Ring homomorphisms
59-706: North American undergraduate mathematics award Not to be confused with De Morgan Medal . [REDACTED] This article relies excessively on references to primary sources . Please improve this article by adding secondary or tertiary sources . Find sources: "Morgan Prize" – news · newspapers · books · scholar · JSTOR ( July 2020 ) ( Learn how and when to remove this message ) The Morgan Prize (full name Frank and Brennie Morgan Prize for Outstanding Research in Mathematics by an Undergraduate Student )
118-503: A binary operation [ ⋅ , ⋅ ] : g × g → g {\displaystyle [\,\cdot \,,\cdot \,]:{\mathfrak {g}}\times {\mathfrak {g}}\to {\mathfrak {g}}} called the Lie bracket, satisfying the following axioms: Given a Lie group, the Jacobi identity for its Lie algebra follows from the associativity of
177-524: A Lie algebra and i {\displaystyle {\mathfrak {i}}} an ideal of g {\displaystyle {\mathfrak {g}}} . If the canonical map g → g / i {\displaystyle {\mathfrak {g}}\to {\mathfrak {g}}/{\mathfrak {i}}} splits (i.e., admits a section g / i → g {\displaystyle {\mathfrak {g}}/{\mathfrak {i}}\to {\mathfrak {g}}} , as
236-683: A Lie algebra over a field means its dimension as a vector space . In physics, a vector space basis of the Lie algebra of a Lie group G may be called a set of generators for G . (They are "infinitesimal generators" for G , so to speak.) In mathematics, a set S of generators for a Lie algebra g {\displaystyle {\mathfrak {g}}} means a subset of g {\displaystyle {\mathfrak {g}}} such that any Lie subalgebra (as defined below) that contains S must be all of g {\displaystyle {\mathfrak {g}}} . Equivalently, g {\displaystyle {\mathfrak {g}}}
295-542: A Lie algebra, which is the tangent space at the identity. (In this case, the Lie bracket measures the failure of commutativity for the Lie group.) Conversely, to any finite-dimensional Lie algebra over the real or complex numbers , there is a corresponding connected Lie group, unique up to covering spaces ( Lie's third theorem ). This correspondence allows one to study the structure and classification of Lie groups in terms of Lie algebras, which are simpler objects of linear algebra. In more detail: for any Lie group,
354-559: A field F determines its Lie algebra of derivations, Der F ( g ) {\displaystyle {\text{Der}}_{F}({\mathfrak {g}})} . That is, a derivation of g {\displaystyle {\mathfrak {g}}} is a linear map D : g → g {\displaystyle D\colon {\mathfrak {g}}\to {\mathfrak {g}}} such that The inner derivation associated to any x ∈ g {\displaystyle x\in {\mathfrak {g}}}
413-826: A field F , a derivation of A over F is a linear map D : A → A {\displaystyle D\colon A\to A} that satisfies the Leibniz rule for all x , y ∈ A {\displaystyle x,y\in A} . (The definition makes sense for a possibly non-associative algebra .) Given two derivations D 1 {\displaystyle D_{1}} and D 2 {\displaystyle D_{2}} , their commutator [ D 1 , D 2 ] := D 1 D 2 − D 2 D 1 {\displaystyle [D_{1},D_{2}]:=D_{1}D_{2}-D_{2}D_{1}}
472-585: A homomorphism of Lie algebras), then g {\displaystyle {\mathfrak {g}}} is said to be a semidirect product of i {\displaystyle {\mathfrak {i}}} and g / i {\displaystyle {\mathfrak {g}}/{\mathfrak {i}}} , g = g / i ⋉ i {\displaystyle {\mathfrak {g}}={\mathfrak {g}}/{\mathfrak {i}}\ltimes {\mathfrak {i}}} . See also semidirect sum of Lie algebras . For an algebra A over
531-408: A result, for any Lie algebra, two elements x , y ∈ g {\displaystyle x,y\in {\mathfrak {g}}} are said to commute if their bracket vanishes: [ x , y ] = 0 {\displaystyle [x,y]=0} . The centralizer subalgebra of a subset S ⊂ g {\displaystyle S\subset {\mathfrak {g}}}
590-435: A vector space V , let g l ( V ) {\displaystyle {\mathfrak {gl}}(V)} denote the Lie algebra consisting of all linear maps from V to itself, with bracket given by [ X , Y ] = X Y − Y X {\displaystyle [X,Y]=XY-YX} . A representation of a Lie algebra g {\displaystyle {\mathfrak {g}}} on V
649-433: Is a Lie algebra homomorphism That is, π {\displaystyle \pi } sends each element of g {\displaystyle {\mathfrak {g}}} to a linear map from V to itself, in such a way that the Lie bracket on g {\displaystyle {\mathfrak {g}}} corresponds to the commutator of linear maps. A representation is said to be faithful if its kernel
SECTION 10
#1733111414358708-576: Is a Lie subalgebra, n g ( S ) {\displaystyle {\mathfrak {n}}_{\mathfrak {g}}(S)} is the largest subalgebra such that S {\displaystyle S} is an ideal of n g ( S ) {\displaystyle {\mathfrak {n}}_{\mathfrak {g}}(S)} . The subspace t n {\displaystyle {\mathfrak {t}}_{n}} of diagonal matrices in g l ( n , F ) {\displaystyle {\mathfrak {gl}}(n,F)}
767-440: Is a linear subspace that satisfies the stronger condition: In the correspondence between Lie groups and Lie algebras, subgroups correspond to Lie subalgebras, and normal subgroups correspond to ideals. A Lie algebra homomorphism is a linear map compatible with the respective Lie brackets: An isomorphism of Lie algebras is a bijective homomorphism. As with normal subgroups in groups, ideals in Lie algebras are precisely
826-525: Is a prize for outstanding contribution to mathematics , awarded by the London Mathematical Society . The Society's most prestigious award, it is given in memory of Augustus De Morgan , who was the first President of the society. It is awarded every three years, usually to a mathematician living and working in the United Kingdom . In 1968, Mary Cartwright became the first woman to receive
885-405: Is again a derivation. This operation makes the space Der k ( A ) {\displaystyle {\text{Der}}_{k}(A)} of all derivations of A over F into a Lie algebra. Informally speaking, the space of derivations of A is the Lie algebra of the automorphism group of A . (This is literally true when the automorphism group is a Lie group, for example when F
944-596: Is an abelian Lie subalgebra. (It is a Cartan subalgebra of g l ( n ) {\displaystyle {\mathfrak {gl}}(n)} , analogous to a maximal torus in the theory of compact Lie groups .) Here t n {\displaystyle {\mathfrak {t}}_{n}} is not an ideal in g l ( n ) {\displaystyle {\mathfrak {gl}}(n)} for n ≥ 2 {\displaystyle n\geq 2} . For example, when n = 2 {\displaystyle n=2} , this follows from
1003-666: Is an annual award given to an undergraduate student in the US, Canada, or Mexico who demonstrates superior mathematics research. The $ 1,200 award, endowed by Mrs. Frank Morgan of Allentown, Pennsylvania , was founded in 1995. The award is made jointly by the American Mathematical Society , the Mathematical Association of America , and the Society for Industrial and Applied Mathematics . The Morgan Prize has been described as
1062-552: Is an ideal in Der F ( g ) {\displaystyle {\text{Der}}_{F}({\mathfrak {g}})} , and the Lie algebra of outer derivations is defined as the quotient Lie algebra, Out F ( g ) = Der F ( g ) / Inn F ( g ) {\displaystyle {\text{Out}}_{F}({\mathfrak {g}})={\text{Der}}_{F}({\mathfrak {g}})/{\text{Inn}}_{F}({\mathfrak {g}})} . (This
1121-533: Is defined by exp ( X ) = I + X + 1 2 ! X 2 + 1 3 ! X 3 + ⋯ {\displaystyle \exp(X)=I+X+{\tfrac {1}{2!}}X^{2}+{\tfrac {1}{3!}}X^{3}+\cdots } , which converges for every matrix X {\displaystyle X} . The same comments apply to complex Lie subgroups of G L ( n , C ) {\displaystyle GL(n,\mathbb {C} )} and
1180-444: Is exactly analogous to the outer automorphism group of a group.) For a semisimple Lie algebra (defined below) over a field of characteristic zero, every derivation is inner. This is related to the theorem that the outer automorphism group of a semisimple Lie group is finite. In contrast, an abelian Lie algebra has many outer derivations. Namely, for a vector space V {\displaystyle V} with Lie bracket zero,
1239-402: Is given by the commutator of matrices, [ X , Y ] = X Y − Y X {\displaystyle [X,Y]=XY-YX} . Given a Lie algebra g ⊂ g l ( n , R ) {\displaystyle {\mathfrak {g}}\subset {\mathfrak {gl}}(n,\mathbb {R} )} , one can recover the Lie group as the subgroup generated by
SECTION 20
#17331114143581298-468: Is not always in t 2 {\displaystyle {\mathfrak {t}}_{2}} ). Every one-dimensional linear subspace of a Lie algebra g {\displaystyle {\mathfrak {g}}} is an abelian Lie subalgebra, but it need not be an ideal. For two Lie algebras g {\displaystyle {\mathfrak {g}}} and g ′ {\displaystyle {\mathfrak {g'}}} ,
1357-642: Is spanned (as a vector space) by all iterated brackets of elements of S . Any vector space V {\displaystyle V} endowed with the identically zero Lie bracket becomes a Lie algebra. Such a Lie algebra is called abelian . Every one-dimensional Lie algebra is abelian, by the alternating property of the Lie bracket. The Lie bracket is not required to be associative , meaning that [ [ x , y ] , z ] {\displaystyle [[x,y],z]} need not be equal to [ x , [ y , z ] ] {\displaystyle [x,[y,z]]} . Nonetheless, much of
1416-709: Is the center z ( g ) {\displaystyle {\mathfrak {z}}({\mathfrak {g}})} . Similarly, for a subspace S , the normalizer subalgebra of S {\displaystyle S} is n g ( S ) = { x ∈ g : [ x , s ] ∈ S for all s ∈ S } {\displaystyle {\mathfrak {n}}_{\mathfrak {g}}(S)=\{x\in {\mathfrak {g}}:[x,s]\in S\ {\text{ for all}}\ s\in S\}} . If S {\displaystyle S}
1475-698: Is the adjoint mapping a d x {\displaystyle \mathrm {ad} _{x}} defined by a d x ( y ) := [ x , y ] {\displaystyle \mathrm {ad} _{x}(y):=[x,y]} . (This is a derivation as a consequence of the Jacobi identity.) That gives a homomorphism of Lie algebras, ad : g → Der F ( g ) {\displaystyle \operatorname {ad} \colon {\mathfrak {g}}\to {\text{Der}}_{F}({\mathfrak {g}})} . The image Inn F ( g ) {\displaystyle {\text{Inn}}_{F}({\mathfrak {g}})}
1534-617: Is the product in the category of Lie algebras. Note that the copies of g {\displaystyle {\mathfrak {g}}} and g ′ {\displaystyle {\mathfrak {g}}'} in g × g ′ {\displaystyle {\mathfrak {g}}\times {\mathfrak {g'}}} commute with each other: [ ( x , 0 ) , ( 0 , x ′ ) ] = 0. {\displaystyle [(x,0),(0,x')]=0.} Let g {\displaystyle {\mathfrak {g}}} be
1593-400: Is the real numbers and A has finite dimension as a vector space.) For this reason, spaces of derivations are a natural way to construct Lie algebras: they are the "infinitesimal automorphisms" of A . Indeed, writing out the condition that (where 1 denotes the identity map on A ) gives exactly the definition of D being a derivation. Example: the Lie algebra of vector fields. Let A be
1652-514: Is the set of elements commuting with S {\displaystyle S} : that is, z g ( S ) = { x ∈ g : [ x , s ] = 0 for all s ∈ S } {\displaystyle {\mathfrak {z}}_{\mathfrak {g}}(S)=\{x\in {\mathfrak {g}}:[x,s]=0\ {\text{ for all }}s\in S\}} . The centralizer of g {\displaystyle {\mathfrak {g}}} itself
1711-489: The product Lie algebra is the vector space g × g ′ {\displaystyle {\mathfrak {g}}\times {\mathfrak {g'}}} consisting of all ordered pairs ( x , x ′ ) , x ∈ g , x ′ ∈ g ′ {\displaystyle (x,x'),\,x\in {\mathfrak {g}},\ x'\in {\mathfrak {g'}}} , with Lie bracket This
1770-530: The Dodecahedral Conjecture , University of Michigan ) Honorable mention: Samit Dasgupta ( Harvard University ) 2000 Winner: Jacob Lurie ( Lie Algebras , Harvard University ) Honorable mention: Wai Ling Yee ( University of Waterloo ) 2001 Winner: Ciprian Manolescu ( Floer Homology , Harvard University ) Honorable mention: Michael Levin ( Massachusetts Institute of Technology ) 2002 Winner: Joshua Greene (Proof of
1829-6203: The Kneser conjecture , Harvey Mudd College ) Honorable mention: None 2003 Winner: Melanie Wood ( Belyi-extending maps and P-orderings , Duke University ) Honorable mention: Karen Yeats ( University of Waterloo ) 2004 Winner: Reid W. Barton ( Packing Densities of Patterns, Massachusetts Institute of Technology ) Honorable mention: Po-Shen Loh ( California Institute of Technology ) 2005 Winner: Jacob Fox ( Ramsey theory and graph theory , Massachusetts Institute of Technology ) Honorable mention: None 2007 Winner: Daniel Kane ( Number Theory , Massachusetts Institute of Technology ) Honorable mention: None 2008 Winner: Nathan Kaplan ( Algebraic number theory , Princeton University ) Honorable mention: None 2009 Winner: Aaron Pixton ( Algebraic topology and number theory , Princeton University ) Honorable mention: Andrei Negut ( Algebraic cobordism theory and dynamical systems , Princeton University ) 2010 Winner: Scott Duke Kominers ( Number theory , computational geometry , and mathematical economics , Harvard University ) Honorable mention: Maria Monks ( Combinatorics and number theory , Massachusetts Institute of Technology ) 2011 Winner: Maria Monks ( Combinatorics and number theory , Massachusetts Institute of Technology ) Honorable mention: Michael Viscardi ( Algebraic geometry , Harvard University ), Yufei Zhao ( Combinatorics and number theory , Massachusetts Institute of Technology ) 2012 Winner: John Pardon (Solving Gromov's problem on distortion of knots , Princeton University ) Honorable mention: Hannah Alpert ( Combinatorics , University of Chicago ), Elina Robeva ( Algebraic geometry , Stanford University ) 2013 Winner: Fan Wei ( Analysis and combinatorics , Massachusetts Institute of Technology ) Honorable mention: Dhruv Ranganathan (Toric Gromov–Witten theory , Harvey Mudd College ), Jonathan Schneider ( Combinatorics , Massachusetts Institute of Technology ) 2014 Winner: Eric Larson ( Algebraic geometry and number theory , Harvard University ) Honorable mention: None 2015 Winner: Levent Alpoge ( Number theory , probability theory , and combinatorics , Harvard University ) Honorable mention: Akhil Mathew ( Algebraic topology , algebraic geometry , and category theory , Harvard University ) 2016 Winner: Amol Aggarwal ( Combinatorics , Massachusetts Institute of Technology ) Honorable mention: Evan O'Dorney ( Number Theory , algebra , and combinatorics , Harvard University ) 2017 Winner: David H. Yang ( Algebraic geometry and geometric representation theory , Massachusetts Institute of Technology ) Honorable mention: Aaron Landesman ( Algebraic geometry , number theory , combinatorics , Harvard University ) 2018 Winner: Ashvin Swaminathan ( Algebraic geometry , number theory , and combinatorics , Harvard University ) Honorable mention: Greg Yang (Homological theory of functions, Harvard University ) 2019 Winner: Ravi Jagadeesan ( Algebraic geometry , mathematical economics , statistical theory , number theory , and combinatorics , Harvard University ) Honorable mention: Evan Chen ( Number theory , Combinatorics , Massachusetts Institute of Technology ), Huy Tuan Pham ( Additive Combinatorics , Stanford University ) 2020 Winner: Nina Zubrilina ( Mathematical analysis and analytic number theory , Stanford University ) Honorable mention: Mehtaab Sawhney ( Combinatorics, Massachusetts Institute of Technology ), Cynthia Stoner ( Combinatorics, Harvard University ), Ashwin Sah ( Combinatorics, Massachusetts Institute of Technology ), Murilo Corato Zanarella ( Princeton University ) 2021 Winner: Ashwin Sah ( Combinatorics , discrete geometry , and probability , Massachusetts Institute of Technology ), Mehtaab Sawhney ( Combinatorics , discrete geometry , and probability , Massachusetts Institute of Technology ) Honorable mention: Noah Kravitz ( Yale University ) 2022 Winner: Travis Dillon ( Number theory , combinatorics , discrete geometry , and symbolic dynamics , Lawrence University ) Honorable mention: Sophie Kriz ( University of Michigan ), Alex Cohen ( Yale University ) 2023 Winner: Letong (Carina) Hong ( Number theory , combinatorics , and probability , Massachusetts Institute of Technology ) Honorable mention: Sophie Kriz ( University of Michigan ), Egor Lappo ( Stanford University ) 2024 Winner: Faye Jakson ( analytic number theory , University of Michigan ) Honorable mention: Rupert Li ( Massachusetts Institute of Technology ), Daniel Zhu ( Massachusetts Institute of Technology ) See also [ edit ] List of mathematics awards LeRoy Apker Award , an award for outstanding undergraduate (experimental) physics References [ edit ] ^ "Churchill student receives prestigious Morgan Prize for Outstanding Research in Mathematics – Churchill College" . www.chu.cam.ac.uk . 17 March 2015. ^ "Prize listing" (PDF) . www.ams.org . Retrieved 2020-07-18 . ^ "Prize listing" (PDF) . www.ams.org . Retrieved 2020-07-18 . ^ "Prize listing" (PDF) . www.ams.org . Retrieved 2020-07-18 . ^ "Prize listing" (PDF) . www.ams.org . Retrieved 2020-07-18 . ^ "Prize listing" (PDF) . www.ams.org . Retrieved 2020-07-18 . ^ "Prize listing" (PDF) . www.ams.org . Retrieved 2020-07-18 . ^ "Prize listing" (PDF) . www.ams.org . Retrieved 2020-07-18 . ^ "Prize listing" (PDF) . www.ams.org . Retrieved 2020-07-18 . ^ "Prize listing" (PDF) . www.ams.org . Retrieved 2020-07-18 . ^ Pardon, John (2011). "On
Morgan Prize - Misplaced Pages Continue
1888-501: The Lie bracket , an alternating bilinear map g × g → g {\displaystyle {\mathfrak {g}}\times {\mathfrak {g}}\rightarrow {\mathfrak {g}}} , that satisfies the Jacobi identity . In other words, a Lie algebra is an algebra over a field for which the multiplication operation (called the Lie bracket) is alternating and satisfies
1947-587: The cross product [ x , y ] = x × y . {\displaystyle [x,y]=x\times y.} This is skew-symmetric since x × y = − y × x {\displaystyle x\times y=-y\times x} , and instead of associativity it satisfies the Jacobi identity: This is the Lie algebra of the Lie group of rotations of space , and each vector v ∈ R 3 {\displaystyle v\in \mathbb {R} ^{3}} may be pictured as an infinitesimal rotation around
2006-796: The kernels of homomorphisms. Given a Lie algebra g {\displaystyle {\mathfrak {g}}} and an ideal i {\displaystyle {\mathfrak {i}}} in it, the quotient Lie algebra g / i {\displaystyle {\mathfrak {g}}/{\mathfrak {i}}} is defined, with a surjective homomorphism g → g / i {\displaystyle {\mathfrak {g}}\to {\mathfrak {g}}/{\mathfrak {i}}} of Lie algebras. The first isomorphism theorem holds for Lie algebras: for any homomorphism ϕ : g → h {\displaystyle \phi \colon {\mathfrak {g}}\to {\mathfrak {h}}} of Lie algebras,
2065-403: The matrix exponential of elements of g {\displaystyle {\mathfrak {g}}} . (To be precise, this gives the identity component of G , if G is not connected.) Here the exponential mapping exp : M n ( R ) → M n ( R ) {\displaystyle \exp :M_{n}(\mathbb {R} )\to M_{n}(\mathbb {R} )}
2124-435: The 2020 Morgan Prize | Mathematical Association of America" . www.maa.org . ^ "Nina Zubrilina to Receive the 2020 Morgan Prize & PhD student Cynthia Stoner receives Honorable Mention" . Stanford Mathematics . 2019-11-20 . Retrieved 2023-09-15 . ^ "The Latest" . American Mathematical Society . ^ "The Latest" . American Mathematical Society . ^ "News from
2183-1298: The AMS" . American Mathematical Society . External links [ edit ] Frank and Brennie Morgan Prize at the American Mathematical Society List of Morgan Prize Recipients at the Mathematical Association of America A brief overview of the career paths of the Morgan Prize winners as of 2015. v t e Society for Industrial and Applied Mathematics Awards John von Neumann Prize SIAM Fellowship Germund Dahlquist Prize George David Birkhoff Prize Norbert Wiener Prize in Applied Mathematics Ralph E. Kleinman Prize J. D. Crawford Prize J. H. Wilkinson Prize for Numerical Software W. T. and Idalia Reid Prize Theodore von Kármán Prize George Pólya Prize Peter Henrici Prize SIAM/ACM Prize in Computational Science and Engineering SIAM Prize for Distinguished Service to
2242-737: The American Mathematical Society Awards established in 1995 Student awards Awards of the Mathematical Association of America Awards of the Society for Industrial and Applied Mathematics North American awards 1995 establishments in North America Hidden categories: Articles with short description Short description is different from Wikidata Articles lacking reliable references from July 2020 All articles lacking reliable references De Morgan Medal The De Morgan Medal
2301-672: The Jacobi identity. The Lie bracket of two vectors x {\displaystyle x} and y {\displaystyle y} is denoted [ x , y ] {\displaystyle [x,y]} . A Lie algebra is typically a non-associative algebra . However, every associative algebra gives rise to a Lie algebra, consisting of the same vector space with the commutator Lie bracket, [ x , y ] = x y − y x {\displaystyle [x,y]=xy-yx} . Lie algebras are closely related to Lie groups , which are groups that are also smooth manifolds : every Lie group gives rise to
2360-444: The Lie algebra Out F ( V ) {\displaystyle {\text{Out}}_{F}(V)} can be identified with g l ( V ) {\displaystyle {\mathfrak {gl}}(V)} . A matrix group is a Lie group consisting of invertible matrices, G ⊂ G L ( n , R ) {\displaystyle G\subset \mathrm {GL} (n,\mathbb {R} )} , where
2419-1048: The Profession Morgan Prize Publications SIAM Review SIAM Journal on Applied Mathematics Theory of Probability and Its Applications SIAM Journal on Control and Optimization SIAM Journal on Numerical Analysis SIAM Journal on Mathematical Analysis SIAM Journal on Computing SIAM Journal on Matrix Analysis and Applications SIAM Journal on Scientific Computing SIAM Journal on Discrete Mathematics Educational programs MathWorks Math Modeling Challenge Related societies Association for Computing Machinery International Council for Industrial and Applied Mathematics Institute of Electrical and Electronics Engineers American Mathematical Society American Statistical Association Retrieved from " https://en.wikipedia.org/w/index.php?title=Morgan_Prize&oldid=1242997496 " Categories : Awards of
Morgan Prize - Misplaced Pages Continue
2478-507: The alternating property together imply It is customary to denote a Lie algebra by a lower-case fraktur letter such as g , h , b , n {\displaystyle {\mathfrak {g,h,b,n}}} . If a Lie algebra is associated with a Lie group, then the algebra is denoted by the fraktur version of the group's name: for example, the Lie algebra of SU( n ) is s u ( n ) {\displaystyle {\mathfrak {su}}(n)} . The dimension of
2537-510: The award. Recipients of the De Morgan Medal include the following: Lie algebra Algebraic structures Related structures Algebraic number theory Noncommutative algebraic geometry Free algebra Clifford algebra In mathematics , a Lie algebra (pronounced / l iː / LEE ) is a vector space g {\displaystyle {\mathfrak {g}}} together with an operation called
2596-457: The axis v {\displaystyle v} , with angular speed equal to the magnitude of v {\displaystyle v} . The Lie bracket is a measure of the non-commutativity between two rotations. Since a rotation commutes with itself, one has the alternating property [ x , x ] = x × x = 0 {\displaystyle [x,x]=x\times x=0} . Lie algebras were introduced to study
2655-897: The calculation: [ [ a b c d ] , [ x 0 0 y ] ] = [ a x b y c x d y ] − [ a x b x c y d y ] = [ 0 b ( y − x ) c ( x − y ) 0 ] {\displaystyle {\begin{aligned}\left[{\begin{bmatrix}a&b\\c&d\end{bmatrix}},{\begin{bmatrix}x&0\\0&y\end{bmatrix}}\right]&={\begin{bmatrix}ax&by\\cx&dy\\\end{bmatrix}}-{\begin{bmatrix}ax&bx\\cy&dy\\\end{bmatrix}}\\&={\begin{bmatrix}0&b(y-x)\\c(x-y)&0\end{bmatrix}}\end{aligned}}} (which
2714-457: The complex matrix exponential, exp : M n ( C ) → M n ( C ) {\displaystyle \exp :M_{n}(\mathbb {C} )\to M_{n}(\mathbb {C} )} (defined by the same formula). Here are some matrix Lie groups and their Lie algebras. Some Lie algebras of low dimension are described here. See the classification of low-dimensional real Lie algebras for further examples. Given
2773-445: The concept of infinitesimal transformations by Sophus Lie in the 1870s, and independently discovered by Wilhelm Killing in the 1880s. The name Lie algebra was given by Hermann Weyl in the 1930s; in older texts, the term infinitesimal group was used. A Lie algebra is a vector space g {\displaystyle \,{\mathfrak {g}}} over a field F {\displaystyle F} together with
2832-437: The diffeomorphism group. An action of a Lie group G on a manifold X determines a homomorphism of Lie algebras g → Vect ( X ) {\displaystyle {\mathfrak {g}}\to {\text{Vect}}(X)} . (An example is illustrated below.) A Lie algebra can be viewed as a non-associative algebra, and so each Lie algebra g {\displaystyle {\mathfrak {g}}} over
2891-1255: The distortion of knots on embedded surfaces". Annals of Mathematics . 174 (1): 637–646. arXiv : 1010.1972 . doi : 10.4007/annals.2011.174.1.21 . S2CID 55567836 . ^ "American Mathematical Society to award prizes" . EurekAlert! . ^ "Prize listing" (PDF) . www.ams.org . Retrieved 2020-07-18 . ^ "The Latest" . American Mathematical Society . ^ "The Latest" . American Mathematical Society . ^ "Prize listing" (PDF) . www.ams.org . Retrieved 2020-07-18 . ^ "The Latest" . American Mathematical Society . ^ "Prize listing" (PDF) . www.ams.org . Retrieved 2020-07-18 . ^ "The Latest" . American Mathematical Society . ^ "Prize listing" (PDF) . www.ams.org . Retrieved 2020-07-18 . ^ "The Latest" . American Mathematical Society . ^ Meetings (JMM), Joint Mathematics. "Joint Mathematics Meetings" . Joint Mathematics Meetings . ^ "The Latest" . American Mathematical Society . ^ "Prize listing" (PDF) . www.ams.org . Retrieved 2020-07-18 . ^ "Nina Zubrilina to Receive
2950-535: The group operation of G is matrix multiplication. The corresponding Lie algebra g {\displaystyle {\mathfrak {g}}} is the space of matrices which are tangent vectors to G inside the linear space M n ( R ) {\displaystyle M_{n}(\mathbb {R} )} : this consists of derivatives of smooth curves in G at the identity matrix I {\displaystyle I} : The Lie bracket of g {\displaystyle {\mathfrak {g}}}
3009-483: The group operation. Using bilinearity to expand the Lie bracket [ x + y , x + y ] {\displaystyle [x+y,x+y]} and using the alternating property shows that [ x , y ] + [ y , x ] = 0 {\displaystyle [x,y]+[y,x]=0} for all x , y {\displaystyle x,y} in g {\displaystyle {\mathfrak {g}}} . Thus bilinearity and
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#17331114143583068-817: The highest honor given to an undergraduate in mathematics. Previous winners [ edit ] 1995 Winner: Kannan Soundararajan ( Analytic Number Theory , University of Michigan ) Honorable mention: Kiran Kedlaya ( Harvard University ) 1996 Winner: Manjul Bhargava ( Algebra , Harvard University ) Honorable mention: Lenhard Ng ( Harvard University ) 1997 Winner: Jade Vinson ( Analysis and Geometry , Washington University in St. Louis ) Honorable mention: Vikaas S. Sohal ( Harvard University ) 1998 Winner: Daniel Biss ( Combinatorial Group Theory and Topology , Harvard University ) Honorable mention: Aaron F. Archer ( Harvey Mudd College ) 1999 Winner: Sean McLaughlin (Proof of
3127-430: The identity give g {\displaystyle {\mathfrak {g}}} the structure of a Lie algebra. It is a remarkable fact that these second-order terms (the Lie algebra) completely determine the group structure of G near the identity. They even determine G globally, up to covering spaces. In physics, Lie groups appear as symmetry groups of physical systems, and their Lie algebras (tangent vectors near
3186-437: The identity) may be thought of as infinitesimal symmetry motions. Thus Lie algebras and their representations are used extensively in physics, notably in quantum mechanics and particle physics. An elementary example (not directly coming from an associative algebra) is the 3-dimensional space g = R 3 {\displaystyle {\mathfrak {g}}=\mathbb {R} ^{3}} with Lie bracket defined by
3245-411: The image of ϕ {\displaystyle \phi } is a Lie subalgebra of h {\displaystyle {\mathfrak {h}}} that is isomorphic to g / ker ( ϕ ) {\displaystyle {\mathfrak {g}}/{\text{ker}}(\phi )} . For the Lie algebra of a Lie group, the Lie bracket is a kind of infinitesimal commutator. As
3304-428: The multiplication operation near the identity element 1 is commutative to first order. In other words, every Lie group G is (to first order) approximately a real vector space, namely the tangent space g {\displaystyle {\mathfrak {g}}} to G at the identity. To second order, the group operation may be non-commutative, and the second-order terms describing the non-commutativity of G near
3363-422: The ring C ∞ ( X ) {\displaystyle C^{\infty }(X)} of smooth functions on a smooth manifold X . Then a derivation of A over R {\displaystyle \mathbb {R} } is equivalent to a vector field on X . (A vector field v gives a derivation of the space of smooth functions by differentiating functions in the direction of v .) This makes
3422-399: The space Vect ( X ) {\displaystyle {\text{Vect}}(X)} of vector fields into a Lie algebra (see Lie bracket of vector fields ). Informally speaking, Vect ( X ) {\displaystyle {\text{Vect}}(X)} is the Lie algebra of the diffeomorphism group of X . So the Lie bracket of vector fields describes the non-commutativity of
3481-426: The terminology for associative rings and algebras (and also for groups) has analogs for Lie algebras. A Lie subalgebra is a linear subspace h ⊆ g {\displaystyle {\mathfrak {h}}\subseteq {\mathfrak {g}}} which is closed under the Lie bracket. An ideal i ⊆ g {\displaystyle {\mathfrak {i}}\subseteq {\mathfrak {g}}}
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