scholarly journals Vanishing theorems for lie algebra cohomology and the cohomology of discrete subgroups of semisimple lie groups

1981 ◽  
Vol 41 (1) ◽  
pp. 78-113 ◽  
Author(s):  
Wilfried Schmid
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Vanishing Theorems ◽  
Semisimple Lie Groups ◽  
Discrete Subgroups ◽  
Lie Algebra Cohomology

On Some q-Analogs of a Theorem of Kostant-Rallis

2000 ◽  
Vol 52 (2) ◽  
pp. 438-448 ◽  
Author(s):  
N. R. Wallach ◽  
J. Willenbring
Keyword(s):  
Conjugacy Class ◽  
Quantum Computation ◽  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Cartan Subgroup ◽  
Semisimple Lie Groups ◽  
Semisimple Lie Group ◽  
Real Group ◽  
Qubit States

AbstractIn the first part of this paper generalizationsof Hesselink’s q-analog of Kostant’smultiplicity formula for the action of a semisimple Lie group on the polynomials on its Lie algebra are given in the context of the Kostant-Rallis theorem. They correspond to the cases of real semisimple Lie groups with one conjugacy class of Cartan subgroup. In the second part of the paper a q-analog of the Kostant-Rallis theorem is given for the real group SL(4, ) (that is SO(4) acting on symmetric 4 × 4 matrices). This example plays two roles. First it contrasts with the examples of the first part. Second it has implications to the study of entanglement of mixed 2 qubit states in quantum computation.

scholarly journals Some complements to the Lazard isomorphism

2010 ◽  
Vol 147 (1) ◽  
pp. 235-262 ◽  
Author(s):  
Annette Huber ◽  
Guido Kings ◽  
Niko Naumann
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Algebraic Groups ◽  
Group Cohomology ◽  
Continuous Group ◽  
Lie Algebra Cohomology ◽  
Seminal Work

AbstractLazard showed in his seminal work (Groupes analytiques p-adiques, Publ. Math. Inst. Hautes Études Sci. 26 (1965), 389–603) that for rational coefficients, continuous group cohomology of p-adic Lie groups is isomorphic to Lie algebra cohomology. We refine this result in two directions: first, we extend Lazard’s isomorphism to integral coefficients under certain conditions; and second, we show that for algebraic groups over finite extensions K/ℚp, his isomorphism can be generalized to K-analytic cochains andK-Lie algebra cohomology.

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