Semisimple group

Humphreys, J. Linear Algebraic Groups (New York Springer, 1975). Much like Borel, going on to classify semisimple groups over algebraically closed fields. 

Scheme 44 Schur s lemma 63 Semi-direct product 19 Semi-invariant element 34 Semisimple group 97 Separable algebra 47 Separable matrix 54 Sheaf 43...

Here, the connection or the gauge potential A assuming values in a(n irreducible) representation R of the compact, semisimple Lie algebra g of the Lie group G is of the form... 

Secondly, the commutator is the Lie product33 of the operators X Xs and Xu this choice of multiplication is particularly appropriate when one realizes that the X XS are the generators of the semisimple compact Lie group U , which is associated with the infinitesimal unitary transformations of the Euclidean vector space R (e.g., the space of the creation operators).34 With the preceding comments, the action of the transformation operator on the creation operators can formally be written in the usual form of the transformation law for covariant vectors,33... 

Up to this moment we have dealt with complex semisimple Lie algebras. But an important role is also played by various real subalgebras contained in complex algebras. One of them is especially remarkable, since the corresponding Lie group is compact. 

Ilyushechkin, N. V. "On the variety of completely integrable left-invariant metrics on a semisimple Lie group. Vestnik Mosk. Cos. Univers. ser. mat. mekh., No. 3 (1979), 35-37. 

Meshcheryakov, M. V. "Integration of the equations for geodesics of left-invariant metrics on semisimple Lie groups by means of special functions. Matem, Sbornik 117 (1982), No. 4, 481-493. 

Wess,J. Realisations of a Compact, Connected, Semisimple Lie Group (Vol. 50)... 

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