Algebraic foundations of non-commutative differential by Ludwig Pittner

By Ludwig Pittner

Quantum teams and quantum algebras in addition to non-commutative differential geometry are vital in arithmetic and thought of to be priceless instruments for version construction in statistical and quantum physics. This ebook, addressing scientists and postgraduates, encompasses a exact and relatively whole presentation of the algebraic framework. Introductory chapters take care of heritage fabric similar to Lie and Hopf superalgebras, Lie super-bialgebras, or formal strength sequence. nice care was once taken to give a competent selection of formulae and to unify the notation, making this quantity an invaluable paintings of reference for mathematicians and mathematical physicists.

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15) Let A be a finite-dimensional complex algebra. Then Derc(A) 30 = Od + On implies, that both the diagonalizable part Od and the nilpotent part on E Derc(A). 1) Let the finite-dimensional complex Lie algebra L be semisimple. The isomorphism of complex Lie algebras: Derc(L) = ad(L) +-----+ L can be used to define an abstract Jordan decomposition, which is unique. "If. n =: ad Xn , Xd + Xn = x, [Xd,X n ] = O. If L ~ gl(m, C), then this decomposition coincides with the usual JordanChevalley decomposition of x E L, due to Weyl's theorem.

M. 2) Let {cl, ... , cm} be the Cartesian basis of Rm, m :::: 2. p in Rm . p+ = {Ci; 1 ~ i ~ m} U {Ci ± Cj; 1 ~ i + = {Qk U {Qk + ~k, l : rkl + Ql; 1 ~ k + Ql-l + = 28kl - (1 -

E Mat (2m,C), D m := {x E gl(2m, C); Lmx + x t L m = O}. One finds, that dim D m = 2m 2 - m, tr D m = {O}. 4) The orthogonal Lie algebra B m "1m EN: K m := [~ ~ o 1m L] = o(2m + 1, C) is defined, mEN. E Mat(2m+ 1,C), 0 + xtKm = = 2m 2 + m. B m := {x E gl(2m + 1, C); Kmx Again tr B m = {O}, and dim B m O}. 5) Due to the isomorphisms of complex Lie algebras: and since dim D 1 = 1, one may restrict the classification to the families {Am;m EN}, {Bm;m;::: 2}, {Cm;m;::: 3}, {Dm;m;::: 4}. 4) Let L be a finite-dimensional Lie algebra over the field C.

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