Irreducible representations of simple Lie algebras with maximum weight multiplicity 2

Hermitian representations play a fundamental role in the study of the representations of simple Lie algebras. We show how this concept generalizes for classical simple graded Lie algebras. Star and grade star representations are defined through adjoint and grade adjoint operations. Each algebra admits at most two adjoint and two grade adjoint operations (we list the various possibilities for all classical simple graded Lie algebras). To each adjoint (grade adjoint) operation corresponds a class of star (grade star) representations. The tensor product of two star representations belonging to one class is completely reducible into irreducible representations belonging to the same class. This property is very useful since in general the finite-dimensional representations of classical simple graded Lie algebras are not completely reducible.

Complete bases are constructed for all finite-dimensional irreducible representations of the simple Lie algebras over C of the types An (n≥1), Bn and Cn (2≤n≤6), Dn (4≤n≤6), and G2. Each basis vector is given as an explicit sequence of weight-lowering generators of the algebra acting on the highest weight vector of the representation space. A similar construction (due to D-N. Verma) for the highest weight representations of all Kac–Moody algebras of rank 2 is presented as well.

In this paper we study the asymptotic of multiplicities of irreducible representations in large tensor products of finite dimensional representations of simple Lie algebras and their statistics with respect to Plancherel and character probability measures. We derive the asymptotic distribution of irreducible components for the Plancherel measure, generalizing results of Biane and Tate and Zelditch. We also derive the asymptotic of the character measure for generic parameters and an intermediate scaling in the vicinity of the Plancherel measure. It is interesting that the asymptotic measure is universal and after suitable renormalization does not depend on which representations were multiplied but depends significantly on the degeneracy of the parameter in the character distribution.

CHAPTER 10 - REPRESENTATIONS OF SEMISIMPLE LIE ALGEBRAS

New combinatorial formulas are obtained for the multiplicities in the decomposition of the tensor product of the representations of simple Lie algebras into irreducible components.

The concept of standard representations of simple Lie algebras was introduced by I. Z. Bouwer [1], One of the difficulties was that of existence. The order zero standard representations are simply those having a dominant weight vector and these have been completely characterized, for example in [2].

A bstract We show how to classify the asymptotically-free gauge theories in four spacetime dimensions, focussing here on the case of purely fermionic matter. The classification depends on the fact (which we prove) that the dimension and Dynkin index of irreducible representations of a simple Lie algebra are both strictly-increasing, integer-valued functions of each Dynkin label. This implies not only that the number of asymptotically-free representations of any one semisimple Lie algebra is finite, but also that they can be written down in a systematic fashion using tables for the asymptotically-free irreducible representations of simple Lie algebras, which we supply. These tables show that at most two out of a possible ten Dynkin labels can be non-zero and that no Dynkin label can exceed four. We also discuss further constraints coming from the need to cancel anomalies, both local and global.

INTEGER CLEBSCH-GORDAN COEFFICIENTS FOR LIE ALGEBRA REPRESENTATIONS

Explicit Constructions of the Fundamental Representations of the Symplectic Lie Algebras

The paper solves the problem of describing the representations V of compact Lie groups G and vectors v ∈ V with single stationary subgroups. It builds the classification of the representation of simple Lie algebras with vectors having a zero stationary subalgebra. If the codimension of the submanifold M 0 of dimension m in an N-dimensional Euclidean space RN is less than the dimension and the group G is simple, then all these cases are described in the corresponding tables in sections 3 and 4.

New combinatorial formulas for multiplicities in the decomposition of the tensor product of representations of simple Lie algebras into irreducible components are obtained. A connection of the constructions arising with the Bethe equations and the representation theory of Yangians is established. Bibliography: 14 titles.

Infinite-dimensional representations of quantum algebras

Two independent algorithms are presented, which together allow the determination of branching rules from an irreducible representation of a compact Lie algebra to those of a subalgebra (or subjoined algebra). The first gives the subalgebra Weyl orbits contained in an algebra orbit. The second gives the irreducible representations of an algebra contained in an orbit, and by inversion of a triangular matrix, the orbits contained in an irreducible representation.

We establish an irreducibility property for the characters of finite dimensional, irreducible representations of simple Lie algebras (or simple algebraic groups) over the complex numbers, i.e., that the characters of irreducible representations are irreducible after dividing out by (generalized) Weyl denominator type factors. For S L ( r ) SL(r) the irreducibility result is the following: let λ = ( a 1 ≥ a 2 ≥ ⋯ ≥ a r − 1 ≥ 0 ) \lambda =(a_1\geq a_2\geq \cdots \geq a_{r-1}\geq 0) be the highest weight of an irreducible rational representation V λ V_{\lambda } of S L ( r ) SL(r) . Assume that the integers a 1 + r − 1 , a 2 + r − 2 , ⋯ , a r − 1 + 1 a_1+r-1, ~a_2+r-2, \cdots , a_{r-1}+1 are relatively prime. Then the character χ λ \chi _{\lambda } of V λ V_{\lambda } is strongly irreducible in the following sense: for any natural number d d , the function χ λ ( g d ) , g ∈ S L ( r , C ) \chi _{\lambda }(g^d), ~g\in SL(r,\mathbb {C}) is irreducible in the ring of regular functions of S L ( r , C ) SL(r,\mathbb {C}) .

This chapter deals with representations of special Lie groups and special Lie algebras. Since a representation of a Lie algebra can be classified with the highest weight, those of a Lie group can be easily treated through those of the corresponding Lie algebra. Also, Lie algebras provide several concepts important for quantum theory. Hence, this chapter is organized so that it constructs a representation of a Lie group via that of the corresponding Lie algebra. This chapter starts with representations of Lie algebras \(\mathop {{\mathfrak {su}}}\nolimits (2)\) and \(\mathop {{\mathfrak {su}}}\nolimits (1,1)\). Since the Lie algebra \(\mathop {{\mathfrak {su}}}\nolimits (2)\) is compact and the Lie algebra \(\mathop {{\mathfrak {su}}}\nolimits (1,1)\) is not compact, they require different treatment caused in this difference. As they have unexpected common features, we handle both in a unified way. Then, we proceed to representation of the Lie algebra \(\mathop {{\mathfrak {su}}}\nolimits (r)\) by using Young diagrams. Especially, the representation of the Lie algebra \(\mathop {{\mathfrak {su}}}\nolimits (r)\) on the tensor product space is closely related to that of the permutation group on the same tensor product space. The relation is called Schur duality. We also consider what a finite subgroup of a Lie group can replace the Lie group when its representation is given. Such a problem is called design, and is discussed in this chapter.