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 very important in arithmetic and regarded to be invaluable instruments for version construction in statistical and quantum physics. This e-book, addressing scientists and postgraduates, features a specified and quite whole presentation of the algebraic framework. Introductory chapters take care of history fabric equivalent to Lie and Hopf superalgebras, Lie super-bialgebras, or formal energy sequence. nice care was once taken to offer 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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QJ = LJ1=l ~k, as disjoint union of subsets of T*, which are mutually orthogonal with respect to the Killing form K. \ifk fl, Va: E ~k,{3 E ~l : r a /3 = Here one uses, that \ifk fl: \if k : T: 3 a:k +----4 O:k o. KILkXLI +----4 t a-k E = O. Obviously Tk' with O:k defined as above. The Weyl group W of qJ is isomorphic to the direct product of the Weyl groups Wk of qJk, k = 1, ... 3) L = R - alg span (UaE~(La U L_ a )), inserting a root basis L1 of qJ. 32) Two maximal toral subalgebras T and T' of L are conjugate under an appropriate automorphism of the complex Lie algebra L.

2) Let E be a finite-dimensional vector space over the field R == K, and use the K-linear bijection GE : E +-----+ E** := (E*)*, such that Then Va E L : ¢**(a) 0 GE = GE 0 ¢(a), denoting ¢** := (¢*)*. 18) The graded version of Ado's theorem states, that every finitedimensional Lie superalgebra over a field of characteristic zero admits a faithful finite-dimensional representation. 19) Consider an irreducible representation ¢J of a Lie superalgebra L on a finite-dimensional vector space E over a field K.

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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