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A Panorama of Algebraic Structures

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A Panorama of Algebraic Structures A panorama of algebraic structures Abdenacer MAKHLOUF University of Haute Alsace 1 CIMPA Casablanca, 1 juin 2021 Abstract The main purpose of this course is to provide an introduction to non- associative algebras and their dualization, together with their applications in various domains of Mathematics and Physics. Moreover we provide some related key structures, working as invariants and playing an important role like representations, cohomology and deformations. The course gives detailed expositions of fundamental concepts along with an introduction to recent developments in the topics using these fun- damental tools. Contents 1 Associative algebras 2 1.1 Examples . 4 1.2 Morphism of algebras . 5 2 Representations and Modules 5 3 Hopf algebras 7 3.1 Coalgebra . 7 3.2 Bialgebra . 7 3.3 Hopf algebra . 8 4 Leibniz algebras 8 5 Lie algebras 9 6 Lie admissible algebras 9 7 Enveloping algebras 12 [email protected] 1ASSOCIATIVE ALGEBRAS 2 8 Poisson structures and quantization 13 8.1 Poisson algebras . 13 8.2 Poisson-Hopf algebras . 15 8.3 Star-product . 16 8.4 Existence of star-product : Kontsevich theorem . 16 9 Pre-Lie algebras 18 9.1 Definition and general properties . 18 9.2 Pre-Lie algebras of vector fields . 19 9.2.1 Flat torsion-free connections . 19 10 Other related algebraic structures 19 10.1 NAP algebras . 19 10.2 Novikov algebras . 20 10.3 Assosymmetric algebras . 20 10.4 Dendriform algebras . 20 10.5 Post-Lie algebras . 21 11 Cohomology 21 11.1 Exact sequences and Complexes . 22 11.2 Homology and Cohomology . 22 11.3 Hochschild Cohomology . 22 11.4 Gerstenhaber algebra . 24 12 Formal power series ring and spaces 25 12.1 Power series properties . 26 12.2 Formal spaces . 27 13 Formal deformations 28 13.1 Definitions . 28 13.2 Deformation equation . 29 13.3 Obstructions . 30 13.4 Equivalent and trivial deformations . 32 13.5 Poisson algebras and Deformations of commutative algebras . 33 13.6 Deformations of unital algebras . 36 13.7 Deformations of Hopf Algebras . 37 1 Associative algebras Definition 1.1 Let K be a field of characteristic 0 and A be a vector space over K. An associative K-algebra structure on A (or simply an algebra) is given by a 1ASSOCIATIVE ALGEBRAS 3 bilinear map A A A satisfying the following conditions ⇥ ! (A1) x, y, z Aµ(µ (x, y) ,z) µ (x, µ (y, z)) = 0. 8 2 − The multiplication µ has a unit. That is, there exists e A such that 2 (A2) x Aµ(x, e)=µ (e, x)=x 8 2 When there is no ambiguity one uses 1 or 1 instead of e. A An algebra is said to be finite dimensional or infinite dimensional according to whether the space A is finite dimensional or infinite dimensional. The dimension of the vector space is called the dimension of the algebra. Remark 1.1 1. In general, let R be a commutative ring. A R-algebra is a unitary R-module on which we define a bilinear map A (Ax,⇥ y)A!x Ay ! · satisfying the associativity and the existence of the unit 1A. 2. Every R-algebra is a unitary ring, the bilinearity of the multiplication is equivalent to the right and left distributivity (xy)a = x(ya)=(xa)y x, y ,a R 8 2A 2 Conversely, if is a unitary ring and a right R-module satisfying the pre- A vious condition then is an R-algebra. A 3. Let be an R-algebra. The following map A R Z( )= x : x y = y x y A ! A { 2a A 1 a· · 8 2 } ! A is a ring homomorphism. Conversely, if is a ring, every homomorphism from R to Z( ) induces on A A an R-module structure which transforms to an R-algebra. A A The previous remark allows to give an alternative definition : An algebra A is a ring and a ring homomorphism A ' : R Z( ) ! A a 1 a. ! A 1ASSOCIATIVE ALGEBRAS 4 4. If the map a 1 a is injective (the R-module is said faithful) then R is ! A identified to a subring of Z( ) (Ker'). Using this identification (xa = ax), A becomes a left R-module. A 5. Using Hopf algebras language and a same notation for the multiplication, an associative algebra A is a triple (A, µ, ⌘) where µ and ⌘ are linear maps: A A A A µ : and ⌘ : K x y⌦ µ!(x y) a ⌘ (a)=! a ⌘ (1) ⌦ ! ⌦ ! · satisfying the associativity condition and the existence of unit. These condi- tions are expressed by the following commuting diagrams µ id ⌘ id id ⌘ A A A ⌦ A V K A ⌦ A A ⌦ A K ⌦ id⌦µ ! ⌦ µ ⌦ != ⌦µ = ⌦ ⌦ ⇠ ⇠ µ # & # . A A A A ? y⌦ ! When there is no ambiguity the multiplication is denoted by a dot " " or just · by concatenation of the elements. 1.1 Examples 1. Cn and Rn 2. The polynomial rings K[x] and K[x1,...,xn]. 3. The matrix algebra Mn(K). The multiplication is a matrix multiplication. It gives an example of a noncommutative algebra. 4. The group algebra. Let G be a group with the elements ↵ . The vector { i}i space KG is generated by the basis e↵i i. The multiplication is defined by { } e↵i e˙↵j = e↵i↵j 5. Tensor product of algebras. Let A and B be two algebras over K We define the tensor product A B as follows : ⌦ the bilinearity implies that for u1,u2 Av1,v2 B, ↵,β K 2 2 2 (↵u + βu ) v = ↵u v + βu v 1 2 ⌦ 1 1 ⌦ 1 2 ⌦ 1 u (↵v + βv )=↵u v + βu v 1 ⌦ 1 2 1 ⌦ 1 1 ⌦ 2 The multiplication on the tensor product A and B is (u1 v1) A B (u2 v2)=u1 A u2 v1 B v2 ⌦ · ⌦ ⌦ · ⌦ · 2REPRESENTATIONS AND MODULES 5 1.2 Morphism of algebras If R is a ring, the class of R-algebra forms a category where the morphisms are simultaneously the ring and module homomorphisms preserving the unit. They are called morphisms of algebras. Let (V,µ,⌘) and (V 0,µ0,⌘0) be two algebras. A linear map f : V V 0 is a ! morphism of algebras if µ0 (f f)=f µ and f ⌘ = ⌘0 ◦ ⌦ ◦ ◦ In particular, (V,µ,⌘) and (V,µ0,⌘0) are isomorphic if there exists a bijective linear map f such that 1 1 µ = f − µ0 (f f) and ⌘ = f − ⌘0 ◦ ◦ ⌦ ◦ 2 Representations and Modules Definition 2.1 A representation of a K-algebra is a homomorphism T of A A into the algebra End(W ) of the linear operators on a K-vector space W . In other words, define a representation T is assign to every element a a linear 2A operator T (a) in such a way that T (a + b)=T (a)+T (b),T(↵a)=↵T (a), T (ab)=T (a)T (b),T(1 )=Id. A For arbitrary a, b , ↵ K and where Id denotes The identity operator. 2A 2 If the vector space W is finite dimensional then its dimension is called the dimension (or degree) of the representation. The image of the representation T forms a subalgebra of End(W ). If T is injective, then this subalgebra is isomor- phic to the algebra . In this case, the representation is said to be faithful. A Theorem 2.1 (Cayley) Every algebra admits a faithful representation. That is every algebra is isomorphic to a subalgebra of the algebra of linear operators. Proof 1 It follows from the axioms of an algebra that, for any a , the map 2A T (a):x xa, x A, is a representation of . If a = b, it follows that the ! 2 A 6 operators T (a) and T (b) are distinct and that T is faithful representation. The representation defined in the proof is called regular representation. In particular, every finite dimensional algebra is isomorphic to a subalgebra of a matrix algebra. In order to view the representation just as operators in W, one needs the concept of module. 2REPRESENTATIONS AND MODULES 6 Definition 2.2 A right module over a K-algebra , or right -module, is a vector A A space M over the field K such that to every pair (m, a),m M, a , there 2 2A corresponds a uniquely determined element ma M satisfying the following 2 conditions (m1 + m2)a = m1a + m2a, m(a1 + a2)=ma1 + ma2, (↵m)a = m(↵a), (↵m)a = ↵(ma), m(ab)=(ma)b, m1 = m. A For arbitrary m1,m2,m M, a1,a2,a and ↵ K 2 2A 2 Remark 2.1 There is a correspondence between representations of an algebra A and left A-modules. In fact, let T (a):x xa, x A be a representation of the algebra . ! 2 A define the product of the element of W by the elements of the algebra by putting wa = wT(a) for any w W and a . On the other hand, if M is right - 2 2A A module, then it follows that for a fixed a , the map T (a):m ma, x A is 2A ! 2 are representation of . A A homomorphism of a right -module M into a right -module N is a linear A A map f : M N for which (ma)f =(mf)a for arbitrary elements m M and ! 2 a . 2A The set of all homomorphisms of a right -module M into a right -module A A N, denoted by Hom(M,N) is a vector space over K with the natural operations. Analogously to the concept of a right module, one can define a left module over the algebra .
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