Let a semisimple complex Lie algebra act faithfully and irreducibly on a space of prime dimension. Complete reducibility for splits into isotypic components. The commuting other ideals preserve these components, so irreducibility leaves just one, of the form . Repeating with the other factors gives , with each an irreducible module for its simple ideal.
Faithfulness makes every factor nontrivial. A nonabelian simple Lie algebra has only the trivial one-dimensional module, so . Their product is the prime , forcing just one simple ideal. Hence is simple. This is a representation-dimension obstruction, independent of the classification of root systems.
Let be a finite-dimensional complex vector space and let be a Lie subalgebra of its traceless endomorphisms acting irreducibly. Its solvable radical is simultaneously triangularizable by Lie's theorem. The resulting finite set of diagonal characters is invariant under conjugation by for , because this conjugation preserves the ideal and gives an equivalent representation. Each character varies continuously with in a finite set, hence is constant; differentiation gives .
A nonzero common -eigenspace exists by Lie's theorem. The displayed vanishing makes it invariant under all of : . Irreducibility therefore makes it all of . Thus acts by scalars, and the zero-trace condition forces those scalars to vanish. The matrix representation is faithful, so . The traceless condition matters: adjoining scalar matrices to an irreducible semisimple algebra gives a reductive algebra with nonzero centre.
A finite-dimensional complex semisimple Lie algebra which is an ideal in a finite-dimensional Lie algebra has the displayed direct complement. The restriction of the Killing form of to equals the Killing form of : the induced adjoint maps of elements of on vanish. This restriction is nondegenerate by the Cartan criterion for semisimplicity. Invariance makes its orthogonal complement an ideal. Nondegeneracy on gives a direct vector-space sum with that complement, and the bracket of the two ideals lies in their zero intersection, so they commute. The ambient algebra need not itself be semisimple.
A Chevalley basis of a complex semisimple Lie algebra consists of suitable coroot elements and one vector in each root space, chosen so all Lie bracket structure constants are integers. In particular when the sum is a root, with where is the last negative-direction root in the root string. For independent roots with no root sum the bracket is zero. The integral lattice makes reduction to other fields possible.
For simple-root generators with , the Serre relations include and for . The off-diagonal Cartan entries are nonpositive integers. These relations restrict which iterated simple-root brackets remain nonzero.
For distinct simple roots, has mixed-sign simple-root coefficients and is not a root. Thus . In the finite-dimensional adjoint representation of the simple-root SU(2) subalgebra, is a lowest-weight vector of weight . Standard SU(2) ladders force with integral and make .
For equal-length roots with a root sum, their angle is . The vectors and have squared length three times a root's squared length, so neither is a root. Thus . Apply the central-commutator exponential identity to and , whose commutator is , to obtain the displayed identity over the complex numbers.
A complex semisimple Lie algebra has an intrinsic Jordan–Chevalley decomposition characterized by , a diagonalizable Adjoint representation operator , and a nilpotent Adjoint representation operator . The Adjoint representation is faithful because the centre is zero. Closure of semisimple matrix Lie algebras are closed under additive Jordan decomposition applied to this faithful representation supplies the parts and their uniqueness. For every finite-dimensional Lie algebra representation , the operators and are the semisimple and nilpotent parts of . Thus the decomposition is independent of a matrix embedding; for a nonfaithful representation it passes to the semisimple quotient by its kernel.
A semisimple Lie algebra over a field of characteristic zero is split if it has a Cartan subalgebra whose adjoint action is simultaneously diagonalizable over that field. Its root-space decomposition then needs no extension of scalars. The sl2R Lie algebra is split, with the real diagonal Cartan subalgebra. The real special unitary Lie algebra is not: its adjoint operators are skew-adjoint for the positive definite inner product obtained by negating its Killing form. Thus a split adjoint operator would have to be zero.
A compact real form of a complex semisimple Lie algebra is a real Lie subalgebra whose complexification of a Lie algebra is and whose Killing form is negative definite. Its simply connected Lie group is compact. The positive internal inner product used in unitary gauge theory is therefore . For the SU(2) Lie algebra is a compact real form, whereas is a different real form with indefinite Killing form.
A Lie algebra is perfect when . Every semisimple Lie algebra over a field of characteristic zero is perfect.
Perfectness is a property of a Lie algebra, defined by equality with its commutator algebra; semisimplicity is a sufficient condition in characteristic zero.
Over an algebraically closed field of characteristic zero, every three-dimensional perfect Lie algebra is isomorphic to the sl2 Lie algebra. Indeed the trace of a commutator gives for all , because each element is a sum of Lie brackets. If every were a nilpotent linear map, the Engel theorem would make the Lie algebra nilpotent, contradicting its constant nonzero lower central series of a Lie algebra. Some therefore has a nonzero eigenvalue . Its other eigenvalues are , since it kills and has zero trace. They are distinct, so its eigenvectors give a basis after scaling to , with and . The Jacobi identity puts in the zero eigenspace . Its coefficient cannot vanish, since otherwise the derived algebra would lie in . Rescaling gives . Thus this basis is an sl2 triple, proving the claimed Lie algebra isomorphism.
Every finite-dimensional representation of a complex semisimple Lie algebra is a direct sum of irreducible representations. Equivalently, every invariant subspace has an invariant complement.
For a finite-dimensional continuous complex representation of a compact group, normalize Haar measure to mass one and average any positive-definite Hermitian inner product:The resulting inner product is positive-definite and invariant. If is an invariant subspace, then is invariant as well, since . Thus . Restriction to a compact real form, followed by integration of a Lie-algebra representation, proves the Weyl complete reducibility theorem.
If a submodule of a Lie algebra representation admits an intertwining map with , then and its kernel is an invariant complement. For a complex semisimple Lie algebra, the Hom representation on maps whose restriction is a scalar identity reduces finding to splitting of a trivial quotient for a semisimple Lie algebra.
In a finite-dimensional short exact sequence of representations of a complex semisimple Lie algebra, the generalized zero eigenspace of the Casimir operator maps onto the trivial quotient. All its irreducible composition factors have zero Casimir eigenvalue, so they are trivial. The action is therefore strictly upper triangular and has solvable image; because a semisimple Lie algebra is a perfect Lie algebra, that image is zero. Every lift in this generalized zero eigenspace is invariant, giving a split short exact sequence.
For a choice of positive roots, the corresponding Borel subalgebra is , the sum of a Cartan subalgebra and all positive root spaces.
An sl2 triple consists of elements of a Lie algebra satisfying the displayed brackets. It defines a homomorphism from by sending its standard generators to . A nonzero homomorphism is injective because is a simple Lie algebra. Root vectors and their coroot give such triples in a semisimple Lie algebra, and their representations supply finite weight strings.
In a finite-dimensional representation of an sl2 triple, start with a nonzero vector satisfying and , and set . The commutator identities give the displayed weight and lowering formulas by induction. If and , then forces , a nonnegative integer. The nonzero terminal vector is essential: choosing a later index after the ladder has already vanished gives no constraint. Applied to the Adjoint representation of a semisimple Lie algebra, this argument proves integrality of Cartan integers.
For and , the representation of is irreducible with highest weight . Its monomial basis supplies weights by adding defining-representation weights, with repeated weights counted separately.
For the defining -dimensional special linear Lie algebra module, with , write . The equivariant contraction is . The map is also equivariant, since is an invariant identity tensor. Direct differentiation gives , so the displayed direct sum follows. Its trace-free part is an irreducible representation of highest weight . Its highest-weight vector is ; the Weyl dimension formula gives dimension , exactly .
// Target: semisimple-lie-algebra.bigb
For the defining three-dimensional special linear Lie algebra representation , contraction maps onto . The kernel contains the highest-weight vector of Dynkin labels . The Weyl dimension formula gives its irreducible module dimension , the entire kernel dimension; the Weyl complete reducibility theorem supplies the displayed splitting.
Let , and let be the dual basis. The equivariant contraction sends to . Its kernel has a weight basis consisting of for , for three distinct indices with , and for . The three families have sizes six, three and six. This is the irreducible fifteen-dimensional highest-weight representation with highest weight .
The formal character identity counts unordered pairs of distinct basis vectors. Substituting and the Weyl character formula gives the displayed multiplicity-free decomposition. Its dimensions sum to . The sl3 interlacing character formula provides an exact weight-enumeration check.
For the special linear Lie algebra , with Dynkin labels indexing irreducible highest-weight representations,The summands have dimensions , summing to . The identity follows from the formal character of a weight module formula and the Weyl character formula.
Over a field of characteristic different from two, any Lie algebra ideal in containing a nonscalar matrix is the whole special linear Lie algebra. Indeed, if for , then extracts a matrix unit. For a nonscalar diagonal matrix, extracts one instead. Its bracket with produces , and extracts the reverse unit. Brackets and spread these units to every row and column; opposite pairs span all trace-zero diagonal matrices. Thus only ideals consisting of scalar matrices can be proper. This proof works even when restrictions of distinct roots of a root system to a Cartan subalgebra coincide in positive characteristic.
The six smallest complex irreducible representations of SU(3) have highest weights and dimensions . Their Young diagram row lengths are . They are realized by scalars, defining vectors, covectors (equivalently antisymmetric tensors of degree two), upper and lower symmetric tensors of degree two, and mixed tensors with zero trace, respectively. The Weyl dimension formula is . For this is at least ten, proving that the list exhausts the six smallest representations.
Split the two dual factors into antisymmetric and symmetric tensors. The invariant Levi-Civita symbol identifies the antisymmetric pair with , giving after multiplying by the remaining triplet. On the symmetric part, contraction maps onto ; the equivariant splitting is . Its trace-free kernel has dimension and highest weight with Dynkin labels . The Weyl dimension formula gives dimension , so this is exactly the irreducible kernel. It is conjugate to the other fifteen-dimensional module with labels ; dimension alone does not distinguish the two.
The tensor product of Lie algebra representations splits into the symmetric and antisymmetric . Then and , as highest-weight character subtraction verifies. The decuplet is fully symmetric under quark permutations, the singlet fully antisymmetric, and the two octets constitute the two-dimensional mixed permutation multiplicity space. Applying the Pauli constraint on three-quark flavour multiplets gives one ground-state spin-one-half octet and one spin-three-halves decuplet.
// Target: physics.bigb
For the special linear Lie algebra with upper triangular positive roots, the simple coroots are and . Their dual fundamental weights are and . In simple-root coordinates, and . The primitive vectors and realize these Fundamental representations of sl3.
This is the instance of the special linear Lie algebra. Classifying its modules is a separate representation-theory problem.
In a finite-dimensional sl2 Lie algebra representation, raising and lowering operators are nilpotent operators, so their exponentials are finite sums. In the defining two-dimensional representation, the displayed product is . It sends to by conjugation, lifting the nontrivial Weyl group reflection. Its square acts on the irreducible highest-weight module by . Thus a Weyl reflection can have a representative of order four on a representation.
// Target: linear-operator-theory.bigb
Let and for . For the sl2 Lie algebra, the raising and lowering operators are , , and has weight . A weight basis of the irreducible representation isA weight basis of the complementary isThe first vector in each list is a highest-weight vector. Applying along the first list gives successive coefficients ; along the second it gives . The final vectors are killed by . At the overlapping weights , the two listed vectors are independent, proving the direct sum.
// Target: semisimple-lie-algebra.bigb
The real traceless two-by-two matrices form a three-dimensional Lie algebra. A basis adapted to conformal mechanics obeys , , .
The SU(2) Lie algebra is the compact real form of . In the Hermitian physics convention its generators satisfy .
In the anti-Hermitian basis , the SU(2) Lie algebra has and Killing form . This is negative definite; the positive internal form for compact Yang-Mills theory is its negative.
With Hermitian angular-momentum generators , the Killing form is and its inverse contraction is . Equivalently the anti-Hermitian basis has Killing form and gives the same operator. In the complex highest-weight basis, . It acts on spin by , so it differs in normalization from the usual angular-momentum quadratic Casimir operator.
For every there is one irreducible -module of dimension , with weights , each of multiplicity one. Every finite-dimensional representation is a direct sum of these modules.
For a finite-dimensional complex sl2 Lie algebra module , the number of simple summands, counted with multiplicity, is . Indeed, the simple module of highest weight has weights , each once. Exactly one of zero and one appears: zero when is even, one when is odd. The classification of finite-dimensional sl2 representations decomposes into these modules, and dimensions of weight spaces add over the direct sum.
For the defining sl2 Lie algebra module with basis , put and . The raising operator has and ; the lowering operator has and . The highest-weight vectors and generate irreducible submodules of highest weights four and two. Bases are respectively and . Their vectors are independent in every weight space, proving the displayed direct sum without only comparing dimensions.
For a finite-dimensional sl2 Lie algebra representation, write . Each irreducible summand has character , so and for nonnegative . Thus each parity subsequence is symmetric and unimodal. If all weights have one parity, removing the unused exponents gives an ordinarily unimodal coefficient sequence. The full sequence including every integer exponent need not be unimodal: has coefficients .
In the sl2 Lie algebra, if and , then the commutator relations give and . Induct using . This controls the raising operator on a string formed by the lowering operator and yields the classification of finite-dimensional sl2 representations.
For a highest-weight vector of weight in an irreducible finite-dimensional sl2 Lie algebra module, the basis diagonalizes , with eigenvalue on . Summing those eigenvalues gives . The formula requires to be a highest-weight vector, not an arbitrary vector of the module.
In the irreducible sl2 Lie algebra module , use , with . The Lie-invariant bilinear form is nondegenerate: its anti-diagonal is nonzero. Invariance follows from , and . Its transpose equals , so it is a symmetric bilinear form when is an even number, and an alternating bilinear form when is an odd integer. The Weyl complete reducibility theorem gives a nondegenerate invariant form on any finite-dimensional module by taking the orthogonal direct sum of these forms.
On any finite-dimensional -module, the lowering operator is injective from the -weight space to the -weight space whenever . This follows on each irreducible summand from its standard weight string.
The Verma module has basis andIt is reducible exactly when ; then its unique proper nonzero submodule is generated by and is isomorphic to .
The Verma module has basis , with , and . Its submodule spanned by for is , and the quotient is the trivial one-dimensional module. A complementary submodule would be trivial and hence lie in the zero-weight space , but . Therefore the displayed extension does not split. This explains why the finite-dimensional hypothesis in the Weyl complete reducibility theorem is essential.
An intermediate-series -module has one-dimensional weight spaces indexed by all integers. In the basis , one useful normalization isIt is irreducible exactly when every raising coefficient is nonzero.
Let be a toral subalgebra and an abelian Lie algebra in the symplectic Lie algebra, over an algebraically closed field of characteristic zero. Its defining representation has a weight decomposition. The symplectic Lie algebra identity gives for , , so unless . Since is a nondegenerate bilinear form, its restrictions give a perfect pairing between and , and a nondegenerate bilinear form on .
An element of the centralizer of a Lie subalgebra preserves each weight space. On it is any element of . On each nonzero pair, its action on is arbitrary, and the perfect pairing uniquely forces the action on to be the negative dual map . These independent choices preserve Lie brackets, proving the displayed direct sum of Lie algebras; the sum runs over one representative per nonzero pair.
This centralizer of a Lie subalgebra is an abelian Lie algebra precisely when and all the nonzero weight spaces are one-dimensional. Indeed, is nonabelian for by , while every nonzero symplectic vector space contains a two-dimensional symplectic subspace on which . In the stated case the surviving summands are copies of and commute. Repeated weights of a representation, including a zero one, are therefore exactly what obstructs this commutativity.
Let be a finite-dimensional symplectic vector space over an algebraically closed field of characteristic zero. For , write its Jordan–Chevalley decomposition as . Applying the symplectic adjoint to this decomposition and using , uniqueness gives and . Thus both parts belong to the symplectic Lie algebra. The operators and commute. The former is diagonalizable on in a matrix unit basis for an eigenbasis of , and the latter is a nilpotent linear map, as follows fromThe restrictions to the invariant symplectic Lie algebra retain these properties. Hence these restrictions form the Jordan–Chevalley decomposition of .
If is diagonalizable, . For every , the rank-one map lies in , since the two terms in cancel by alternation. For its image is , so commuting with forces to preserve . A linear map preserving every line is scalar: compare its action on two linearly independent vectors and on their sum. Since is a nilpotent linear map, that scalar is zero. Thus is diagonalizable. Conversely, if is diagonalizable on , its operator commutator action is diagonalizable on and therefore on its invariant symplectic Lie algebra. This proves both directions, including .
Relative to , a symplectic Lie algebra matrix has blocks with symmetric. Its diagonal Cartan subalgebra has basis . Write . A root-space decomposition is obtained fromThe matrix unit rule proves . These root vectors and the diagonal basis span all the permitted blocks and are independent. There are roots and diagonal basis elements, giving dimension .
For , the defining Lie algebra representation of has the displayed tensor product decomposition. The crystal of the defining symplectic representation and the crystal tensor-product rule give exactly these three highest vertices, each occurring once. The symmetric square is of dimension . The exterior square splits into its invariant inverse-form line and the primitive exterior square of dimension . Their dimensions sum to . At , the exterior square is only the invariant line, giving .
The Lie algebra of the compact symplectic group consists of and . Equivalently with and . Its real dimension is , and its complexification is the symplectic Lie algebra.
In its block form, a matrix-unit basis of the unitary Lie algebra spans the anti-Hermitian block , with the corresponding conjugate block below. Independent real and imaginary symmetric matrix units span , with lower block . These give independent real generators, including diagonal symmetric entries.
For the symplectic Lie algebra preserving , write and . Its positive roots of a root system have sl2 subalgebra associated with a root triplesThe matrix unit commutator identity verifies and the two eigenvalue relations , .
Over , the symplectic Lie algebra on a four-dimensional symplectic vector space acts faithfully on its five-dimensional primitive exterior square, preserving its symmetric bilinear form. Faithfulness follows because an operator killing all wedges has all off-diagonal coefficients zero and satisfies for every pair, hence is zero. The source and target Lie algebras both have dimension ten, so this faithful map is an isomorphism to the Special orthogonal Lie algebra .
For the defining four-dimensional representation of the symplectic Lie algebra , with the short simple root numbered first, its highest weight is . Its tensor square decomposes aswith dimensions . The first summand is the symmetric square; the other two form the exterior square, split by symplectic contraction of an exterior square.
For , the finite Dynkin diagram has a chain of short roots with single bonds, followed by a double bond to the long root , pointing toward . The Extended Dynkin diagram adds the long root , doubly bonded to with arrow toward . All other bonds remain. For , the finite diagram is a single vertex; the affine diagram has two equal-length vertices joined by a double bond without an arrow, with Cartan matrix .
Choose positive roots of a root system , for , and . The simple roots are for and . Their coroots are and . Therefore the fundamental weights and Weyl vector areThe highest root is . These formulas include rank one.
The Weyl group of the Cn root system is the signed symmetric group. A Weyl reflection in changes the sign of coordinate ; one in interchanges coordinates ; one in interchanges and negates both coordinates. These generate all independent sign changes and all permutations, giving order .
In an orthonormal basis , the roots are , , , and . Choose simple roots and , with a short root. The fundamental weights are and .
The roots are for and for . One root basis is , , . With , its Cartan matrix isThe fundamental representation of highest weight is six-dimensional, with weights .
In the realization , each coordinate two-plane cuts out a C2 root system. Each plane cuts out an A2 root system; there are four such planes after identifying opposite normals. Thus the irreducible full rank-two root subsystems comprise three of type and four of type . For an simple pair , equal lengths and angle degrees give roots . For a mixed-length simple pair with and , the B2 root system has roots . These lists also verify that the selected pairs are simple, rather than simply independent.
The C3 root system consists of and in . Its Weyl group is the signed symmetric group on three coordinates, of order . The reflecting planes are and . The open Weyl chamber has closure generated by , and . Sign choices and absolute-coordinate orderings give its translates. On the unit sphere the chamber is a spherical triangle with angles , , , and hence area ; the areas of all triangles sum to .
For the C3 root system, take , , . With columns indexing coroots, the Cartan matrix isThe convention with rows indexing coroots gives . Both describe a double bond between the short root and long root , with arrow toward .
The special orthogonal Lie algebra consists of the infinitesimal transformations preserving a nondegenerate symmetric bilinear form.
For a real inner product space , the displayed map identifies with its Special orthogonal Lie algebra. It sends a wedge of distinct orthonormal basis vectors to the corresponding elementary skew-symmetric matrix, so it is an isomorphism. For an orthogonal transformation , invariance of the inner product gives . Thus the exterior square of the defining representation is the Adjoint representation of the orthogonal group. Complexification identifies their weight spaces, allowing the root system to be read from the exterior-square weights and the zero-weight multiplicity.
For a complex vector space with a nondegenerate symmetric bilinear form , the map identifies the exterior square of the defining representation with the Adjoint representation of a Lie algebra of . Equivariance follows from invariance of ; an orthogonal basis verifies bijectivity. It is irreducible when is simple. In dimension four it splits into the adjoint modules of the two commuting sl2 Lie algebras.
For the complexified Special orthogonal Lie algebra , , choose positive roots of a root system and for . Its Weyl vector is . Dominant coordinates satisfy . The Weyl dimension formula becomes . The case is two independent A1 root systems, while is isomorphic to the A3 root system.
The Weyl group of the Dn root system consists of coordinate permutations together with an even number of sign changes. Reflection in swaps coordinates ; reflection in swaps them and negates both. These generate all even signed permutations, giving order .
With simple roots for and , the fundamental weights are for , followed by and . All roots have squared length two, so their pairings with coroots verify the dual-basis identities. The last two weights are the half-spin highest weights; for , the defining highest weight is .
For the antidiagonal symmetric matrix in dimension , put and . The diagonal Cartan subalgebra has coordinates . The root spaces of weights are spanned by ; those of weights by , with negative-root spaces given by transposes. The condition gives these generators directly and excludes roots .
For antisymmetric Maurer–Cartan forms with , let . Direct expansion gives and cyclic analogues, with no mixed terms. The dual triples form two commuting ideals, each with the SO(3) Lie algebra brackets. This is a real compact splitting, distinct from the splitting of the complexified algebra into two copies of .
Differentiating gives . The generators satisfy by the cross product identity. This identifies it with the SU(2) Lie algebra by matching a skew-Hermitian Pauli basis.
For , the Bn root system has simple roots for and . Its Dynkin diagram is a chain with a double terminal bond pointing toward short . The highest root is . In the Extended Dynkin diagram, attaches by a single bond to for . For , both long nodes attach by double bonds to short node . Rank one is the rank-one root system; its affine diagram has two equal-length nodes joined by a double bond without a short-root arrow.
The roots are for and for . Its root basis , , exhibits it as the dual root system of the C3 root system. With , its Cartan matrix isThe fundamental representation of highest weight is the seven-dimensional vector representation, with weights .
The complexified so4 Lie algebra consists of four-by-four complex skew-symmetric matrices and has dimension six. Its root system is , reflecting its decomposition into two simple rank-one ideals.
The self-dual and anti-self-dual combinations of the skew generators form two commuting copies of . Consequentlyand its Adjoint representation decomposes as .
For , , the two complex half-spin irreducible representations have dimension . Their weights are with respectively even and odd numbers of minus signs, after a chirality convention is chosen; every weight multiplicity is one. A model is the even or odd part of the exterior algebra of , where an occupied mode reverses the corresponding sign. Each is a minuscule representation. These are representations of the spin group and its Lie algebra, but do not descend through the double covering to the special orthogonal group.
Fix a volume form on a four-dimensional real symplectic vector space. Wedge pairing on its exterior square has signature . The symplectic group fixes the bivector inverse to the symplectic form. Its orthogonal complement is five-dimensional with signature after choosing the appropriate overall sign. The induced representation has kernel : identity action on the entire exterior square preserves every two-plane and thus every line, forcing a scalar transformation with square one. Its derivative is injective and both group dimensions are ten, so its image is the identity component of .
On two-by-two Hermitian matrices, the determinant of is . The action preserves it for . A kernel element is unitary and commutes with every Hermitian matrix, hence is . The derivative is injective and both real dimensions are six, giving the full proper orthochronous Lorentz group as the connected image.
Congruence on real symmetric two-by-two matrices preserves the quadratic form of signature when . Its kernel is and its derivative is injective: for every symmetric implies . Both Lie groups have dimension three, so the connected image is the full identity component. Also identifies with .
A fixed volume form makes wedge pairing on a symmetric form of signature . The exterior-square action of special linear group matrices preserves it. Its kernel is : a transformation acting trivially on every bivector preserves every two-plane and hence every line, so is scalar, with scalar square one. The derivative is injective and both groups have dimension fifteen, so the connected image is the full identity component. This is a concrete real spin-group cover.
Two unit quaternions act on Euclidean four-space by . This norm-preserving action maps onto the SO(4) group. Its kernel is precisely and : a kernel pair must have equal entries commuting with every quaternion. The two simply connected SU(2) factors therefore form the universal spin cover.
A spinor representation is a representation of a Spin group, or infinitesimally of a Special orthogonal Lie algebra, constructed from a Clifford algebra. In even dimensions it can split into two chiral spinor representations.
The Spin group double covers SO(7). Its central kernel element acts trivially in the vector representation and as in the spinor representation. Thus a spinor Lie algebra representation does not define an ordinary SO(7) group representation. Under the same obstruction is visible because acts trivially on but negatively on .
Choose five Hermitian four-by-four matrices obeying the Euclidean Clifford algebra relations . For , the displayed anti-Hermitian matrices satisfy the Special orthogonal Lie algebra Lie bracket relations. A direct proof first establishes , then applies the commutator product rule to . One explicit choice is , , , , , with the Pauli matrices. The Cartan subalgebra operators and have simultaneous eigenvalues , identifying the four-dimensional spinor representation. A full rotation acts as , proving it does not descend to .
For the B2 root system with long simple root first, the four-dimensional spinor representation has highest weight . Its symmetric square has highest weight and dimension ten, so it is the Adjoint representation. The exterior square has the four nonzero defining weights and two zero weights; it splits into the five-dimensional vector representation of highest weight and a trivial representation. Thus the tensor square decomposes as displayed. The invariant alternating bilinear form reflects the pseudoreal representation structure. The spinor representation is a group representation of the Spin group , while its tensor square descends to .
An irreducible complex Clifford algebra module in even spacetime dimension has dimension and splits under the connected Spin group into two Weyl spinor modules of dimension . In odd dimension , an irreducible spinor still has dimension but has no Weyl splitting. Reality conditions depend on signature and are additional to these complex counts.
Lorentz generators produce infinitesimal Lorentz transformations. In four dimensions they split into three rotation and three boost generators. Their action on supercharges uses the appropriate spinor representation, while their action on translations is the mixed Poincare algebra commutator. Signs in component formulas must agree with whether the generators act on states, operator columns or dual rows.
For the Lorentz algebra convention , choose , , , , and . The resulting rotation bracket is as well as the two displayed brackets. Changing the Minkowski metric signature while holding this generator convention fixed reverses all coefficients. A simultaneous redefinition of the generators can instead preserve the familiar form. Conventions must be specified before comparing algebra signs.
Articles were limited to the first 100 out of 324 total.
Articles by others on the same topic
There are currently no matching articles.