Lessons log


Day Start time End time Hours Kind Title Description References
13/03/2019 16:15 17:15 1 0001 Lesson Presentation of the course: themes and objectives, timetable, course site on moodle2, books and references. Introduction to 'Geometric (Clifford) Algebra', David Hestenes' "Vectorial Virus". A bit of history: Hamilton, Grassman, Clifford and the "vector-algebra war". Quick definition of algebra; Clifford algebra and the basic properties of its product, the two cases of orthogonal and collinear vectors, the algebra of R^2 and R^3, corresponding matrix algebra with Pauli matrices. Hestenes' paper, D. Ceravolo thesis
14/03/2019 16:00 17:30 2 0001 Lesson General properties of Clifford algebra: grade, contraction and expansion, pseudoscalar, Hodge duality. The Clifford algebra of the euclidean space R^3, identification of the vector product as the Hodge dual of the corresponding bivector. The Clifford algebra of Minkowski spacetime with metric (1,-1,- 1,-1), elements of the algebra, spatio-temporal split of a generic bivector F and similarity with Riemann Silberstein complex formulation of the electromagnetic tensor as F = E + i B. Generalisation of the gradient, evaluation of the generalised gradient on a generic bivector written in split form that produces differential equations that have the exact form of the Maxwell equations in vacuum. Hestenes' paper, G. D'auria thesis
19/03/2019 17:15 18:00 1 0001 Lesson Introduction and notation: maps and their general properties; linear spaces over a field and its dimensions; linear maps, kernel and images; linear subspaces; rank and nullity of a linear map. Direct product of linear spaces and decomposition of a linear space in the direct sum of linear subspaces; basis; representation of a linear map with a matrix (also in basis independent form). [Porteous 1995] chapter 1
20/03/2019 17:15 18:00 1 0001 Lesson The space of linear maps and its 'natural' linear structure. The dual of a linear space and the dual function induced by a linear function; bilinear functions. Endomorphism space of a linear space; the automorphism group; relations with matrices via isomorphism with the F^n linear space. Ring modules as generalizations of a linear space, the particular case of the double field and its correspondence with a direct sum of subspaces of a linear space; submodules and module maps. [Porteous 1995] chapter 1
27/03/2019 17:15 18:00 1 0001 Lesson Algebra: a linear space with a product: simple examples, full matrix algebras, the case of the double field. Subalgebras, algebra maps and algebra reversing maps, centre and central algebras. The full matrix algebra R(2) and its subalgebras isomorphic to field C and to ring R + R. [Porteous 1995] chapter 2
28/03/2019 17:15 18:00 1 0001 Lesson Algebra ideals and minimal ideals, the case of the minimal left ideals (MLI) of F(n) and their identification with F^n, rank of a MLI. The other side of the medal: MLI as vector spaces on which the algebra elements act; MLI as carriers of the regular representations of an algebra. Algebra homomorphisms and anti-homomorphisms, characterization by restriction of the map to a basis of the algebra and to a set of generators. Automorphisms and involutions, automorphisms of R and C (algebra and field). [Porteous 1995] chapter 2
01/04/2019 18:15 19:00 1 0001 Lesson Still on automorphisms of fields and double fields, similar automorphisms. Idempotents as markers of the automorphisms, reducible and irreducible automorphisms; the cases of a double field. Quadratic forms defined by means of a symmetric scalar products on a real linear space; typical examples of scalar products, null (isotropic) subspaces, positive and negative definite spaces, neutral spaces, orthogonal vectors. [Porteous 1995] chapter 2 and 4
03/04/2019 17:15 18:00 1 0001 Lesson Orthogonal and invertible vectors and basic properties. Dual of a linear space and correlations as maps from a linear space to its dual, relations of correlations with scalar products. Definition of real correlated linear space vs orthogonal space; non degenerate spaces, all R^{p,q} are non degenerate. Orthogonal maps and definition with the dual map. [Porteous 1995] chapter 4
09/04/2019 17:30 18:15 1 0001 Lesson Always on quadratic or correlated spaces: orthogonal maps, conditions and properties, orthogonal iso and automorphisms, orthogonal groups O() and SO(). Definition of the adjoint f* of a map f:X->Y for a non degenerate space X. Properties of the adjoint of a linear map, the adjoint of orthogonal maps. The map f->f* is an anti-automorphism of the algebra End X: a first step towards the connection between transformations of a linear space and automorphisms of its Clifford algebra. Differences between the adjoint and the dual of a linear map. Coincidence of the adjoint with the transpose for map between Euclidean spaces. [Porteous 1995] chapter 4
11/04/2019 17:30 18:15 1 0001 Lesson Always on the adjoint map, Euclidean spaces, the case R^2- >R^2; generalization to the case of real spaces with arbitrary signature. Orthogonal annihilator of a subspace, orthogonal decomposition of a quadratic subspace, totally null planes and their maximal dimensions. Basis theorems; reflections with respect to a subspace. [Porteous 1995] chapter 4 and 5
16/04/2019 17:30 18:15 1 0001 Lesson Always on real linear quadratic spaces: rotations, anti-rotations and automorphisms. Reflections and the particular case of hyperplane reflections (antirotations); reflections are involutions and their orthogonality; mapping any two non null vectors by one or two reflections. Cartan Dieudonnè theorem and its proof, applications to the standard Euclidean plane and space. [Porteous 1995] chapter 5
30/04/2019 17:30 18:15 1 0001 Lesson Always on Cartan Dieudonnè theorem and its applications, unification of views between "continuous" and "discrete" transformations, e.g. C, P and T. Basis theorems and space signatures. Hints on Witt decomposition of a linear space, on neutral spaces and on totally null subspaces with examples in Minkowski space. Remind of some basic properties of Euclidean spaces. [Porteous 1995] chapter 5
02/05/2019 18:00 18:45 1 0001 Lesson Remind of the antinvolution in EndX induced by the adjoint of a linear automorphism of linear spaces X. Reflexive inner products and the case of a skew inner product: hints on real symplectic spaces and maps. Emergence of a one to one relation between the adjoint map of a real linear correlated space X and its induced anti-involutions on EndX. Extensions of linear spaces to arbitrary fields (of characteristic different from 2) and in particular to the complex field. [Porteous 1995] chapter 6 and 7
07/05/2019 17:15 18:00 1 0001 Lesson Always on complex linear spaces, semilinear maps, the case of a complex conjugate (c.c.) linear map, definition of its dual map. Semilinear correlations, symmetric and skew inner products referred to the two C involutions; Hermitian spaces, equivalent correlations, adjoint map for a Hermitian space. Relations between adjoints and anti-involutions of the endomorphisms in the case of Hermitian spaces. Hints on the classification of Hermitian spaces and unitary maps. [Porteous 1995] chapter 7
09/05/2019 17:30 18:15 1 0001 Lesson Tensor product of algebras compared to the direct sum of vector spaces, the case of matrices. General properties of the tensor product and applications to linear spaces endomorphisms and in particular to the case of real spaces; tensoring R, C and H by R, C and H. [Porteous 1995] chapter 11
14/05/2019 17:30 18:15 1 0001 Lesson Hints on complexification as a tensor product and possible issues in the non real cases. Recognition of subalgebras in a few simple cases for the determination of the factors of a tensor product. Axiomatic definition of the Clifford (geometric) algebra for a real linear space V and its general properties. [Porteous 1995] chapter 11 and 15
16/05/2019 17:30 18:15 1 0001 Lesson Again on axiomatic definition of the Clifford algebra; examples of Clifford algebras in simple cases: R^{2,0} and R^{1,1} => R(2); R^{0,1} => C; R^{0,2} => H etc. Definition of the Clifford map and of the injections of field and linear space in the algebra, natural emergence of spinor spaces, hints on Wedderburn theorem. Scalar product in the linear space as anticommutator of the corresponding elements of the algebra; inverse of a vector. [Porteous 1995] chapter 15
21/05/2019 17:15 18:00 1 0001 Lesson Again on axiomatic definition of the Clifford algebra; Clifford algebra of the subspace of a linear space. Orthonormal subsets of an algebra, signature of an orthonormal subset. Base of the linear space and orthonormal subsets of the algebra and their role in identifying the Clifford algebra; vector space structure of the Clifford algebra and its dimensions. Universal Clifford algebra and outline of the case of non universality. [Porteous 1995] chapter 15
23/05/2019 17:30 18:15 1 0001 Lesson Constructive proof ot the existence of the universal Clifford algebra for real neutral spaces R^{n,n} of any dimension and, consequently, for real quadratic spaces of any signature. Construction of an orthonormal subset of type (q+1,p) from one of type (p+1,q) and isomorphism of the relative Clifford algebras. Hints on periodicity theorems for Clifford algebras; table of the real Clifford algebras for dimensions up to 7. Hints of spinor spaces and quick description of Clifford algebras of complex spaces. [Porteous 1995] chapter 15
28/05/2019 17:30 18:15 1 0001 Lesson Automorphisms and antiautomorphisms induced in Clifford algebra from orthogonal automorphisms of the vector space. Involutions and anti-involutions generated by involutions 1 and -1 of the vector space. The fundamental automorphisms of Clifford algebras: identity, grade (main) involution, reversion and conjugation; simple examples. Grading of the Clifford algebra induced by the main involution, definition of the even sub algebra, construction of a base for the even subalgebra, even subalgebras isomorphic to Clifford algebras. [Porteous 1995] chapter 15
30/05/2019 17:15 18:00 1 0001 Lesson Definition of the Clifford group in the Clifford algebra; orthogonal automorphisms of the vector space induced by elements of the Clifford group; the particular case of automorphisms induced by non null vectors in the Clifford group (reflections). The Clifford group of a universal Clifford algebra. Surjective group map between the Clifford group and the group of orthogonal automorphism of the vector space O(X) and its kernel. Hints on representations in general and to regular representations of an algebra. [Porteous 1995] chapter 16
04/06/2019 17:30 18:15 1 0001 Lesson Again on representations: contragradient and complex representations of a Clifford algebra and definition of the maps B: S -> S* and C: S -> S(c.c.) intertwining the representations. Proof that all the automorphisms of a Clifford algebra of an even dimensional space are inner: exhibition of the elements \omega and \tau responsible of the main automorphism and of the automorphism that maps every generator to its inverse; general properties of blades and of their inverses, inverses of the generators; hints to the relation with similar properties of the representations. Papers by A.Trautman and M.Budinich on Moodle site
06/06/2019 17:30 18:45 2 0001 Lesson Complex representations of Clifford algebra seen as complexification of a linear space and of its relative Clifford algebra, remind of the Clifford algebras for complex linear spaces. Linear maps between complex (spinor) spaces that intertwine equivalent complex representations induce automorphisms on the regular representations of Clifford algebra and thus on the Clifford algebra itself. Exhibition of the inner elements of the algebra giving the automorphisms and recognition that the linear maps that intertwine equivalent complex representations are nothing else that the representation of these elements. Hints to the study of the properties of these linear maps done by means of the algebra elements that they represent. Hints to the complex representation of real Clifford algebras, e.g. Cl(R^{3,1}), and to its reducibility that give rise to Majorana spinors. Hints to the possibility that, similarly, also complex representations of quaternionic Clifford algebras, e.g. Cl(R^{1,3}), are 'reducible' in the sense that the representations itself is a subalgebra of the complex matrix algebra. Papers by A.Trautman and M.Budinich on Moodle site
Last modified: Thursday, 6 June 2019, 7:21 PM