By Ludwig Pittner
Quantum teams and quantum algebras in addition to non-commutative differential geometry are vital in arithmetic and regarded to be necessary instruments for version development in statistical and quantum physics. This publication, addressing scientists and postgraduates, incorporates a designated and particularly entire presentation of the algebraic framework. Introductory chapters take care of heritage fabric akin to Lie and Hopf superalgebras, Lie super-bialgebras, or formal energy sequence. nice care used to be taken to offer 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
Obviously, AO is some subalgebra of A. An element a E A is called homogenous of degree p E Z2, if and only if a E AP. Then A is called superalgebra over R. Moreover let A be associative. With the super-commutator, Va = aO + aI, b = bO + bl E A: [a, b]:= L (aPb'1 - (-l)pqb'1a P), p,'1EZ~ the super-commutator algebra A L of A is constructed; A L is some Lie superalgebra over R. Furthermore one finds, that Va,b,cE A: [a,bc] = [a,b]c+ L (-l)pqb'1[aP,c]. p,'1EZ~ An associative superalgebra A over R is called graded-commutative, if and only if Va, b E A : [a, b] = O.
If D is an ideal of L, then x:IDXD is the Killing form of D. The kernel of x: is some ideal of L. 1) Vo E Derc(L) ,Vx, y E L : x:(o(x), y) + x:(x, o(y)) = O. 5) Consider again a finite-dimensional Lie algebra Lover C. An element x E L is called ad-nilpotent, if and only if 3n EN: (ad x)n = 0 in the sense of Endc(L). 1) L is nilpotent, if and only if all the elements of L are ad-nilpotent. 2) If all the elements of L ~ gl(m, C) are nilpotent in the sense of M at( m, C), then L is nilpotent; moreover in this case one can choose some t E Mat(m, C) : det t -# 0, Vx E L : txt- l E n(m, C).
The easy proofs of these statements are similar to those for free Lie algebras over a field. 4) Of course the above defined homomorphism of Lie superalgebras 5. can be lifted to an according homomorphism >. of unital associative superalgebras over K, with the universal enveloping superalgebra V of L, as is shown in the next diagram. 5 - _ . K(5) I_A. L(5) 1X • L ---;----po. envelope T( K(5)) Ji -,-_ _.... 5) Assume char K = 0, and let the set 8 be finite. Denote by 6 : T -----+ T~T the unique homomorphism of unital associative superalgebras over K, such that 'Vs E 8 : 6(s) = s ~ eT + eT ~ s, with the unit eT of T.
Algebraic foundations of non-commutative differential geometry and quantum groups by Ludwig Pittner