Recursive calculation of nonprimitive coupling and recoupling brakets

2003 ◽  
Vol 81 (8) ◽  
pp. 1051-1066 ◽  
Author(s):  
P H Butler ◽  
W P Joyce ◽  
L F McAven ◽  
B G Searle
Keyword(s):  
Category Theory ◽  
Group Representation ◽  
Rotation Group ◽  
Coupling Coefficients ◽  
Unified Approach ◽  
Vector Coupling ◽  
Recursion Scheme ◽  
Recursive Calculation ◽  
The University ◽  
Recoupling Coefficients

The concept of coupling coefficients of angular momentum for the rotation group chain [Formula: see text] can be extended to representations of any group-chain factorisation by defining the generalised notion of a braket. We give a unified approach to recoupling coefficients (rccs), vector-coupling coefficients (vccs), and 3j phases for all group-chain transformations. Category theory is the appropriate tool for studying the representations of groups and algebras. The explicit use of category theory leads to a new recursion scheme for the calculation of brakets. We derive specialisations for calculating nonprimitive rccs and nonprimitive vccs from primitive rccs, primitive vccs, and 3j phases. This new recursion scheme forms the algorithmic core of Racah v4. Racah v4 is a software package developed at the University of Canterbury to calculate group representation coefficients (brakets). PACS Nos.: 02.20.Mp, 03.65.Fd, 31.15.–p, 31.15.Hz, 02.10.Ws

The Racah–Wigner category

2002 ◽  
Vol 80 (6) ◽  
pp. 613-632 ◽  
Author(s):  
W P Joyce ◽  
P H Butler ◽  
H J Ross
Keyword(s):  
Category Theory ◽  
Irreducible Representations ◽  
Coupling Coefficients ◽  
Sum Rule ◽  
Chain Decomposition ◽  
Direct Sum ◽  
Direct Sum Decomposition ◽  
Recoupling Coefficients ◽  
Diagram Techniques

The Racah–Wigner calculus is formulated using category theory. The notion of recoupling requires a consistent choice of isomorphisms corresponding to regrouping of irreducible representations. This is the essential content of a ring category. Hence the Racah–Wigner calculus is inherently a ring category. Category theory places the emphasis on maps between representations. The diagrammatic approach of category theory lays bare underlying relationships in the Racah–Wigner calculus. Direct sum decomposition, coupling coefficients, recoupling coefficients, and group chain decomposition are simplified and clarified when represented by diagrams. The diagram techniques of category theory are unlike diagram techniques used for group calculations. The Biedenharn–Elliott sum rule, Racah backcoupling, Racah factorisation lemma, and the Wigner–Eckart theorem are derived from properties of this category. PACS Nos.: 02.10Ws, 02.20Qs, 02.20Fh, 02.20Df, 03.65Fd, 31.15Hz

scholarly journals OnSU 3 vector coupling coefficients

1966 ◽  
Vol 41 (4) ◽  
pp. 639-639
Author(s):  
G. Ponzano
Keyword(s):  
Coupling Coefficients ◽  
Vector Coupling

scholarly journals On the uniqueness of cellular injectives

2018 ◽  
Vol 167 (3) ◽  
pp. 489-504 ◽  
Author(s):  
J. ROSICKÝ
Keyword(s):  
Banach Spaces ◽  
Category Theory ◽  
Existence And Uniqueness ◽  
Boolean Algebras ◽  
Small Object ◽  
Unified Approach ◽  
Basic Tool ◽  
Saturated Models ◽  
Small Object Argument

AbstractA. Avilés and C. Brech proved an intriguing result about the existence and uniqueness of certain injective Boolean algebras or Banach spaces. Their result refines the standard existence and uniqueness of saturated models. They express a wish to obtain a unified approach in the context of category theory. We provide this in the framework of weak factorisation systems. Our basic tool is the fat small object argument.

Unified Approach to Physics at the University Level

1958 ◽  
Vol 26 (3) ◽  
pp. 179-182
Author(s):  
J. R. Shepanski ◽  
H. F. Pollard ◽  
L. G. Parry
Keyword(s):  
Unified Approach ◽  
University Level ◽  
The University
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