Lie Groups, Lie Algebras, and Representations

Brian C. Hall, L. G., & Algebras, L. (2015). Lie Groups, Lie Algebras, and Representations: An Elementary Introduction. Springer.

Type

Book

Intellectual base

Review or monograph

Year

2015

DOI

10.1007/978-3-319-13467-3

ISBN

9783319134666

Link

https://doi.org/10.1007/978-3-319-13467-3

Cited by

11 citations across 11 artifacts.

Each citation links to the sentence it supports in the citing article.

Primes

• Jacobi identity
  • This conserved decision sequence is the basis for Prime status.

Domain-specific

• Abelian Lie group
  • … demonstrates that the finite-dimensional smooth group has smooth multiplication and inverse and every pair of elements commutes, with connectedness and real or complex convention stated. to An applied instance preserves the same invariant under a changed scale, notation, jurisdiction, dataset, or implementation..
• Cayley Transform
  • Conversely, suitable orthogonal matrices map back to skew-symmetric matrices by the inverse fractional formula.
• Exponential map (Lie theory)
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Infinitesimal transformation
  • An infinitesimal transformation replaces a finite symmetry motion by its local tangent action.
• Kostant partition function
  • … vector, nonnegative integer coefficient vector, equality to the root sum, order-insensitive count, generating-function product, support cone and role in Kostant’s multiplicity formula are explicit. to An applied instance preserves the invariant under changed notation, scale, dataset, jurisdiction, or implementation..
• Linear group
  • A group \(G\) is linear over a field \(K\) if there is a finite integer \(d\) and an injective homomorphism \(\rho:G\to\operatorname{GL}_d(K)\); equivalently, \(G\) admits a faithful finite-dimensional representation over \(K\).
• Matrix exponential
  • Numerical `expm` output should be assessed by backward error and conditioning rather than by naive truncation alone.
• One-parameter group
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Symplectic Representation
  • In Lie theory, symplectic representations are homomorphisms into classical type-$C$ groups and algebras.

    This sourceModern source for group/Lie-algebra representation relations and matrix classical groups.

• Trivial Representation
  • Tensor singlet. A tensor product can contain a trivial summand even when neither factor is trivial; this summand consists of invariant tensors and is central to coupling representations.

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