The special linear Lie algebra sl_2

Depends on

• Lie algebras over a field
• Lie subalgebras, ideals, and center
• Representations of Lie algebras

Used by

• A nondominant integral highest-weight module can be infinite-dimensional Counterexample
• An intersection of two primitive ideals need not be primitive Counterexample
• Omitting the exterior root-weight shifts gives the wrong dot weight Counterexample
• SL₂ and PGL₂ have the same Lie algebra but differ globally Counterexample
• The central character does not determine the primitive ideal Counterexample
• The full weight lattice need not integrate through a central quotient Counterexample
• Two nonconjugate real cartan subalgebras Counterexample
• Whitehead vanishing does not apply to the nilpotent radical Counterexample
• Iwasawa and minimal-parabolic data for SL2(R) Definition
• Smooth and K-finite vectors for SL2(R), and the (g,K)-module Definition
• A point-supported SL2 module gives the shifted Verma module Example
• All irreducible finite-dimensional sl2 modules Example
• Borel-Weil-Bott on the projective line for SL2 Example
• Cartan subalgebra and roots of sl₂ Example
• Clebsch–Gordan decomposition for sl2 Example
• Compact and split cartan subalgebras of sl two r Example
• Compact and split real forms of sl two c Example
• Diagonal Cartan subalgebra and roots of slₙ Example
• Kostant cohomology for sl2 Example
• Primitive ideals of U(sl2) at a generic central character Example
• Root-space brackets for matrix units Example
• The annihilator of the trivial sl(2)-module Example
• The critical-level sl2 affine vertex algebra and its current OPE Example
• The one-strand degenerate action on the trivial module has scalar dot Example
• The regular integral sl2 block Example
• The root sl₂ triple inside slₙ Example
• The SL2 flag charts realize the stated rho shift Example
• The two projectives in the principal sl2 block Example
• The Weyl reflection in sl₂ Example
• Verma modules for sl2 Example
• Weyl character and dimension formulas for sl2 Example
• A plain dynkin diagram classifies real forms False statement
• A tensor-product top weight does not determine all constituents False statement
• All cartan subalgebras of a real semisimple lie algebra are conjugate False statement
• All integer multiples of a root are roots False statement
• All real forms of a complex semisimple lie algebra are isomorphic False statement
• Dominance depends on a positive system False statement
• Every element of a complex semisimple Lie algebra is semisimple False statement
• Finite-dimensionality requires dominance integrality False statement
• Global cartan and iwasawa decompositions hold for every nonlinear cover without modified k False statement

…and 14 more results.

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Pavel Etingof, MIT 18.745 Lie Groups and Lie Algebras I, Lectures 19–24 (standard reference, not scraped)