Download Algebraic foundations of non-commutative differential by Ludwig Pittner PDF

By Ludwig Pittner

Quantum teams and quantum algebras in addition to non-commutative differential geometry are vital in arithmetic and regarded to be valuable instruments for version development in statistical and quantum physics. This booklet, addressing scientists and postgraduates, includes a special and quite whole presentation of the algebraic framework. Introductory chapters care for heritage fabric akin to Lie and Hopf superalgebras, Lie super-bialgebras, or formal energy sequence. nice care was once taken to give a competent number of formulae and to unify the notation, making this quantity an invaluable paintings of reference for mathematicians and mathematical physicists.

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Additional resources for Algebraic foundations of non-commutative differential geometry and quantum groups

Example text

Yf3p:= Tkl 0 ••• 0 Tk p_1(Yk p) E L_/3 p' The simple finite-dimensional complex Lie algebra L is then C-spanned by the following basis. L = C({hk,X/3j,Y/3j;k = 1, ... ,p}). Of course one could also insert here the inverse automorphisms Tk 1, k= 1, ... ,m. 38) The braid group B n + ll n E N, is defined as the free group over the set {b 1 , ... , bn }, factorized by the relations {bkbl = blbk for Ik -ll ~ 2, bkblbk = blbkbl for Ik - lj = 1; k, l = 1,00 . , n}. ,n}, one obtains the symmetric group Pn +!

M, are defined such that Vl'k, l : Tt- for k 1 : i- l, hi --+ hi - rklhk, Xk --+ -Yk, Yk --+ -Xk; one conveniently defines Tk such that Classification of Semisimple Lie Algebras Their braid-like relations read 'V'{"k 39 :F l: Tk 0 Tl = 1/ 0 Tk, if rkl = 0; Tk 0 Tl 0 Tk = 11 0 Tk 011, if rkl = Ilk = -1; Tk 0 Tl 0 Tk 011 = 11 0 Tk 011 0 Tk, if rkl = -2; Tk 01/ 0 Tk 011 0 Tk 011 = 11 0 Tk 011 0 Tk 011 0 Tk, if rkl = -3. The last possibility only occurs in the exceptional case of G 2 with r12 = -3.

Lie Superalgebras Here 8 =: L1 is called the diagonal homomorphism of the universal enveloping superalgebra V of L. } + D. 7) Every representation ¢ : L ---+ (EndK(E),L ofthe Lie superalgebra L, on the direct sum of vector spaces E = EO EB E over the field K, induces an according representation ¢ : V ---+ EndK(E) of the universal enveloping superalgebra V of L, such that ¢ 0 v = ¢, with the universal mapping v: L ---+ V. 1) The previously defined tensor product ofrepresentations ¢, t/J of L is just the composition t(¢, t/J) = T(¢, t/J) 06, inserting the skew-symmetric tensor product of K-linear mappings.

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