Generalized Casimir operators

2019 ◽  
Vol 18 (11) ◽  
pp. 1950219 ◽  
Author(s):  
S. Eswara Rao
Keyword(s):  
Lie Algebra ◽  
Casimir Operator ◽  
New Class ◽  
Highest Weight ◽  
Casimir Operators

Let [Formula: see text] be symmetrizable Kac–Moody Lie algebra. In this paper, we describe a new class of central operators generalizing the Casimir operator. We also prove some properties of these operators and show that these operators move highest weight vectors to new highest weight vectors.

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.

Algebraic structure of Dirac–Pauli equation with account to AMM of neutron in the presence of magnetic field with cylindrical symmetry

2021 ◽  
pp. 2150121
Author(s):  
Masoud Seidi
Keyword(s):  
Magnetic Field ◽  
Lie Algebra ◽  
Algebraic Structure ◽  
Anomalous Magnetic Moment ◽  
Casimir Operator ◽  
Cylindrical Symmetry ◽  
Pauli Equation ◽  
Eigenvalues And Eigenfunctions ◽  
Ladder Operators ◽  
Nu Method

The eigenvalues and eigenfunctions of Dirac–Pauli equation have been obtained for a neutron with anomalous magnetic moment (AMM) in the presence of a strong magnetic field with cylindrical symmetry. In our calculations, the Nikiforov and Uvarov (NU) method has been used. Using the eigenfunctions and construction of the ladder operators, we show that these generators satisfy su(2) Lie algebra and computed the second-order Casimir operator of the lie algebra.

scholarly journals Isomorphisms of Twisted Hilbert Loop Algebras

2017 ◽  
Vol 69 (02) ◽  
pp. 453-480
Author(s):  
Timothée Marquis ◽  
Karl-Hermann Neeb
Keyword(s):  
Root System ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Positive Energy ◽  
Algebra Structure ◽  
Affine Lie Algebras ◽  
Subsequent Work ◽  
Highest Weight ◽  
Loop Algebras ◽  
Highest Weight Representations

Abstract The closest infinite-dimensional relatives of compact Lie algebras are Hilbert-Lie algebras, i.e., real Hilbert spaces with a Lie algebra structure for which the scalar product is invariant. Locally affine Lie algebras (LALAs) correspond to double extensions of (twisted) loop algebras over simple Hilbert-Lie algebras , also called affinisations of . They possess a root space decomposition whose corresponding root system is a locally affine root system of one of the 7 families for some infinite set J. To each of these types corresponds a “minimal ” affinisation of some simple Hilbert-Lie algebra , which we call standard. In this paper, we give for each affinisation g of a simple Hilbert-Lie algebra an explicit isomorphism from g to one of the standard affinisations of . The existence of such an isomorphism could also be derived from the classiffication of locally affine root systems, but for representation theoretic purposes it is crucial to obtain it explicitly as a deformation between two twists that is compatible with the root decompositions. We illustrate this by applying our isomorphism theorem to the study of positive energy highest weight representations of g. In subsequent work, this paper will be used to obtain a complete classification of the positive energy highest weight representations of affinisations of .

scholarly journals A note on the zeroth products of Frenkel–Jing operators

2017 ◽  
Vol 16 (03) ◽  
pp. 1750053 ◽  
Author(s):  
Slaven Kožić
Keyword(s):  
Lie Algebra ◽  
Vertex Operator ◽  
Vertex Operator Algebra ◽  
Vertex Algebra ◽  
Infinite Dimensional ◽  
Enveloping Algebra ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Quantum Vertex ◽  
Algebra Theory

Let [Formula: see text] be an untwisted affine Kac–Moody Lie algebra. The top of every irreducible highest weight integrable [Formula: see text]-module is the finite-dimensional irreducible [Formula: see text]-module, where the action of the simple Lie algebra [Formula: see text] is given by zeroth products arising from the underlying vertex operator algebra theory. Motivated by this fact, we consider zeroth products of level [Formula: see text] Frenkel–Jing operators corresponding to Drinfeld realization of the quantum affine algebra [Formula: see text]. By applying these products, which originate from the quantum vertex algebra theory developed by Li, on the extension of Koyama vertex operator [Formula: see text], we obtain an infinite-dimensional vector space [Formula: see text]. Next, we introduce an associative algebra [Formula: see text], a certain quantum analogue of the universal enveloping algebra [Formula: see text], and construct some infinite-dimensional [Formula: see text]-modules [Formula: see text] corresponding to the finite-dimensional irreducible [Formula: see text]-modules [Formula: see text]. We show that the space [Formula: see text] carries a structure of an [Formula: see text]-module and, furthermore, we prove that the [Formula: see text]-module [Formula: see text] is isomorphic to the [Formula: see text]-module [Formula: see text].

Generalized Casimir Operators

1969 ◽  
Vol 21 ◽  
pp. 1496-1505
Author(s):  
A. J. Douglas
Keyword(s):  
Finite Group ◽  
Casimir Operator ◽  
Identity Element ◽  
Ring Homomorphism ◽  
Fixed Ring ◽  
Injective Modules ◽  
Casimir Operators ◽  
Maschke’S Theorem ◽  
A Finite Group ◽  
Special Case

Throughout this paper, S will be a ring (not necessarily commutative) with an identity element ls ≠ 0s. We shall use R to denote a second ring, and ϕ: S→ R will be a fixed ring homomorphism for which ϕ1S = 1R.In (7), Higman generalized the Casimir operator of classical theory and used his generalization to characterize relatively projective and injective modules. As a special case, he obtained a theorem which contains results of Eckmann (3) and of Higman himself (5), and which also includes Gaschütz's generalization (4) of Maschke's theorem. (For a discussion of some of the developments of Maschke's idea of averaging over a finite group, we refer the reader to (2, Chapter IX).) In the present paper, we define the Casimir operator of a family of S-homomorphisms of one R-module into another, and we again use this operator to characterize relatively projective and injective modules.

scholarly journals Canonical bases and standard monomials

1998 ◽  
Vol 41 (3) ◽  
pp. 611-623
Author(s):  
R. J. Marsh
Keyword(s):  
Lie Algebra ◽  
Canonical Basis ◽  
Monomial Basis ◽  
Fundamental Weight ◽  
Enveloping Algebra ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Standard Monomials ◽  
Standard Monomial ◽  
Quantized Enveloping Algebra

Let U be the quantized enveloping algebra associated to a simple Lie algebra g by Drinfel'd and Jimbo. Let λ be a classical fundamental weight for g, and ⋯(λ) the irreducible, finite-dimensional type 1 highest weight U-module with highest weight λ. We show that the canonical basis for ⋯(λ) (see Kashiwara [6, §0] and Lusztig [18, 14.4.12]) and the standard monomial basis (see [11, §§2.4 and 2.5]) for ⋯(λ) coincide.

scholarly journals Verma Modules over a Block Lie Algebra

2008 ◽  
Vol 15 (02) ◽  
pp. 235-240 ◽  
Author(s):  
Qifen Jiang ◽  
Yuezhu Wu
Keyword(s):  
Lie Algebra ◽  
Verma Module ◽  
Total Order ◽  
Block Type ◽  
Verma Modules ◽  
Additive Subgroup ◽  
Highest Weight ◽  
Proper Quotient

Let [Formula: see text] be the Lie algebra with basis {Li,j, C|i, j ∈ ℤ} and relations [Li,j, Lk,l] = ((j + 1)k - i(l + 1))Li+k, j+l + iδi, -kδj+l, -2C and [C, Li,j] = 0. It is proved that an irreducible highest weight [Formula: see text]-module is quasifinite if and only if it is a proper quotient of a Verma module. An additive subgroup Γ of 𝔽 corresponds to a Lie algebra [Formula: see text] of Block type. Given a total order ≻ on Γ and a weight Λ, a Verma [Formula: see text]-module M(Λ, ≻) is defined. The irreducibility of M(Λ, ≻) is completely determined.

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