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Or, Playing Havoc with the Product Rule) DERIVATIONS An introduction to non associative algebra (or, Playing havoc with the product rule) Series 2|Part 5 The linking algebra of a triple system Colloquium Fullerton College Bernard Russo University of California, Irvine June 28, 2016 Bernard Russo (UCI) DERIVATIONS An introduction to non associative algebra (or, Playing havoc with the product rule) 1 / 30 HISTORY Series 1 • PART I FEBRUARY 8, 2011 ALGEBRAS; DERIVATIONS • PART II JULY 21, 2011 TRIPLE SYSTEMS; DERIVATIONS • PART III FEBRUARY 28, 2012 MODULES; DERIVATIONS • PART IV JULY 26, 2012 COHOMOLOGY (ASSOCIATIVE ALGEBRAS) • PART V OCTOBER 25, 2012 THE SECOND COHOMOLOGY GROUP • PART VI MARCH 7, 2013 COHOMOLOGY (LIE ALGEBRAS) • PART VII JULY 25, 2013 COHOMOLOGY (JORDAN ALGEBRAS) • PART VIII SEPTEMBER 17, 2013 VANISHING THEOREMS IN DIMENSIONS 1 AND 2 (ASSOCIATIVE ALGEBRAS) • PART IX FEBRUARY 18, 2014 VANISHING THEOREMS IN DIMENSIONS 1 AND 2 (JORDAN ALGEBRAS) Bernard Russo (UCI) DERIVATIONS An introduction to non associative algebra (or, Playing havoc with the product rule) 2 / 30 Series 2 • PART I JULY 24, 2014 THE REMARKABLE CONNECTION BETWEEN JORDAN ALGEBRAS AND LIE ALGEBRAS (Two theorems relating different types of derivations) • PART II NOVEMBER 18, 2014 THE REMARKABLE CONNECTION BETWEEN JORDAN ALGEBRAS AND LIE ALGEBRAS (Two theorems embedding triple systems in Lie algebras) • (digression) FEBRUARY 24, 2015 GENETIC ALGEBRAS • PART III JULY 15, 2015 LOCAL DERIVATIONS • (Fall 2015 missed due to the flu) • PART IV FEBRUARY 23, 2016 2-LOCAL DERIVATIONS • PART V (today) JUNE 28, 2016 THE LINKING ALGEBRA OF A TRIPLE SYSTEM "Slides" for all series 1 and series 2 talks available at http://www.math.uci.edu/INSERT a \∼" HERE brusso/undergraduate.html Bernard Russo (UCI) DERIVATIONS An introduction to non associative algebra (or, Playing havoc with the product rule) 3 / 30 The Derivative f (x + h) − f (x) f 0(x) = lim h!0 h DIFFERENTIATION IS A LINEAR PROCESS (f + g)0 = f 0 + g 0 (cf )0 = cf 0 Product Rule THE SET OF DIFFERENTIABLE FUNCTIONS FORMS AN ALGEBRA (fg)0 = fg 0 + f 0g If f and g are differentiable, so are f + g, fg and cf where c is a constant. Bernard Russo (UCI) DERIVATIONS An introduction to non associative algebra (or, Playing havoc with the product rule) 4 / 30 HEROS OF CALCULUS #1 Sir Isaac Newton (1642-1727) Isaac Newton was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, and is considered by many scholars and members of the general public to be one of the most influential people in human history. Bernard Russo (UCI) DERIVATIONS An introduction to non associative algebra (or, Playing havoc with the product rule) 5 / 30 HEROS OF CALCULUS #2 Gottfried Wilhelm Leibniz (1646-1716) Gottfried Wilhelm Leibniz was a German mathematician and philosopher. He developed the infinitesimal calculus independently of Isaac Newton, and Leibniz's mathematical notation has been widely used ever since it was published. Bernard Russo (UCI) DERIVATIONS An introduction to non associative algebra (or, Playing havoc with the product rule) 6 / 30 Bernard Russo (UCI) DERIVATIONS An introduction to non associative algebra (or, Playing havoc with the product rule) 7 / 30 LEIBNIZ RULE (fg)0 = f 0g + fg 0 ******************************************************************* (fgh)0 = f 0gh + fg 0h + fgh0 (Will be very important for us) ******************************************************************* 0 0 0 (f1f2 ··· fn) = f1 f2 ··· fn + ··· + f1f2 ··· fn The chain rule, (f ◦ g)0(x) = f 0(g(x))g 0(x) plays no role in this talk. Neither does the quotient rule gf 0 − fg 0 (f =g)0 = g 2 Bernard Russo (UCI) DERIVATIONS An introduction to non associative algebra (or, Playing havoc with the product rule) 8 / 30 RECTANGULAR MATRICES Mp;q = all p by q real matrices 2 3 a11 a12 ··· a1q 6a21 a22 ··· a2q 7 a = [aij ]p×q = 6 7 (aij 2 R) 4········· 5 ap1 ap2 ··· apq Matrix Multiplication Mp;q × Mq;r ⊂ Mp;r q X ab = [aij ]p×q[bkl ]q×r = [cij ]p×r where cij = aik bkj k=1 Example 2 3 2 3 2 3 a11 a12 a11b11 + a12b21 c11 b11 4a21 a225 = [aij ]3×2[bij ]2×1 = 4a21b11 + a22b215 = 4c215 = [cij ]3×1 b21 a31 a32 a31b11 + a32b21 c31 Bernard Russo (UCI) DERIVATIONS An introduction to non associative algebra (or, Playing havoc with the product rule) 9 / 30 SQUARE MATRICES Mp = Mp;p = all p by p real matrices 2 3 a11 a12 ··· a1p 6a21 a22 ··· a2p7 a = [aij ]p×p = 6 7 (aij 2 R) 4········· 5 ap1 ap2 ··· app Matrix Multiplication Mp × Mp ⊂ Mp p X ab = [aij ]p×p[bkl ]p×p = [cij ]p×p where cij = aik bkj k=1 Examples p = 1; 2 • M1 = f[a11]: a11 2 Rg (Behaves exactly as R) a11 a12 b11 b12 a11b11 + a11b12 a21b11 + a22b21 • = [aij ]2×2[bij ]2×2 = a21 a22 b21 b22 a21b11 + a22b21 a21b12 + a22b22 Bernard Russo (UCI) DERIVATIONS An introduction to non associative algebra (or, Playing havoc with the product rule)10 / 30 Important special cases Mp;q is a linear space [aij ] + [bij ] = [aij + bij ], c[aij ] = [caij ] p M1;p = R (row vectors) p Mp;1 = R (column vectors) Mp is an algebra Pn [aij ] + [bij ] = [aij + bij ], c[aij ] = [caij ], [aij ] × [bij ] = [ k=1 aik bkj ] Mp;q is a triple system You need the transpose of a matrix; details forthcoming. Bernard Russo (UCI) DERIVATIONS An introduction to non associative algebra (or, Playing havoc with the product rule)11 / 30 Review of Algebras|Axiomatic approach AN ALGEBRA IS DEFINED TO BE A SET (ACTUALLY A VECTOR SPACE OVER A FIELD) WITH TWO BINARY OPERATIONS, CALLED ADDITION AND MULTIPLICATION|we are downplaying multiplication by scalars (=numbers=field elements) ADDITION IS DENOTED BY a + b AND IS REQUIRED TO BE COMMUTATIVE a + b = b + a AND ASSOCIATIVE (a + b) + c = a + (b + c) MULTIPLICATION IS DENOTED BY ab AND IS REQUIRED TO BE DISTRIBUTIVE WITH RESPECT TO ADDITION (a + b)c = ac + bc; a(b + c) = ab + ac AN ALGEBRA IS SAID TO BE ASSOCIATIVE (RESP. COMMUTATIVE) IF THE MULTIPLICATION IS ASSOCIATIVE (RESP. COMMUTATIVE) (RECALL THAT ADDITION IS ALWAYS COMMUTATIVE AND ASSOCIATIVE) Bernard Russo (UCI) DERIVATIONS An introduction to non associative algebra (or, Playing havoc with the product rule)12 / 30 Table 1 (FASHIONABLE) ALGEBRAS commutative algebras ab = ba (Real numbers, Complex numbers, Continuous functions) ***************************************** * associative algebras a(bc) = (ab)c * * (Matrix multiplication) * ***************************************** Lie algebras a2 = 0 , (ab)c + (bc)a + (ca)b = 0 (Bracket multiplication on associative algebras: [x; y] = xy − yx) Jordan algebras ab = ba, a(a2b) = a2(ab) (Circle multiplication on associative algebras: x ◦ y = (xy + yx)=2) Bernard Russo (UCI) DERIVATIONS An introduction to non associative algebra (or, Playing havoc with the product rule)13 / 30 DERIVATIONS ON MATRIX ALGEBRAS THE SET Mn(R) of n by n MATRICES IS AN ALGEBRA UNDER MATRIX ADDITION A + B AND MATRIX MULTIPLICATION A × B (or AB) WHICH IS ASSOCIATIVE BUT NOT COMMUTATIVE. For the Record: (square matrices) Pn A + B = [aij ] + [bij ] = [aij + bij ] A × B = [aij ] × [bij ] = [ k=1 aik bkj ] DEFINITION A DERIVATION ON Mn(R) WITH RESPECT TO MATRIX MULTIPLICATION IS A LINEAR PROCESS δ: δ(A + B) = δ(A) + δ(B) WHICH SATISFIES THE PRODUCT RULE δ(A × B) = δ(A) × B + A × δ(B) or δ(AB) = δ(A)B + Aδ(B) Compare with (fg)0 = f 0g + fg 0 Bernard Russo (UCI) DERIVATIONS An introduction to non associative algebra (or, Playing havoc with the product rule)14 / 30 PROPOSITION FIX A MATRIX A in Mn(R) AND DEFINE δA(X ) = A × X − X × A (= AX − XA) THEN δA IS A DERIVATION WITH RESPECT TO MATRIX MULTIPLICATION (WHICH IS CALLED AN INNER DERIVATION) THEOREM 1 Finite dimensional 1920-1940: Noether,Wedderburn,Hochschild,Jacobson,. Infinite dimensional 1950-1970: Kaplansky,Kadison,Sakai,. EVERY DERIVATION ON Mn(R) WITH RESPECT TO MATRIX MULTIPLICATION IS INNER, THAT IS, OF THE FORM δA FOR SOME A IN Mn(R). We gave a proof of this theorem for n = 2 in part 8 of series 1. Bernard Russo (UCI) DERIVATIONS An introduction to non associative algebra (or, Playing havoc with the product rule)15 / 30 Here is that proof Matrix units 1 0 0 1 0 0 0 0 Let E = ; E = ; E = ; E = 11 0 0 12 0 0 21 1 0 22 0 1 LEMMA I E11 + E22 = I t I Eij = Eji I Eij Ekl = δjk Eil THEOREM 1 (restated) Let δ : M2 ! M2 be a derivation: δ is linear and δ(AB) = Aδ(B) + δ(A)B. Then there exists a matrix K such that δ(X ) = XK − KX for X in M2. Bernard Russo (UCI) DERIVATIONS An introduction to non associative algebra (or, Playing havoc with the product rule)16 / 30 PROOF OF THEOREM 1 0 = δ(1) = δ(E11 + E22) = δ(E11) + δ(E22) = δ(E11E11) + δ(E21E12) = E11δ(E11) + δ(E11)E11 + E21δ(E12) + δ(E21)E12 = E11δ(E11) + E21δ(E12) + δ(E11)E11 + δ(E21)E12: Let K = E11δ(E11) + E21δ(E12) = −δ(E11)E11 − δ(E21)E12. Then I KE11 = −δ(E11)E11 ; E11K = E11δ(E11) I KE12 = −δ(E11)E12 ; E12K = E11δ(E12) I KE21 = −δ(E21)E11 ; E21K = E21δ(E11) I KE22 = −δ(E21)E12 ; E22K = E21δ(E12) I E11K − KE11 = E11δ(E11) + δ(E11)E11 = δ(E11E11) = δ(E11) I E12K − KE12 = E11δ(E12) + δ(E11)E12 = δ(E11E12) = δ(E12) I E21K − KE21 = E21δ(E11) + δ(E21)E11 = δ(E21E11) = δ(E21) I E22K − KE22 = E21δ(E12) + δ(E21)E12 = δ(E21E12) = δ(E22) Q.E.D. Bernard Russo (UCI) DERIVATIONS An introduction to non associative algebra (or, Playing havoc with the product rule)17 / 30 Transpose (to the rescue) t t t If a = [aij ] 2 Mp;q then a = [aij ] 2 Mq;p where aij = aji 2 3 a11 a12 ··· a1q 6a21 a22 ··· a2q 7 a = [aij ]p×q = 6 7 4········· 5 ap1 ap2 ··· apq 2 t t t 3 2 3 a11 a12 ··· a1p a11 a21 ··· ap1 t t t t t 6a21 a22 ··· a2p7 6a12 a22 ··· ap27 a = [a ]q×p = 6 7 = 6 7 ij 4········· 5 4········· 5 t t t aq1 aq2 ··· aqp a1q a2q ··· apq (ab)t = bt at Pq Proof: If a = [aij ], b = [bij ] and c = ab = [cij ], so cij = k=1 aik bkj , t t t t Pq t t b a = [bij ][aij ] = [ k=1 bik akj ] and t t Pq Pq t t (ab) = [cij ] = [ k=1 ajk bki ] = [ k=1 akj bik ] Q:E:D: Bernard Russo (UCI) DERIVATIONS An introduction to non associative algebra (or, Playing havoc with the product rule)18 / 30 DERIVATIONS ON RECTANGULAR MATRICES MULTIPLICATION DOES NOT MAKE SENSE ON Mm;n(R) if m 6= n.
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