Decomposition algorithms in Clifford algebras

Transcription

1 Decomposition algorithms in Clifford algebras G. Stacey Staples 1 Department of Mathematics and Statistics Southern Illinois University Edwardsville Combinatorics and Computer Algebra Fort Collins, Joint work with David Wylie

2 Cl Q (V ) Preliminaries Cayley graphs & hypercubes Hyperplanes & Reflections Vector space V over R. n = dim(v ). Nondegenerate quadratic form Q, inducing, Q. 2 n dim l algebra obtained by associative linear extension of xy = x, y Q + x y

3 Cl Q (V ) Preliminaries Cayley graphs & hypercubes Hyperplanes & Reflections Vector space V over R. n = dim(v ). Nondegenerate quadratic form Q, inducing, Q. 2 n dim l algebra obtained by associative linear extension of xy = x, y Q + x y Familiar special cases: C, H.

4 Cayley graphs & hypercubes Hyperplanes & Reflections Example: Clifford algebra Cl p,q 1 Real, associative algebra of dimension 2 n. 2 Generators {e i : 1 i n} along with the unit scalar e = 1 R. 3 Generators satisfy: [e i, e j ] := e i e j + e j e i = 0 for 1 i j n, { e 2 1 if 1 i p, i = 1 ifp + 1 i p + q.

5 Multi-index notation Preliminaries Cayley graphs & hypercubes Hyperplanes & Reflections Let [n] = {1, 2,..., n} and denote arbitrary, canonically ordered subsets of [n] by capital Roman characters. 2 [n] denotes the power set of [n]. Elements indexed by subsets: γ J = j J γ j. Natural binary representation Generators: any orthonormal basis for V.

6 Cayley graphs & hypercubes Hyperplanes & Reflections Hypercube Q 4

7 Modified hypercubes Cayley graphs & hypercubes Hyperplanes & Reflections

8 Example: Cl n,0 (a.k.a. Cl n ) Cayley graphs & hypercubes Hyperplanes & Reflections [n] = {1, 2,..., n} For j [n] and I [n], define µ j (I) = {i I : i > J} counting measure I J = (I J) \ (I J) set symmetric difference Basis blade multiplication: e I e J = ( 1) P j J µ j (I) e I J

9 Cayley graphs & hypercubes Hyperplanes & Reflections A randomly-generated grade-5 element of Cl 8 { } + { } + { } + { } - { } - { } - { } + { } - { } - { } + { } - { } - { } + { } - { } + { } + { } + { } - { } - { } - { } - { } + { } + { } + { } - { } - { } + { } + { } - { } + { } + { } + { } - { } - { } - { } - { } - { } + { } + { } - { } - { } + { } + { } - { } - { } - { } + { } + { } + { } - { } - { } + { } + { } - { } + { } - { } - { } + { } + { } + { } - { } - { } + { } - { } + { } + { } - { } - { } - { } + { } + { } - { } + { } + { } + { } - { } + { } - { } - { } - { } + { } + { } - { } + { } - { } + { } + { } - { } - { } - { } + { } + { } - { } + { } - { } - { } + { } + { } - { } - { } + { } + { } - { } + { } - { } + { } + { } - { } + { } - { } - { } + { } + { } + { } - { } - { } + { } + { } + { }

10 Geometric algebra Preliminaries Cayley graphs & hypercubes Hyperplanes & Reflections Consider the Euclidean signature. Quadratic form is positive definite; generators satisfy e i 2 = 1. Given unit vector u R 2, x uxu represents reflection across hyperplane u. Given unit vectors u, v with uv = α, x vuxuv is 2α rotation in uv-plane.

11 Motivation: the Euclidean case x vuxuv Cayley graphs & hypercubes Hyperplanes & Reflections

12 Automorphisms Preliminaries Cayley graphs & hypercubes Hyperplanes & Reflections Arbitrary element u = u I e I. I 2 [n] Grade involution: û = ( 1) I u I e I. I 2 [n] Reversion: ũ = I ( I 1) ( 1) 2 u I e I. I 2 [n] Clifford Conjugate: u = I ( I +1) ( 1) 2 u I e I. I 2 [n]

13 Cayley graphs & hypercubes Hyperplanes & Reflections The conformal orthogonal group CO Q (V ) Q is a nondegenerate quadratic form. V is an n-dimensional R-vector space with inner product, Q induced by Q. Cl Q (V ) is the Clifford algebra of this space. Conformal orthogonal group CO Q (V ) is the direct product of dilations and Q-orthogonal linear transformations of V.

14 Hyperplane Reflections in V Cayley graphs & hypercubes Hyperplanes & Reflections 1 Product of orthogonal vectors is a blade. 2 Given unit blade u Cl Q (V ), where Q is positive definite. 3 The map x uxû 1 represents a composition of orthogonal hyperplane reflections. 4 Each vertex of the hypercube underlying the Cayley graph corresponds to a hyperplane arrangement.

15 Historical Framework Arbitrary Signatures Examples Complexity of Representations Versor factorization algorithms: Christian Perwass 2. Efficient blade factorization algorithms: Dorst and Fontijne 3, 4. 2 Geometric Algebra with Applications in Engineering, Springer-Verlag, Berlin, L. Dorst, D. Fontijne, Efficient Algorithms for Factorization and Join of Blades, Geometric Algebra Computing, E. Bayro-Corrochano, G. Scheuermann, Eds., Springer, London, 2010, pp D. Fontijne, Efficient Implementation of Geometric Algebra, Ph.D. thesis, University of Amsterdam, 2007.

16 Helmstetter s work Preliminaries Arbitrary Signatures Examples Complexity of Representations Factorization in Clifford algebras of arbitrary signature was considered by Helmstetter 5. Lipschitz monoid: multiplicative monoid generated in Cl Q (V ) over a field k by all scalars in k, all vectors in V, and all 1 + xy where x and y are vectors that span a totally isotropic plane. Elements called Lipschitzian elements. Given a Lipschitzian element a in Cl Q (V ) over k containing at least three scalars. If a is not in the subalgebra generated by a totally isotropic subspace of V, then it is a product of linearly independent vectors of V. 5 J. Helmstetter, Factorization of Lipschitzian elements, Advances in Applied Clifford Algebras, 24 (2014),

17 Arbitrary Signatures Examples Complexity of Representations Definition An invertible element u Cl Q (V ) of grade k is said to be decomposable if there exists a linearly independent collection {w 1,..., w k } of anisotropic vectors in V such that u = w 1 w k and u k is invertible a. In this case, u is referred to as a decomposable k-element. a Requiring invertibility of the top form makes the decomposable elements of Cl Q (V ) a proper subset of the Lipschitz group of Cl Q (V ).

18 Motivation: the Euclidean case Arbitrary Signatures Examples Complexity of Representations Consider b = 4 + 8e {1,2} + 6e {1,3} 6e {2,3} Cl 3. The action of x bxb is the composition of a plane rotation and dilation by factor b b = 152 in R 3. Letting p = e 1 serve as a probing vector, we compute p = bp b 1 and obtain p = 6 19 e e e 3. Letting b 1 = (p p )/ p p, we obtain the normalized projection b 1 of p into the plane of rotation. In particular, b 1 = 5 e e e 3.

19 Arbitrary Signatures Examples Complexity of Representations

20 Arbitrary Signatures Examples Complexity of Representations Computing u = bb 1 b 1, we obtain u = 275e e e Computing the unit vector b 2, which lies halfway between b 1 and its image, we obtain b 2 = (b 1 + u)/ b 1 + u = e e e 3, The rotation induced by b now corresponds to the composition of two reflections across the orthogonal complements of b 1 and b 2, respectively. Note that b 2 is the normalization of w in the previous figure. The factorization of b is then given by b = 152 b 2 b 1 = 4 + 8e {1,2} + 6e {1,3} 6e {2,3}.

21 Arbitrary Signatures Examples Complexity of Representations Extending to non-euclidean signatures Negative definite signatures. Indefinite signatures.

22 Arbitrary Signatures Examples Complexity of Representations Extending to non-euclidean signatures Negative definite signatures. No problem! Indefinite signatures.

23 Arbitrary Signatures Examples Complexity of Representations Extending to non-euclidean signatures Negative definite signatures. No problem! Indefinite signatures. Be careful!

24 Existence Theorem Preliminaries Arbitrary Signatures Examples Complexity of Representations Given a decomposable k-element u = w 1 w k Cl Q (V ), let n = dim V and define ϕ u O Q (V ) by ϕ u (v) = uvû 1. Then ϕ u has an eigenspace E of dimension n k with corresponding eigenvalue 1.

25 CliffordDecomp Algorithm Arbitrary Signatures Examples Complexity of Representations Input: b, a decomposable k-element. Output: {b k,..., b 1 } such that b = b k b 1. l 1 u b/ b while u > 1 do Let x V such that x u 0 and x 2 0 x uxû 1 if (x x ) 2 0 then b u (x x )/ x x if u b u 1 is decomposable then u u b u 1 end end end return {b k,..., b 2, b u}

26 Arbitrary Signatures Examples Complexity of Representations A randomly-generated grade-5 element of Cl 8 { } + { } + { } + { } - { } - { } - { } + { } - { } - { } + { } - { } - { } + { } - { } + { } + { } + { } - { } - { } - { } - { } + { } + { } + { } - { } - { } + { } + { } - { } + { } + { } + { } - { } - { } - { } - { } - { } + { } + { } - { } - { } + { } + { } - { } - { } - { } + { } + { } + { } - { } - { } + { } + { } - { } + { } - { } - { } + { } + { } + { } - { } - { } + { } - { } + { } + { } - { } - { } - { } + { } + { } - { } + { } + { } + { } - { } + { } - { } - { } - { } + { } + { } - { } + { } - { } + { } + { } - { } - { } - { } + { } + { } - { } + { } - { } - { } + { } + { } - { } - { } + { } + { } - { } + { } - { } + { } + { } - { } + { } - { } - { } + { } + { } + { } - { } - { } + { } + { } + { }

27 Arbitrary Signatures Examples Complexity of Representations Decomposition via CliffordDecomp [ = [ ]] { { { } + { } - { } + { } - { } + { } + { } + { } { } - { } - { } + { } + { } - { } + { } - { } { } + { } + { } + { } - { } + { } + { } - { } { } + { } + { } + { } + { } - { } - { } + { } { } + { } - { } - { } + { } + { } + { } - { }}}

28 Decomposing the 5-blade Arbitrary Signatures Examples Complexity of Representations [ = [ ]] { { { } - { } - { } + { } - { } - { } - { } - { } { } + { } + { } - { } - { } - { } - { } { } + { } + { } - { } + { } + { } { } + { } + { } + { } + { } - { } - { } - { } + { }}}

29 FastBladeFactor Preliminaries Arbitrary Signatures Examples Complexity of Representations Multi indices are well ordered by f I f J 2 i 1 2 j 1. i I j J Define ( ) FirstTerm α I f I I := min {f X :α X 0} α X f X.

30 FastBladeFactor Preliminaries Arbitrary Signatures Examples Complexity of Representations Input: Blade b Cl Q (V ) of grade k as a sum I α If I. Output: Scalar α and set of vectors {b 1,..., b k } such that b = αb k b 1. α M f M FirstTerm(b) Say M = {m 1,..., m k }. for l 1 to k do u f M\{ml } b l bu 1 1 end return {α M, b 1,..., b k }

31 ... via FastBladeFactor Arbitrary Signatures Examples Complexity of Representations [ = [ ]] { } { } + { } { } { } - - { } { } { } { } { } { } { } { } { } { } { } { } { } + - { } { }

32 Arbitrary Signatures Examples Complexity of Representations 4-elements and 4-blades in Cl 6 and Cl 7. Time(s) 1.5 Cl 7 Cl 7Blade 1.0 Cl 6 Cl 6Blade Cl 7FBF Cl 6FBF Trial #

33 Complexity of representations Arbitrary Signatures Examples Complexity of Representations Blades are simple compared to more general elements. FBF is fast. Number of terms in canonical expansion is understood. Can we do better?

34 Lemma (Wylie) Preliminaries Arbitrary Signatures Examples Complexity of Representations If v Cl Q (V ) is a decomposable k-element for k dim V, then c k,j, as defined below, gives an upper bound on the number of blades required to express v j as a sum of blades. This upper bound satisfies the following recurrence: ( 1) k j +1 2 if j = 0 or 1 1 if j = k c k,j = c k 1,j 1 + c k 1,j+1 if 1 < j < k 0 if j > k

35 Arbitrary Signatures Examples Complexity of Representations k\j T k Table: Values of c k,j

36 Arbitrary Signatures Examples Complexity of Representations Open questions & avenues for further research How to efficiently break up into blades? Improvement by simple change of basis? Decompositions in combinatorially interesting Clifford subalgebras? Near-blade approximations?

37 Blade conjugation 1 u Cl Q (V ) a blade. 2 Φ u (x) := uxû 1 is a Q-orthogonal transformation on V. 3 The operators are self-adjoint w.r.t., Q ; i.e., they are quantum random variables. 4 Characteristic polynomial of Φ u generates Kravchuk polynomials: χ(t) = (t + ( 1) k ) k (t ( 1) k ) n k

38 Induced operators 1 Φ u induces ϕ u on Cl Q (V ). 2 Conjugation operators allow decomposition of blades. Eigenvalues ±1 Basis for each eigenspace provides factorization of corresponding blade. 3 Quantum random variables obtained at every level of induced operators. ϕ (l) is self-adjoint w.r.t. Q-inner product for each l = 1,..., n. 4 Kravchuk polynomials appear in traces at every level. 5 Kravchuk matrices represent blade conjugation operators (in most cases 6 ). 6 G.S. Staples, Kravchuk Polynomials & Induced/Reduced Operators on Clifford Algebras, Complex Analysis and Operator Theory (2014).

39 Idea: Graph-Induced Operators Let A be the adjacency matrix or combinatorial Laplacian of a graph. View A as a linear operator on V. A naturally induces an operator A on the Clifford algebra Cl Q (V ) according to action (multiplication, conjugation, etc.) on Cl Q (V ).

40 Idea: Graph-Induced Operators Particular subalgebras: fermions and zeons. Induced operators reveal information about the graph. Enumeration of Hamiltonian cycles and spanning trees 7. 7 G.S. Staples. Graph-induced operators: Hamiltonian cycle enumeration via fermion-zeon convolution. Preprint.

41 Operator Calculus (OC) 1 Lowering operator Λ differentiation annihilation deletion 2 Raising operator Ξ integration creation addition/insertion

42 OC & Clifford multiplication 1 Left lowering Λ x : u x u 2 Right lowering ˇΛ x : u u x 3 Left raising Ξ x : u x u 4 Right raising ˇΞ x : u u x 5 Clifford product satisfies xu = Λ x u + Ξ x u ux = ˇΛ x u + ˇΞ x u

43 Raising & Lowering

44 OC on Graphs - Schott & Staples Walks on hypercubes Walks on Clifford algebras (homogeneous and dynamic) Graph enumeration problems Routing in networks More 8 8 R. Schott, G.S. Staples, Operator Calculus on Graphs (Theory and Applications in Computer Science), Imperial College Press, London, 2012.

45 To be continued... THANKS FOR YOUR ATTENTION!

46 Selected readings G.S. Staples, D. Wylie. Clifford algebra decompositions of conformal orthogonal group elements. Preprint, C. Cassiday, G. S. Staples. On representations of semigroups having hypercube-like Cayley graphs. Clifford Analysis, Clifford Algebras and Their Applications, 4 (2015), G.S. Staples, Kravchuk polynomials & induced/reduced operators on Clifford algebras, Complex Analysis and Operator Theory (2014), dx.doi.org/ /s z. G. Harris, G.S. Staples. Spinorial formulations of graph problems, Advances in Applied Clifford Algebras, 22 (2012),

47 More on Clifford algebras, operator calculus, and graph theory R. Schott, G.S. Staples, Operator Calculus on Graphs (Theory and Applications in Computer Science), Imperial College Press, London, 2012.