. Here, this subgroup is denoted byA2because it is isomorphic to the Weyl group of the simple Lie algebraA2. Eight of theA2orbits are hexagons and four are triangles. Only two of the hexagons appear as part of the C60surface shell. The orbits form a stack of parallel layers centered on the axis of C60passing through the centers of two opposite hexagons on the surface of C60. By inserting into the middle of the stack twoA2orbits of six points each and twoA2orbits of three points each, one can match the structure of C78. Repeating the insertion, one gets C96; multiple such insertions generate nanotubes of any desired length. Five different polytopes with 78 carbon-like vertices are described; only two of them can be augmented to nanotubes.

AUTOMORPHISMS OF CERTAIN NILPOTENT LIE ALGEBRAS OVER COMMUTATIVE RINGS

2007 ◽  
Vol 17 (03) ◽  
pp. 527-555 ◽  
Author(s):  
YOU'AN CAO ◽  
DEZHI JIANG ◽  
JUNYING WANG
Keyword(s):  
Automorphism Group ◽  
Commutative Ring ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Commutative Rings ◽  
Nilpotent Lie Algebras ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Positive Roots ◽  
Nilpotent Subalgebra

Let L be a finite-dimensional complex simple Lie algebra, Lℤ be the ℤ-span of a Chevalley basis of L and LR = R⊗ℤLℤ be a Chevalley algebra of type L over a commutative ring R. Let [Formula: see text] be the nilpotent subalgebra of LR spanned by the root vectors associated with positive roots. The aim of this paper is to determine the automorphism group of [Formula: see text].

scholarly journals THE UNIVERSAL LINK POLYNOMIAL

1992 ◽  
Vol 07 (05) ◽  
pp. 877-945 ◽  
Author(s):  
E. GUADAGNINI
Keyword(s):  
Field Theory ◽  
Closed Form ◽  
Lie Algebra ◽  
Wilson Line ◽  
Expectation Values ◽  
Simple Lie Algebra ◽  
Chern Simons

The solution of the non-Abelian SU (N) quantum Chern–Simons field theory defined in R3 is presented. It is shown how to compute the expectation values of the Wilson line operators, associated with oriented framed links, in closed form. The main properties of the universal link polynomial, defined by these expectation values, are derived in the case of a generic real simple Lie algebra. The resulting polynomials for some simple examples of links are reported.

scholarly journals ELEMENTARY PARABOLIC TWIST

2002 ◽  
Vol 01 (04) ◽  
pp. 413-424 ◽  
Author(s):  
V. D. LYAKHOVSKY ◽  
M. E. SAMSONOV
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Borel Subalgebra ◽  
Main Element ◽  
Two Dimensional ◽  
Parabolic Subalgebra ◽  
Simple Lie Algebras ◽  
Simple Lie Algebra

The twist deformations for simple Lie algebras [Formula: see text] whose twisting elements ℱ are known explicitly are usually defined on the carrier subspace injected in the Borel subalgebra [Formula: see text]. We consider the case where the carrier of the twist intersects nontrivially with both [Formula: see text] and [Formula: see text]. The main element of the new deformation is the parabolic twist ℱ℘ whose carrier is the minimal parabolic subalgebra of simple Lie algebra [Formula: see text]. It has the structure of the algebra of two-dimensional motions, contains [Formula: see text] and intersects nontrivially with [Formula: see text]. The twist ℱ℘ is constructed as a composition of the extended jordanian twist [Formula: see text] and the factor [Formula: see text]. The latter can be considered as a special deformed version of the jordanian twist. The twisted costructure is found for [Formula: see text] and the corresponding universal ℛ-matrix is presented. The parabolic twist can be composed with certain types of chains of extended jordanian twists for algebras A2(n-1). The chains enlarged by the parabolic factor ℱ℘ perform the explicit quantization of the new set of classical r-matrices.

scholarly journals Weyl modules and Weyl functors for hyper-map algebras

2020 ◽  
pp. 2140007
Author(s):  
Angelo Bianchi ◽  
Samuel Chamberlin
Keyword(s):  
Lie Algebra ◽  
Finitely Generated ◽  
Weyl Modules ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Universal Properties ◽  
Definition Of ◽  
Unitary Algebra

We investigate the representations of the hyperalgebras associated to the map algebras [Formula: see text], where [Formula: see text] is any finite-dimensional complex simple Lie algebra and [Formula: see text] is any associative commutative unitary algebra with a multiplicatively closed basis. We consider the natural definition of the local and global Weyl modules, and the Weyl functor for these algebras. Under certain conditions, we prove that these modules satisfy certain universal properties, and we also give conditions for the local or global Weyl modules to be finite-dimensional or finitely generated, respectively.

Sign in / Sign up

Export Citation Format

Share Document