Basis states for equivalent electrons. I. States for the Lie group U(2l + 1) × SU(2)

1980 ◽  
Vol 58 (12) ◽  
pp. 1718-1723
Author(s):  
William R. Ross
Keyword(s):  
Irreducible Representation ◽  
Lie Group ◽  
Spin Operator ◽  
Total Spin ◽  
Slater Basis

The antisymmetric Slater basis states for N equivalent electrons form the basis for an irreducible representation of U(4l + 2). When we consider the subgroup U(2l + 1) × SU(2) we obtain states which are eigenstates of the total spin operator. The basis states for the irreducible representation of U(2l + 1) × SU(2) are expressed in terms of the Slater basis states. General expressions are obtained which can easily be applied regardless of the number of electrons, the value of l, or the irreducible representation that is considered.

Basis states for equivalent electrons. II. States for the Lie group Sp(4l + 2)

1980 ◽  
Vol 58 (12) ◽  
pp. 1724-1728
Author(s):  
William R. Ross
Keyword(s):  
Angular Momentum ◽  
Irreducible Representation ◽  
Orbital Angular Momentum ◽  
Lie Group ◽  
Total Spin ◽  
Irreducible Representations ◽  
Total Orbital Angular Momentum ◽  
Intermediate Group ◽  
Slater Basis

The Slater basis states for N equivalent electrons form the basis for the irreducible representation (1N) of the Lie group U(4l + 2). States which are eigenfunctions of the total spin and total orbital angular momentum form the basis for irreducible representations of SO(3) × SU(2). In this paper the intermediate group Sp(4l + 2) is studied. The basis states for irreducible representations of Sp(4l + 2) are expressed in terms of the Slater basis states.

Basis states for equivalent electrons. III. States for the Lie group SO(2l + 1) × SU(2) using Sp(4l + 2) states

1981 ◽  
Vol 59 (2) ◽  
pp. 207-212
Author(s):  
William R. Ross
Keyword(s):  
Irreducible Representation ◽  
Lie Group ◽  
Unitary Group ◽  
Symplectic Group ◽  
Irreducible Representations ◽  
Slater Basis

The Lie group SO(2l + 1) × EU(2) is a subgroup of the symplectic group Sp(4l + 2), which in turn is a subgroup of the unitary group U(4l + 2). The Slater basis states for N equivalent electrons form the basis for the irreducible representation (1N) of U(4l + 2). The basis states for the irreducible representations of SO(2l + 1) × SU(2) are expressed in terms of the states for irreducible representations of Sp(4l + 2). The basis states for SO(2l + 1) × SU(2) are also expressed in terms of the Slater basis states.

Basis states for equivalent electrons. IV. States for the Lie group SO(2l + 1) × SU(2) using U(2l + 1) × SU(2) states

1981 ◽  
Vol 59 (3) ◽  
pp. 315-324
Author(s):  
William R. Ross
Keyword(s):  
Irreducible Representation ◽  
Lie Group ◽  
Irreducible Representations ◽  
Intermediate Group ◽  
Slater Basis

In this paper we examine the irreducible representations of SO(2l + 1) × SU(2) that are present when we have equivalent electrons. Basis states for these irreducible representations are defined using the basis states for U(2l + 1) × SU(2). Using earlier results, where the U(2l + 1) × SU(2) states were defined using the U(4l + 2) states, we are able to express the SO(2l + 1) × SU(2) states in terms of the Slater basis states associated with the irreducible representation (1N) of U(4l + 2). The SO(2l + 1) × SU(2) states obtained in this paper using the intermediate group U(2l + 1) × SU(2) are compared with those obtained earlier using the intermediate group Sp(4l + 2).

Représentations irréductibles bornées des groupes de Lie exponentiels

2001 ◽  
Vol 53 (5) ◽  
pp. 944-978 ◽  
Author(s):  
J. Ludwig ◽  
C. Molitor-Braun
Keyword(s):  
Banach Space ◽  
Irreducible Representation ◽  
Lie Group ◽  
Simple Extension ◽  
Induced Representations ◽  
New Type ◽  
Exponential Lie Group

AbstractLet G be a solvable exponential Lie group. We characterize all the continuous topologically irreducible bounded representations (T, ) of G on a Banach space by giving a G-orbit in n* (n being the nilradical of g), a topologically irreducible representation of L1(ℝn, ω), for a certain weight ω and a certain n ∈ ℕ, and a topologically simple extension norm. If G is not symmetric, i.e., if the weight ω is exponential, we get a new type of representations which are fundamentally different from the induced representations.

scholarly journals THE 8V CSOS MODEL AND THE sl2 LOOP ALGEBRA SYMMETRY OF THE SIX-VERTEX MODEL AT ROOTS OF UNITY

2002 ◽  
Vol 16 (14n15) ◽  
pp. 1899-1905 ◽  
Author(s):  
TETSUO DEGUCHI
Keyword(s):  
Transfer Matrix ◽  
Lattice Model ◽  
Algebraic Method ◽  
System Size ◽  
Spin Operator ◽  
Total Spin ◽  
Coupling Constants ◽  
Roots Of Unity ◽  
Vertex Model ◽  
Elliptic Quantum

We review an algebraic method for constructing degenerate eigenvectors of the transfer matrix of the eight-vertex Cyclic Solid-on-Solid lattice model (8V CSOS model), where the degeneracy increases exponentially with respect to the system size. We consider the elliptic quantum group Eτ,η(sl2) at the discrete coupling constants: 2N η = m1 + im2τ, where N, m1 and m2 are integers. Then we show that degenerate eigenvectors of the transfer matrix of the six-vertex model at roots of unity in the sector SZ ≡ 0 ( mod N) are derived from those of the 8V CSOS model, through the trigonometric limit. They are associated with the complete N strings. From the result we see that the dimension of a given degenerate eigenspace in the sector SZ ≡ 0 ( mod N) of the six-vertex model at Nth roots of unity is given by [Formula: see text], where [Formula: see text] is the maximal value of the total spin operator SZ in the degenerate eigenspace.

Infinitesimal Generators on the Quantum Group SUq(2)

1998 ◽  
Vol 01 (04) ◽  
pp. 573-598 ◽  
Author(s):  
Michael Schürmann ◽  
Michael Skeide
Keyword(s):  
Irreducible Representation ◽  
Quantum Group ◽  
Lie Group ◽  
Lévy Processes ◽  
Levy Processes ◽  
Infinite Dimensional ◽  
Infinitesimal Generators ◽  
Dimensional Irreducible Representation

Quantum Lévy processes on a quantum group are, like classical Lévy processes with values in a Lie group, classified by their infinitesimal generators. We derive a formula for the infinitesimal generators on the quantum group SU q(2) and decompose them in terms of an infinite-dimensional irreducible representation and of characters. Thus we obtain a quantum Lévy–Khintchine formula.

Local Central Λ(p) Dual Objects

1977 ◽  
Vol 20 (4) ◽  
pp. 515-515
Author(s):  
Willard A. Parker
Keyword(s):  
Irreducible Representation ◽  
Compact Group ◽  
Lie Group ◽  
Compact Lie Group ◽  
Irreducible Characters ◽  
Dual Object

The dual object T of a compact group is called a local central A(p) set if there is a constant K such that ‖X‖P < K ‖X‖1 for all irreducible characters X of G. For each γ∊Γ, Dr is an irreducible representation of G of dimension dγ. Several authors [1, 2, 3, 4] have observed that Γ is a local central Λ(p) set for p<l provided sup{dγ:γ∊Γ}>∞, and some of them [2, 3] conjectured the converse. Cecchini [1] showed that Γ is not a local central Λ(4) set if G is a compact Lie group.

scholarly journals Symmetry-resolved entanglement entropy in Wess-Zumino-Witten models

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
Pasquale Calabrese ◽  
Jérôme Dubail ◽  
Sara Murciano
Keyword(s):  
Irreducible Representation ◽  
Lie Group ◽  
Entanglement Entropy ◽  
Leading Order ◽  
Rényi Entropies ◽  
Leading Term ◽  
Renyi Entropies

Abstract We consider the problem of the decomposition of the Rényi entanglement entropies in theories with a non-abelian symmetry by doing a thorough analysis of Wess-Zumino-Witten (WZW) models. We first consider SU(2)k as a case study and then generalise to an arbitrary non-abelian Lie group. We find that at leading order in the subsystem size L the entanglement is equally distributed among the different sectors labelled by the irreducible representation of the associated algebra. We also identify the leading term that breaks this equipartition: it does not depend on L but only on the dimension of the representation. Moreover, a log log L contribution to the Rényi entropies exhibits a universal prefactor equal to half the dimension of the Lie group.

HARDY’S THEOREM FOR GABOR TRANSFORM

2018 ◽  
Vol 106 (2) ◽  
pp. 143-159
Author(s):  
ASHISH BANSAL ◽  
AJAY KUMAR ◽  
JYOTI SHARMA
Keyword(s):  
Irreducible Representation ◽  
Lie Group ◽  
Discrete Group ◽  
Irreducible Representations ◽  
Gabor Transform ◽  
Nilpotent Lie Group ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Hardy’S Theorem ◽  
Compact Abelian Groups

Hardy’s uncertainty principle for the Gabor transform is proved for locally compact abelian groups having noncompact identity component and groups of the form$\mathbb{R}^{n}\times K$, where$K$is a compact group having irreducible representations of bounded dimension. We also show that Hardy’s theorem fails for a connected nilpotent Lie group$G$which admits a square integrable irreducible representation. Further, a similar conclusion is made for groups of the form$G\times D$, where$D$is a discrete group.

Condition that All Irreducible Representations of a Compact Lie Group, if Restricted to a Subgroup, Contain no Representation More than Once

1972 ◽  
Vol 24 (3) ◽  
pp. 432-438 ◽  
Author(s):  
Fredric E. Goldrich ◽  
Eugene P. Wigner
Keyword(s):  
Irreducible Representation ◽  
Lie Groups ◽  
Finite Groups ◽  
Lie Group ◽  
Unitary Group ◽  
Unit Vector ◽  
Doctoral Dissertation ◽  
Irreducible Representations ◽  
Compact Lie Group ◽  
Compact Lie Groups

One of the results of the theory of the irreducible representations of the unitary group in n dimensions Un is that these representations, if restricted to the subgroup Un-1 leaving a vector (let us say the unit vector e1 along the first coordinate axis) invariant, do not contain any irreducible representation of this Un-1 more than once (see [1, Chapter X and Equation (10.21)]; the irreducible representations of the unitary group were first determined by I. Schur in his doctoral dissertation (Berlin, 1901)). Some time ago, a criterion for this situation was derived for finite groups [3] and the purpose of the present article is to prove the aforementioned result for compact Lie groups, and to apply it to the theory of the representations of Un.

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