DERIVATIONS Introduction to non-associative algebra OR Playing havoc with the product rule? PART VI—COHOMOLOGY OF LIE ALGEBRAS BERNARD RUSSO University of California, Irvine FULLERTON COLLEGE DEPARTMENT OF MATHEMATICS MATHEMATICS COLLOQUIUM MARCH 7, 2013
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DERIVATIONS
Introduction to non-associative algebra
OR
Playing havoc with the product rule?
PART VI—COHOMOLOGY OF LIE
ALGEBRAS
BERNARD RUSSO
University of California, Irvine
FULLERTON COLLEGE
DEPARTMENT OF MATHEMATICS
MATHEMATICS COLLOQUIUM
MARCH 7, 2013
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HISTORY OF THESE LECTURES
PART IALGEBRAS
FEBRUARY 8, 2011
PART IITRIPLE SYSTEMS
JULY 21, 2011
PART IIIMODULES AND DERIVATIONS
FEBRUARY 28, 2012
PART IVCOHOMOLOGY OF ASSOCIATIVE
ALGEBRASJULY 26, 2012
PART VMEANING OF THE SECOND
COHOMOLOGY GROUPOCTOBER 25, 2012
PART VICOHOMOLOGY OF LIE ALGEBRAS
MARCH 7, 2013
OUTLINE OF TODAY’S TALK
1. DERIVATIONS ON ALGEBRAS
(FROM FEBRUARY 8, 2011)
2. SET THEORY and GROUPS
(EQUIVALENCE CLASSES and QUOTIENT
GROUPS)
(FROM OCTOBER 25, 2012)
3. FIRST COHOMOLOGY GROUP
(FROM JULY 26, 2012)
4. SECOND COHOMOLOGY GROUP
5. COHOMOLOGY OF LIE ALGEBRAS
PART I: REVIEW OF ALGEBRAS
AXIOMATIC APPROACH
AN ALGEBRA IS DEFINED TO BE A SET
(ACTUALLY A VECTOR SPACE) WITH
TWO BINARY OPERATIONS, CALLED
ADDITION AND MULTIPLICATION
ADDITION IS DENOTED BY
a+ b
AND IS REQUIRED TO BE
COMMUTATIVE AND ASSOCIATIVE
a+ b = b+ a, (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)
Table 2
ALGEBRAS
commutative algebras
ab = ba
associative algebras
a(bc) = (ab)c
Lie algebras
a2 = 0
(ab)c+ (bc)a+ (ca)b = 0
Jordan algebras
ab = ba
a(a2b) = a2(ab)
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DERIVATIONS ON THE SET OF
MATRICES
THE SET Mn(R) of n by n MATRICES IS
AN ALGEBRA UNDER
MATRIX ADDITION
A+B
AND
MATRIX MULTIPLICATION
A×B
WHICH IS ASSOCIATIVE BUT NOT
COMMUTATIVE.
DEFINITION 2
A DERIVATION ON Mn(R) WITH
RESPECT TO MATRIX MULTIPLICATION
IS A LINEAR PROCESS δ WHICH
SATISFIES THE PRODUCT RULE
δ(A×B) = δ(A)×B +A× δ(B)
.
PROPOSITION 2
FIX A MATRIX A in Mn(R) AND DEFINE
δA(X) = A×X −X ×A.
THEN δA IS A DERIVATION WITH
RESPECT TO MATRIX MULTIPLICATION
(WHICH CAN BE NON-ZERO)
THEOREM 2(1942 Hochschild)
EVERY DERIVATION ON Mn(R) WITHRESPECT TO MATRIX MULTIPLICATION
IS OF THE FORM δA FOR SOME A INMn(R).
Gerhard Hochschild (1915–2010)
(Photo 1968)Gerhard Paul Hochschild was an Americanmathematician who worked on Lie groups,algebraic groups, homological algebra and
algebraic number theory.
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THE BRACKET PRODUCT ON THE
SET OF MATRICES
THE BRACKET PRODUCT ON THE SET
Mn(R) OF MATRICES IS DEFINED BY
[X,Y ] = X × Y − Y ×X
THE SET Mn(R) of n by n MATRICES IS
AN ALGEBRA UNDER MATRIX ADDITION
AND BRACKET MULTIPLICATION,
WHICH IS NOT ASSOCIATIVE AND NOT
COMMUTATIVE.
DEFINITION 3
A DERIVATION ON Mn(R) WITH
RESPECT TO BRACKET MULTIPLICATION
IS A LINEAR PROCESS δ WHICH
SATISFIES THE PRODUCT RULE
δ([A,B]) = [δ(A), B] + [A, δ(B)]
.
PROPOSITION 3
FIX A MATRIX A in Mn(R) AND DEFINE
δA(X) = [A,X] = A×X −X ×A.
THEN δA IS A DERIVATION WITH
RESPECT TO BRACKET
MULTIPLICATION
THEOREM 3
(1942 Hochschild, Zassenhaus)EVERY DERIVATION ON Mn(R) WITH
RESPECT TO BRACKETMULTIPLICATION IS OF THE FORM δA
FOR SOME A IN Mn(R).
Hans Zassenhaus (1912–1991)
Hans Julius Zassenhaus was a Germanmathematician, known for work in many parts
of abstract algebra, and as a pioneer ofcomputer algebra.
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THE CIRCLE PRODUCT ON THE SET
OF MATRICES
THE CIRCLE PRODUCT ON THE SET
Mn(R) OF MATRICES IS DEFINED BY
X ◦ Y = (X × Y + Y ×X)/2
THE SET Mn(R) of n by n MATRICES IS
AN ALGEBRA UNDER MATRIX ADDITION
AND CIRCLE MULTIPLICATION, WHICH IS
COMMUTATIVE BUT NOT ASSOCIATIVE.
DEFINITION 4
A DERIVATION ON Mn(R) WITH
RESPECT TO CIRCLE MULTIPLICATION
IS A LINEAR PROCESS δ WHICH
SATISFIES THE PRODUCT RULE
δ(A ◦B) = δ(A) ◦B +A ◦ δ(B)
PROPOSITION 4
FIX A MATRIX A in Mn(R) AND DEFINE
δA(X) = A×X −X ×A.
THEN δA IS A DERIVATION WITH
RESPECT TO CIRCLE MULTIPLICATION
THEOREM 4
(1972-Sinclair)
EVERY DERIVATION ON Mn(R) WITH
RESPECT TO CIRCLE MULTIPLICATION
IS OF THE FORM δA FOR SOME A IN
Mn(R).
REMARK
(1937-Jacobson)
THE ABOVE PROPOSITION AND
THEOREM NEED TO BE MODIFIED FOR
THE SUBALGEBRA (WITH RESPECT TO
CIRCLE MULTIPLICATION) OF
SYMMETRIC MATRICES.
Alan M. Sinclair (retired)
Nathan Jacobson (1910–1999)
Nathan Jacobson was an Americanmathematician who was recognized as one ofthe leading algebraists of his generation, andhe was also famous for writing more than a
dozen standard monographs.
Table 1
Mn(R) (ALGEBRAS)
matrix bracket circleab = a× b [a, b] = ab− ba a ◦ b = ab+ ba
Th. 2 Th.3 Th.4δa(x) δa(x) δa(x)
= = =ax− xa ax− xa ax− xa
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PART 2 OF TODAY’S TALK
A partition of a set X is a disjoint class {Xi}of non-empty subsets of X whose union is X
4. Show that if h : A1 → A2 is a homomor-phism of algebras, then A1/ker h is isomor-phic to h(A1)Hint: Show that the map [x] 7→ h(x) is anisomorphism of A1/ker h onto h(A1)
5. Show that the algebra Mh in Example 2 isassociative.Hint: You use the fact that A is associa-tive AND the fact that, since h ∈ ker T2,h(a, b)c+ h(ab.c) = ah(b, c) + h(a, bc)
6. Show that equivalence of extensions is ac-tually an equivalence relation.Hint:• reflexive: ψ : M →M is the identity map• symmetric: replace ψ : M → M ′ by its
inverse ψ−1 : M ′ →M
• transitive: given ψ : M → M ′ and ψ′ :M ′ →M ′′ let ψ′′ = ψ′ ◦ ψ : M →M ′′
7. Show that in example 2, if h1 and h2 areequivalent bilinear maps, that is, h1−h2 =T1f for some linear map f , then Mh1
andMh2
are equivalent extensions of {0}×A byA. Hint: ψ : Mh1