Lie theory · Representation

$A_{1}:\quad V_{(2)}$ Adjoint of SU(2)

Highest weight (2) of the specified simply connected compact group: complex dimension 3, real type, with exact dimension, Casimir, and center checks.

Executable coverageHighest weightCompactIrreducible

Definition

The finite-dimensional continuous complex irreducible representation of SU(2) with Dynkin labels (2) in the displayed Bourbaki/Sage node order.

Definition sources

Complex dimension: $3$
Coefficient field: ℂ; characteristic zero

Representation identity

Domain

\mathrm{SU}(2)

Categoryfinite-dimensional-continuous

Coefficient field

\mathbb C

Conventionlie-bourbaki-long-root-2-v1

This record specifies an abstract representation. A matrix basis and normalized invariant tensors are separate choices.

Reference properties

Property	Value	Evidence
Complex dimension	$3$	Sources
Coefficient field	ℂ; characteristic zero	Sources
Quadratic Casimir C₂	$2$	Sources
Quadratic index T(R)	$2$	Sources
Reality type	Real	Sources
Conjugate highest weight	(2)	Sources
Descends to the adjoint global form	Yes	Sources
Order of central kernel	$2$	Sources

Highest weight and conventions

Nodes 1,…,r run along the chain. Fundamental weight 1 is the defining representation of SU(r+1). Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. Complex dimension counts the underlying complex vector space; reality type is an additional invariant. [sources]

Global form and center

The domain is the stated simply connected compact group. The test for descent to its adjoint global form gives yes. This divides out the entire center; a different central quotient must be checked against its particular embedded kernel. [sources]

Use in a calculation

Choose the exact domain and highest weight when entering a multiplet. A dimension label can be ambiguous, especially for conjugate spinors or representations related by triality. A tensor multiplicity or Casimir does not determine a basis of matrices or intertwiners. [sources]

Related objects and maps

$\mathrm{SU}(2)$
Special unitary group
Representation domain

Claims and sources

Each assertion has a stable identifier within this record and a source locator or derivation. Cite the dataset edition when referring to a corrected or historical claim.

Definition

The finite-dimensional continuous complex irreducible representation of SU(2) with Dynkin labels (2) in the displayed Bourbaki/Sage node order.

Lie Groups and Lie Algebras ISections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations.Open source
SageMath: Cartan matricesFinite Cartan matrices and root-system conventions; documentation checked 2026-11-08.Open source
SageMath · Weyl charactersCoroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.Open source
Math 210C. Centers of simply connected semisimple groupsPages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.Open source
The Real Chevalley InvolutionLemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.Open source

representation:su-2-weight-2#definition

Link to claim Suggest a correction

Complex dimension

Lie Groups and Lie Algebras ISections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations.Open source
SageMath: Cartan matricesFinite Cartan matrices and root-system conventions; documentation checked 2026-11-08.Open source
SageMath · Weyl charactersCoroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.Open source
Math 210C. Centers of simply connected semisimple groupsPages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.Open source
The Real Chevalley InvolutionLemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.Open source

Derivation: Exact Weyl dimension product over positive coroots

Premises: group:su-2 · source:workbench-etingof-lie · source:workbench-sage-roots · source:sage · source:conrad-centers · source:workbench-adams-indicator

representation:su-2-weight-2#dimension

Link to claim Suggest a correction

Coefficient field

ℂ; characteristic zero

Lie Groups and Lie Algebras ISections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations.Open source
SageMath: Cartan matricesFinite Cartan matrices and root-system conventions; documentation checked 2026-11-08.Open source
SageMath · Weyl charactersCoroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.Open source
Math 210C. Centers of simply connected semisimple groupsPages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.Open source
The Real Chevalley InvolutionLemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.Open source

representation:su-2-weight-2#field

Link to claim Suggest a correction

Quadratic Casimir C₂

Lie Groups and Lie Algebras ISections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations.Open source
SageMath: Cartan matricesFinite Cartan matrices and root-system conventions; documentation checked 2026-11-08.Open source
SageMath · Weyl charactersCoroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.Open source
Math 210C. Centers of simply connected semisimple groupsPages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.Open source
The Real Chevalley InvolutionLemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.Open source

Derivation: (lambda,lambda+2rho)/2 with squared long-root length two

Premises: group:su-2 · source:workbench-etingof-lie · source:workbench-sage-roots · source:sage · source:conrad-centers · source:workbench-adams-indicator

representation:su-2-weight-2#casimir

Link to claim Suggest a correction

Quadratic index T(R)

Lie Groups and Lie Algebras ISections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations.Open source
SageMath: Cartan matricesFinite Cartan matrices and root-system conventions; documentation checked 2026-11-08.Open source
SageMath · Weyl charactersCoroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.Open source
Math 210C. Centers of simply connected semisimple groupsPages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.Open source
The Real Chevalley InvolutionLemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.Open source

Derivation: T(R) dim(g) = C2(R) dim(R)

Premises: group:su-2 · source:workbench-etingof-lie · source:workbench-sage-roots · source:sage · source:conrad-centers · source:workbench-adams-indicator

representation:su-2-weight-2#index

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Reality type

Real

Lie Groups and Lie Algebras ISections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations.Open source
SageMath: Cartan matricesFinite Cartan matrices and root-system conventions; documentation checked 2026-11-08.Open source
SageMath · Weyl charactersCoroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.Open source
Math 210C. Centers of simply connected semisimple groupsPages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.Open source
The Real Chevalley InvolutionLemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.Open source

representation:su-2-weight-2#reality

Link to claim Suggest a correction

Conjugate highest weight

(2)

Lie Groups and Lie Algebras ISections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations.Open source
SageMath: Cartan matricesFinite Cartan matrices and root-system conventions; documentation checked 2026-11-08.Open source
SageMath · Weyl charactersCoroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.Open source
Math 210C. Centers of simply connected semisimple groupsPages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.Open source
The Real Chevalley InvolutionLemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.Open source

representation:su-2-weight-2#conjugate

Link to claim Suggest a correction

Descends to the adjoint global form

Yes

Lie Groups and Lie Algebras ISections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations.Open source
SageMath: Cartan matricesFinite Cartan matrices and root-system conventions; documentation checked 2026-11-08.Open source
SageMath · Weyl charactersCoroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.Open source
Math 210C. Centers of simply connected semisimple groupsPages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.Open source
The Real Chevalley InvolutionLemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.Open source

Derivation: Highest weight lies in the root lattice exactly when the full center acts trivially

Premises: group:su-2 · source:workbench-etingof-lie · source:workbench-sage-roots · source:sage · source:conrad-centers · source:workbench-adams-indicator

representation:su-2-weight-2#adjoint-descent

Link to claim Suggest a correction

Order of central kernel

Lie Groups and Lie Algebras ISections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations.Open source
SageMath: Cartan matricesFinite Cartan matrices and root-system conventions; documentation checked 2026-11-08.Open source
SageMath · Weyl charactersCoroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.Open source
Math 210C. Centers of simply connected semisimple groupsPages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.Open source
The Real Chevalley InvolutionLemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.Open source

Derivation: Order of the center divided by the order of this central character

Premises: group:su-2 · source:workbench-etingof-lie · source:workbench-sage-roots · source:sage · source:conrad-centers · source:workbench-adams-indicator

representation:su-2-weight-2#central-kernel-order

Link to claim Suggest a correction

Highest weight and conventions

Lie Groups and Lie Algebras ISections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations.Open source
SageMath: Cartan matricesFinite Cartan matrices and root-system conventions; documentation checked 2026-11-08.Open source
SageMath · Weyl charactersCoroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.Open source
Math 210C. Centers of simply connected semisimple groupsPages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.Open source
The Real Chevalley InvolutionLemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.Open source

representation:su-2-weight-2#representations-1

Link to claim Suggest a correction

Global form and center

Lie Groups and Lie Algebras ISections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations.Open source
SageMath: Cartan matricesFinite Cartan matrices and root-system conventions; documentation checked 2026-11-08.Open source
SageMath · Weyl charactersCoroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.Open source
Math 210C. Centers of simply connected semisimple groupsPages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.Open source
The Real Chevalley InvolutionLemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.Open source

representation:su-2-weight-2#global-form-1

Link to claim Suggest a correction

Use in a calculation

Lie Groups and Lie Algebras ISections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations.Open source
SageMath: Cartan matricesFinite Cartan matrices and root-system conventions; documentation checked 2026-11-08.Open source
SageMath · Weyl charactersCoroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.Open source
Math 210C. Centers of simply connected semisimple groupsPages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.Open source
The Real Chevalley InvolutionLemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.Open source

representation:su-2-weight-2#physics-1

Link to claim Suggest a correction

Representation domain

group:su-2

Lie Groups and Lie Algebras ISections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations.Open source
SageMath: Cartan matricesFinite Cartan matrices and root-system conventions; documentation checked 2026-11-08.Open source
SageMath · Weyl charactersCoroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.Open source
Math 210C. Centers of simply connected semisimple groupsPages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.Open source
The Real Chevalley InvolutionLemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.Open source

representation:su-2-weight-2#relation-1

Link to claim Suggest a correction

A1:V(2)A_{1}:\quad V_{(2)}A1​:V(2)​Adjoint of SU(2)

Representation identity

Reference properties

Highest weight and conventions

Global form and center

Use in a calculation

Related objects and maps

Claims and sources

$A_{1}:\quad V_{(2)}$ Adjoint of SU(2)