Orbit–orbit branching rules for families of classical Lie algebra–subalgebra pairs

1996 ◽  
Vol 37 (9) ◽  
pp. 4750-4757 ◽  
Author(s):  
M. Thoma ◽  
R. T. Sharp
Keyword(s):  
Lie Algebra ◽  
Branching Rules ◽  
Classical Lie Algebra

scholarly journals FUSION RULES AND COMPLETE REDUCIBILITY OF CERTAIN MODULES FOR AFFINE LIE ALGEBRAS

2013 ◽  
Vol 13 (01) ◽  
pp. 1350062 ◽  
Author(s):  
DRAŽEN ADAMOVIĆ ◽  
OZREN PERŠE
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Operator Algebras ◽  
Affine Lie Algebras ◽  
Fusion Rules ◽  
Type Formula ◽  
Complete Reducibility ◽  
Affine Lie Algebra ◽  
Branching Rules ◽  
Level 1

We develop a new method for obtaining branching rules for affine Kac–Moody Lie algebras at negative integer levels. This method uses fusion rules for vertex operator algebras of affine type. We prove that an infinite family of ordinary modules for affine vertex algebra of type A investigated in our previous paper J. Algebra319 (2008) 2434–2450, is closed under fusion. Then, we apply these fusion rules on explicit bosonic realization of level -1 modules for the affine Lie algebra of type [Formula: see text], obtain a new proof of complete reducibility for these representations, and the corresponding decomposition for ℓ ≥ 3. We also obtain the complete reducibility of the associated level -1 modules for affine Lie algebra of type [Formula: see text]. Next, we notice that the category of [Formula: see text] modules at level -2ℓ + 3 has the isomorphic fusion algebra. This enables us to decompose certain [Formula: see text] and [Formula: see text]-modules at negative levels.

scholarly journals Cohomology of simple modules for sl3(k) in characteristic 3

2021 ◽  
Vol 103 (3) ◽  
pp. 36-43
Author(s):  
A.A. Ibrayeva ◽  
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Quotient Algebra ◽  
Characteristic P ◽  
One Dimensional ◽  
Simple Modules ◽  
Trivial Module ◽  
Characteristic 3 ◽  
Simple Quotient ◽  
Classical Lie Algebra

In this paper we calculate cohomology of a classical Lie algebra of type A2 over an algebraically field k of characteristic p = 3 with coefficients in simple modules. To describe their structure we will consider them as modules over an algebraic group SL3(k). In the case of characteristic p = 3, there are only two peculiar simple modules: a simple that module isomorphic to the quotient module of the adjoint module by the center, and a one-dimensional trivial module. The results on the cohomology of simple nontrivial module are used for calculating the cohomology of the adjoint module. We also calculate cohomology of the simple quotient algebra Lie of A2 by the center.

scholarly journals On a conjecture of Naito-Sagaki: Littelmann paths and Littlewood-Richardson Sundaram tableaux

2021 ◽  
pp. 305-319
Author(s):  
Jacinta Torres
Keyword(s):  
Fixed Point ◽  
Irreducible Representation ◽  
Recent Work ◽  
Lie Algebra ◽  
Fixed Point Set ◽  
New Approach ◽  
Branching Rule ◽  
Branching Rules ◽  
Point Set ◽  
Littelmann Paths

In recent work with Schumann we have proven a conjecture of Naito-Sagaki giving a branching rule for the decomposition of the restriction of an irreducible representation of the special linear Lie algebra to the symplectic Lie algebra, therein embedded as the fixed-point set of the involution obtained by the folding of the corresponding Dyinkin diagram. It provides a new approach to branching rules for non-Levi subalgebras in terms of Littelmann paths. In this paper we motivate this result, provide examples, and give an overview of the combinatorics involved in its proof.

A CLASSICAL REALIZATION OF QUANTUM ALGEBRAS

1990 ◽  
Vol 05 (28) ◽  
pp. 2325-2333 ◽  
Author(s):  
ALEXIOS P. POLYCHRONAKOS
Keyword(s):  
Hopf Algebra ◽  
Lie Algebra ◽  
Group Homomorphism ◽  
Quantum Algebras ◽  
Quantum Deformation ◽  
Standard Quantum ◽  
Algebra For All ◽  
Special Case ◽  
Classical Lie Algebra ◽  
Construction Works

We construct a realization of a deformation of the Lie algebra of a group in terms of the generators of the classical Lie algebra of the group. The construction works for arbitrary (odd) deforming functions and, as a special case, it reproduces the standard quantum deformation of the algebra. For all these functions it gives a co-multiplication, that is, a group homomorphism, and provides an antipode and a co-unit. It therefore promotes any arbitrary deformation into a Hopf algebra.

scholarly journals Characterizing the Absolute Continuity of the Convolution of Orbital Measures in a Classical Lie Algebra

2016 ◽  
Vol 68 (4) ◽  
pp. 841-875 ◽  
Author(s):  
Sanjiv Kumar Gupta ◽  
Kathryn Hare
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Absolute Continuity ◽  
Classical Result ◽  
Continuous Measure ◽  
Empty Interior ◽  
Absolutely Continuous ◽  
Absolutely Continuous Measure ◽  
Orbital Measure ◽  
Classical Lie Algebra

AbstractLet 𝓰 be a compact simple Lie algebra of dimension d. It is a classical result that the convolution of any d non-trivial, G-invariant, orbitalmeasures is absolutely continuous with respect to Lebesgue measure on 𝓰, and the sum of any d non-trivial orbits has non-empty interior. The number d was later reduced to the rank of the Lie algebra (or rank +1 in the case of type An). More recently, the minimal integer k = k(X) such that the k-fold convolution of the orbital measure supported on the orbit generated by X is an absolutely continuous measure was calculated for each X ∈ 𝓰.In this paper 𝓰 is any of the classical, compact, simple Lie algebras. We characterize the tuples (X1 , . . . , XL), with Xi ∊ 𝓰, which have the property that the convolution of the L-orbital measures supported on the orbits generated by the Xi is absolutely continuous, and, equivalently, the sum of their orbits has non-empty interior. The characterization depends on the Lie type of 𝓰 and the structure of the annihilating roots of the Xi. Such a characterization was previously known only for type An.

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