Revision #506 → #1083 · back to history
addedClifford algebra24d3a564d062
addedNotation Cl_{p,q}(R)f3bfe2a74edd
addedConstruction as tensor algebra quotient30de7bd8b719
addedDistinguished subspace4a4853565ea6
addedFundamental identity via bilinear form (2 invertible)aec91965cb52
addedCharacteristic 2 exceptional casef2318a69c38a
addedQ=0 gives exterior algebrab370aa52218f
addedCanonical linear isomorphism with exterior algebra502830466651
addedFiltered algebra with exterior associated gradedf77f64995527
addedClifford algebra by universal property4e8e857fe29a
addedReplacing Q by a bilinear form1f8115076bda
addedEquivalent requirement in characteristic not 2b9c7c7633321
addedExistence and construction of Clifford algebrae28fadc1c3fd
addedUniqueness up to unique isomorphism5f419ffc9d9f
addedFunctoriality of Cl9da8e55b9bc9
addedExistence of orthogonal basesf87507300235
addedClifford identity for orthogonal basis465e5674f207
addedFree basis of the Clifford algebrad86ad31124e5
addedDimension is 2^n5b7e1588a56d
addedReal/complex Clifford algebras as matrix rings46799c8280fc
addedReal nondegenerate form is standard diagonalc1937a49791b
addedSignature of a quadratic formbb4fbeccbc76
addedStandard basis for R^{p,q}8874f3bc1e0c
addedCl_{0,0}(R) ≅ R7d2ae787d7df
addedCl_{0,1}(R) ≅ C05c4f71b12c7
addedCl_{1,0}(R) ≅ split-complex numbers63bea9d82123
addedCl_{0,2}(R) ≅ H76d3ed41c1ed
addedCl_{2,0}(R) ≅ split-quaternionseb9d00a0c7ef
addedCl_{0,3}(R) ≅ H ⊕ Haae9c567df1b
addedCl_{3,0}(R) Pauli algebra ≅ biquaternionsae2f85d07017
addedComplex nondegenerate form is standard diagonalae4111feffc9
addedUnique complex Clifford algebra per dimensionf189ffe9fce4
addedCl_0(C) ≅ C798f34868661
addedCl_1(C) ≅ C ⊕ C336d17f4ccfd
addedCl_2(C) ≅ M_2(C)0465a286ea67
addedQuaternions as even subalgebra of Cl_{3,0}(R)bb69806bd2db
addedDual quaternions as even subalgebra2f463cca631b
addedDimension 1 Clifford algebra848b5b6d9beb
addedDimension 1, a=0: dual numbers4df0808472e8
addedDimension 1, a a nonzero square8b55a1b91588
addedDimension 1, quadratic field extension1e0f314a8698
addedDimension 2 Clifford algebra2402f03caa7c
addedDimension 2 as quaternion algebra4399e2b7e491
addedDimension 2 special case M_2(K)cd661bae11e1
addedNatural vector-space isomorphism with exterior algebraaae568a05f80
addedAlgebra isomorphism iff Q=02698404fc794
addedIsomorphism independent of orthogonal basisdb180532b546
addedIsomorphism by antisymmetrizing in characteristic 0bbb74e81d1c3
addedAssociated graded is exterior algebra93036c08e230
addedClifford algebras are Z2-gradedb076db61e16e
addedGrade automorphism α2efc44e1e997
addedEigenspace decomposition under α4106697d21d5
addedEven subalgebraa7c649aec699
addedMain involution / grade involution180447e2d256
addedNot Z-graded but Z-filterede0c6524c8f7a
addedEven subalgebra is itself a Clifford algebra7f46e0ec17b3
addedEven subalgebra isomorphism formulaf0fa1df4bd80
addedComplex even subalgebra of Cl_n(C)e0f8640a4f95
addedTranspose / reversal antiautomorphismb47bbae34ef8
addedClifford conjugationb6e85bd5c5f5
addedOperations depend only on degree mod 473408665f6a7
addedExtension of Q to the Clifford algebra70167ca9868f
addedAssociated symmetric bilinear form on Claeca0929949b
addedNondegeneracy criterion0507f578b498
addedTranspose as adjoint of Clifford multiplication0f33e6208d8a
addedCentral simple algebraddf6655e18b3
addedEven dimension: central simpled367526554b3
addedEven dimension: even subalgebra structure4da44af2fc3d
addedOdd dimension: structure of Cl9d9fb0a64678
addedOdd dimension: even subalgebra central simple896bd17964d0
addedTensor product decomposition of Cl(U+V)599c04e0f25c
addedBott periodicity of Morita class2406114aa303
addedTwisted conjugation7fbf7de2fd58
addedLipschitz group9c60c0ac29ba
addedHomomorphism to orthogonal group967d1a912aa0
addedLipschitz group contains reflections23a2fb284165
addedLipschitz maps onto orthogonal group (Cartan–Dieudonné)1fbea884f74c
addedSpinor norm59a6c2b3d47b
addedSpinor norm is a homomorphismd19005c298fe
addedInduced spinor norm on orthogonal groupaa7179523d04
addedSpinor norm of a reflection706eb347e396
addedSpinor norm as connecting homomorphism3c357c2a6918
addedPin and spin groups65be204c5aca
addedSpecial orthogonal group6173e3c2cc19
addedHomomorphism from pin groupbdddd304dbff
addedSpin maps onto SO and is simply connected for dim ≥ 3e4b27ed3db81
addedSpin(n) is a double cover of SO(n)6d1e6c49286b
addedSpin not simply connected for R^{p,q}e9e507db8ffc
addedEven complex Clifford algebra as matrix algebrae78af9cf6f9c
addedSpin representationc5e453902210
addedSplitting into half spin representations4c234ee052be
addedOdd-dimensional complex Clifford algebraf43448781733
addedPin group as products of unit vectorsdf30219e9757
addedSpin group as even products of unit vectors883e205f885f
addedSpin as α-fixed subgroupcf387b02691f
addedIrreducible reps restrict to pin group90ce685d9bdc
addedClifford bundle4a325f3cd91f
addedDirac matrices10f4e362871a
addedCl_{1,3}(R) complexification ≅ M_4(C)c5356c4fa73b
addedDefinition extends to modules over a commutative ring3c7f7003e990
addedGeneralization to higher-degree forms7c5e96077497