Properties of the g-invariant Bilinear Form on the Spin Representations of the Simple Lie Algebras of Type Dn and Bn

Transcription

1 Uiversity of Colorado, Boulder CU Scholar Udergraduate Hoors Theses Hoors Program Sprig 013 Properties of the g-ivariat Biliear Form o the Spi Represetatios of the Simple Lie Algebras of Type D ad B Sergey Lozisky Uiversity of Colorado Boulder Follow this ad additioal works at: Recommeded Citatio Lozisky, Sergey, "Properties of the g-ivariat Biliear Form o the Spi Represetatios of the Simple Lie Algebras of Type D ad B" (013). Udergraduate Hoors Theses This Thesis is brought to you for free ad ope access by Hoors Program at CU Scholar. It has bee accepted for iclusio i Udergraduate Hoors Theses by a authorized admiistrator of CU Scholar. For more iformatio, please cotact cuscholaradmi@colorado.edu.

2 Properties of the g-ivariat Biliear Form o the Spi Represetatios of the Simple Lie Algebras of Type D ad B Sergey Lozisky Uiversity of Colorado Boulder Defeded o April 9th, 013 Defese Committee Dr. Richard M. Gree, Departmet of Mathematics, Dr. David Grat, Departmet of Mathematics, Dr. Nathaiel Thiem, Departmet of Mathematics, Dr. Daiel L. Feldheim, Departmet of Chemistry & Biochemistry. 1

3 Abstract The spi modules for the Lie algebras of type D ad B are costructed through miuscule systems. It is show that the uique (up to multiplicatio by a ozero scalar) g-ivariat biliear form o the modules is odegeerate. By defiig a equatio for the height of weights of the modules, it is show that the biliear form is symplectic whe = 1 or (mod 4), ad orthogoal whe = 0 or 3 (mod 4).

4 1 Itroductio Lie algebras were origially itroduced by S. Lie for the study of Lie groups. The focus of this paper is o the fiite dimesioal simple Lie algebras over the complex field, which were classified betwee with the work of E. Carta ad W. Killig. We costruct the spi modules o the Lie algebras of type D ad B, ad our mai result defies a equatio for the height of weights o these modules. We the use these equatios to prove that the g ivariat biliear form o the modules alterates betwee orthogoal ad smyplectic depedig o the the choice of. For the simple Lie algebra of type B, the biliear form o the spi module is orthogoal whe = 0 or 3 (mod 4) ad symplectic whe = 1 or (mod 4). For the simple Lie algebra of type D, there are two spi modules, ad the biliear form o both is orthogoal whe = 0 (mod 4) ad symplectic whe = (mod 4). Both the equatios ad the proofs are origial to the best kowldge of the author. The strategy of the paper is as follows. I Sectio 1 we itroduce geeral results i Lie algebra. We defie the simple Lie algebras B ad D through matrices uder matrix multiplicatio, as subalgebras of the geeral liear algebra gl(, C), ad derive their root systems ad bases via the root space decompositio. We the show that these algebras ca be costructed usig oly geerators, relatios, ad their associated simple roots. I Sectio, we begi by itroducig geeral results of modules. We the itroduce miuscule systems ad show that they have the structure of weights of irreducible modules. We use these to costruct the spi modules for the Lie algebras D ad B, based o the method itroduced i [1]. I Sectio 3, we defie a g-ivariat biliear form o the spi modules, ad show that it is uique up to multiplicatio by a ozero scalar. Usig the costructio of the spi modules from Sectio, we the provide a equatio for the height of weights of these modules. Fially, we use these equatios to prove that the biliear form alterates from orthogoal to symplectic as discussed above. Defiitio 1.1. A Lie algebra is a k-vector space g equipped with a k-biliear map called the Lie bracket, g g g (x, y) [xy] satisfyig the coditios: [xx] = 0 for all x g, [x[yz]] + [z[xy]] + [y[zx]] = 0 for all x, y, z g. These are kow as atisymmetry ad the Jacobi idetity respectively. A Lie algebra g is said to be abelia if [xy] = 0 for all x, y g. Defiitio 1.. Let A be a associative k-algebra ad let x, y A. We defie a ew k-biliear operatio [, ] such that [x, y] = xy yx uder the usual operatios o the right. 3

5 Lemma 1.3. Let M (k) deote the associative k-algebra of all matrices uder matrix multiplicatio, ad let gl(v ) deote the algebra of all k-liear edomorphisms of a k-vector space V uder compositio of fuctios. (i) The algebra M (k) edowed with the operatio [, ] from Defiitio 1. is a Lie algebra over k, deoted gl(, k). (ii) The algebra gl(v ) edowed with [, ] is a Lie algebra over k. Proof. (i) Let x, y, z M (k). The [x, y] = xy yx M (k) so the set is closed uder the operatio. We have: [x, x] = xx xx = 0, ad [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = x(yz zy) (yz zy)x + y(zx xz) (zx xz)y + z(xy yx) (xy yx)z = xyz xzy yzx + zyx + yzx yxz zxy xzy + zxy zyx xyz + yxz = (xyz xyz) + (xzy xzy) + (yxz yxz) + (yzx yzx) = 0 + (zxy zxy) + (zyx zyx) satisfyig both coditios of Defiitio 1.1. The proof for (ii) is idetical whe cosiderig multiplicatio as compositio of fuctios. Defiitio 1.4. A subspace h of a Lie algebra g is called a subalgebra of g if [xy] h wheever x, y h. We call h a ideal if [xy] h for all x h, y g. Lemma 1.5. Let h 1,..., h k V V be diagoalizable liear trasformatios such that h 1,..., h k commute. The the maps h 1,..., h k ca be simultaeously diagoalized. Proof. This is a geeral result i liear algebra, ad is proved i [3, Lemma 16.7]. Defiitio 1.6. Let g be a Lie algebra. We say that a elemet h g is semisimple if it ca be represeted by a diagoal matrix. Defiitio 1.7. Let g ad h be Lie algebras over k. A Lie algebra homomorphism is a k-liear map ϕ g h such that ϕ([xy]) = [ϕ(x)ϕ(y)] for all x, y g. A Lie algebra isomorphism is a bijective homomorphism. Defiitio 1.8. Let g be a Lie algebra. We defie the adjoit homomorphism as the liear map ad g gl(g) ad(x)(y) = [x, y] 4

6 Lemma 1.9. (i) Suppose h is a semisimple elemet of g. The adh is semisimple. (ii) Suppose the elemets h 1,... h form a abelia subalgebra of a Lie algebra g. The the maps adh 1,..., adh form a abelia subalgebra of gl(g). Proof. The proof for this is cotaied i [4, 6.1]. Propositio Let s gl(, k). The set h = {x sx = x t s, x gl(, k)} edowed with [, ] is a subalgebra of g; x t deotes the traspose of x. Proof. The set h is a subspace of gl(, k). Satisfactio of the two axioms of Defiitio 1.1 is iherited from gl(, k). Therefore, it is sufficiet to show that h is closed uder [, ]. Let x, y h. The s[x, y] = s(xy yx) = sxy syx = x t sy + y t sx = x t y t s y t x t s = (x t y t y t x t )s = ((yx) t (xy) t )s = (yx xy) t s = [x, y] t s. Defiitio Let I l be the l l idetity matrix. Let s d ad s b be matrices of dimesio l l ad l + 1 l + 1 respectively, such that s d = ( 0 I l I l 0 ) ad s b = 0 0 I l. 0 I l 0 (i) The subalgebra so(l, C) of gl(l + 1, C) is defied as the followig: so(l, C) = {x gl(l, k) x t s d = s d x}, ad is kow as the orthogoal Lie algebra of type D. (ii) The subalgebra so(l + 1, C) of gl(l, C) is defied as the followig: so(l + 1, C) = {x gl(l + 1, k) x t s b = s b x}, ad is kow as the orthogoal Lie algebra of type B. 5

7 Lemma 1.1. We have: so(l, C) = {( m p q m t ) p = p t ad q = q t }, (1) 0 b t c t so(l + 1, C) = b m p p = p t ad q = q t. () c q m t Proof. For (1), s d x = x t s d implies ( 0 I l I l 0 ) (m p q ) = ( mt q t p t t ) ( 0 I I 0 ) ( q m p ) = ( qt m t t p t ), showig that q = q t, p = p t, ad = m t as required. For (), s b x = x t s b implies a d e a t b t c t I l b m p = d t m t q t 0 0 I l 0 I 0 c q e t p t t 0 I 0 a d e a t c t b t c q = d t q t m t, b m p e t t p t showig as required that c = d t, b = e t, q = q t, p = p t, m = t, ad a = a t = 0 as it is a 1 1 matrix. Defiitio A Lie algebra is said to be simple if it has o ideals other tha 0 ad itself, ad it is ot abelia. A Lie algebra is semisimple if it ca be writte as a direct sum of simple Lie algebras. It is well kow that the orthogoal algebras are simple. Formal proofs of this may be foud i [4, 11] or [3, 10]. Defiitio Let g be a Lie algebra. A Lie subalgebra h of g is said to be a Carta subalgebra if h is abelia, every elemet h h is semisimple, ad moreover h is maximal with these properties. Propositio Let g = so(, C). The subspace of g cosistig of all diagoal matrices i g is a Carta subalgebra of g. Proof. By Defiitio 1.14 we must show (i) that every elemet of h is semisimple, (ii) that h is abelia, (iii) that it is a subalgebra of g, ad (iv) that h is maximal with these properties. Sice h cosists etirely of diagoal matrices, every elemet is obviously semisimple by Defiitio 1.6, satisfyig (i). 6

8 Sice multiplicatio of diagoal matrices is commutative, for h 1, h h we have [h 1, h ] = h 1 h h h 1 = 0, showig that it is abelia ad satisfyig (ii). Furthermore, this satisfies (iii) by Defiitio 1.4, sice 0 h. Now for (iv), suppose that h is ot maximal ad is cotaied i a maximal Carta subalgebra H. Let a H ad h h. Sice H is abelia, we have [h, a] = ha ah = 0 which implies ha = ah. Suppose a is a matrix with etries (a i,j ), ad h has diagoal etries (h 0,..., h 1 ). The ha = (h j a i,j ) ad ah = (h i a i,j ). Sice this is true for all h h, this implies that i = j ad a is a diagoal matrix. Therefore, H = h, completig the proof. Let g be a Lie algebra ad let h be the Carta subalgebra of g. We deote the dual space of h by h. Sice every elemet h h is diagoalizable ad h is abelia, by Lemma 1.9 the maps adh = {adh h h} are both abelia ad diagoalizable. Therefore, as a cosequece of Lemma 1.5, the map ad h gl(g) iduces the followig decompositio o g: g = g 0 g α α Φ where Φ is the set of α h such that α 0 ad g α 0, ad g α = {v g [h, v] = α(h)v for all h h}, g 0 = {z g [h, z] = 0 for all h h}. It is a well kow result that g 0 = h, ad this is proved i [4, Propositio 8.]. We call this the Carta decompositio. We say that a elemet α Φ is a root of g with associated root space g α. There is a atural isomorphism betwee a fiite dimesioal euclidea dot product space V with its dual space V, give by ϕ V V v v,. The kerel of the map is zero sice the dot product is odegeerate. Sice dim(v ) = dim(v ), it follows immediately from the first isomorphism theorem of liear algebra that the map is a isomorphism. By imposig a dot product o the elemets of h, we obtai the followig isomorphism: ϕ h h (e i, e j ) (e j, e j ) ε i, ε j, where e i, e j ad ε i, ε j are the stadard basis elemets of h ad the stadard basis elemets of h with respect to h, respectively. We omit the details of 7

9 this derivatio, but the reader is referred to [3, 10.5] or [4, 8.5] for further discussio. It is a major result that the Carta decompositio produces roots that form a root system as defied below i Defiitio 1.17, ad we refer the reader to [4, 8] for further discussio. Defiitio Let E be a fiite-dimesioal euclidea space with the regular biliear dot product writte (, ). Let λ, α E. We defie λ, α = (λ, α) (α, α) α, where λ, α is oly liear i the first variable. We let σ α deote the reflectio i the hyperplae ormal to α. That is, for some λ E, σ α (λ) = λ λ, α α. Defiitio A subset Φ of a euclidea dot product space E is called a root system i E if the followig axioms are satisfied: (R1) Φ is fiite, spas E, ad does ot cotai 0; (R) if α Φ, the oly multiples of α i Φ are ±α; (R3) if α Φ, σ α is ivariat uder Φ; (R4) if α, β Φ, the β, α Z. Defiitio A subset of Φ is called a base if: (B1) is a basis of E; (B) each root β Φ ca be writte as β = k α α with k α Z, where the α coefficiets k α are all oegative or all opositive. The elemets of are called simple roots. Defiitio We say that a root α Φ is positive with respect to, if the coefficiets give i (B) of Defiitio 1.18 are positive, ad otherwise it is egative with respect to. Though it is ot obvious, every root system has a base. However, a base is ot uique. This is ot proved ad the reader is referred to [3, Theorem 11.10]. Propositio 1.0. Let g = so(l + 1, C). A root system of g is give by the followig set: Φ = b c m p q, where b = {ε i h 0 i l 1} c = { ε i h 0 i l 1} m = {(ε i ε j ) h 0 i < j l 1} = {(ε i ε j ) h 0 j < i l 1} p = {(ε i + ε j ) h 0 i < j l 1} q = { (ε i ε j ) h 0 i < j l 1}. 8

10 Proof. This is derived i detail i [3, 1.3]. Propositio 1.1. Let g = so(l, C). The root system of g is give by the followig set: Φ = m p q, where m = {(ε i ε j ) h 0 i < j l 1} = {(ε i ε j ) h 0 j < i l 1} p = {(ε i + ε j ) h 0 i < j l 1} q = { (ε i ε j ) h 0 i < j l 1}. Proof. This is derived i detail i [3, 1.4]. Propositio 1.. Let g = so(l + 1, C), ad let Φ be the correspodig root system as i Propositio 1.0. A base for Φ is give by the followig set: where = {α i h 0 i l 1}, α i = ε i ε i+1 α l 1 = ε l 1. for 0 i l, ad The positive roots with respect to Φ are the elemets of the sets b, p, m Φ of Propositio 1.0. Proof. We must show that axiom (B1) of Defiitio 1.18 is satisfied, ad that the elemets of b, p, ad m ca be writte as oegative iteger liear combiatios of. We first observe that the elemets of are liearly idepedet ad that dim( ) = l =dim(h ). This satisfies (B1). Now, the elemets of b, m ad p ca be writte as follows, respectively: ε i = (ε i ε i+1 ) + (ε i+1 ε i+ ) + + (ε l ε l 1 ) + ε l 1 = α i + α i α l 1 + α l ε i ε j = (ε i ε i+1 ) + (ε i+1 ε i+ ) + + (ε j ε j 1 ) + (ε j 1 ε j ) = α i + α i α j 1 ε i + ε j = (α i + α i α l + α l 1 ) + (α j + α j α l + α l 1 ) This shows that the elemets of the sets b, p, m are positive roots by Defiitio As the elemets of the sets c,, q are all egatives of those of b, p, m, this satisfies coditio (B), completig the proof. 9

11 Propositio 1.3. Let g = so(l, C), ad let Φ be the correspodig root system as i Propositio 1.1. A base for Φ is give by the followig set: where = {α i h 0 i l 1}, α i = ε i ε i+1 for 0 i l, ad α l 1 = ε l + ε l 1. The positive roots with respect to Φ are the elemets of the sets b, m Φ of Propositio 1.1. Proof. We proceed as i Propositio 1.. The elemets of are liearly idepedet ad dim( ) = l =dim(h ), satisfyig coditio (B1) of Defiitio The elemets of the sets b ad m ca be expressed as follows, respectively: ε i ε j = (ε i ε i+1 ) + (ε i+1 ε i+ ) + + (ε j ε j 1 ) + (ε j 1 ε j ) = α i + α i α j 1 + α j 1 ε i + ε j = ((ε i ε i+1 ) + (ε i+1 ε i+ ) + + (ε j ε j 1 ) + (ε j 1 ε j )) + ((ε j ε j+1 ) + (ε j+1 ε j+ ) + + (ε l ε l 1 ) + (ε l + ε l 1 )) = (α i + α i α j + α j 1 ) + (α j + α j α l + α l 1 ), which satisfies Defiitio As the elemets of the sets ad p are egatives of the elemets of b ad m, this satisfies coditio (B) of Defiitio 1.18, ad completes the proof. Defiitio 1.4. (i) Let g = so(l + 1, C). We deote Φ B to be the root system of g as i Propositio 1.0. We deote B to be the base of Φ B, with the fixed order of elemets as i Propositio 1.. (ii) Let g = so(l, C). We deote Φ D to be the root system of g as i Propositio 1.1. We deote D to be the base of Φ D, with the fixed order of elemets as i Propositio 1.3. Defiitio 1.5. Let be a base for a root system Φ. Fix a order o the elemets of, say (α 0,..., α l 1 ). The Carta matrix of is defied as the l l matrix with (i, j)th etry α i, α j. Lemma 1.6. (i) The Carta matrix of B is

12 (ii) The Carta matrix of D is Proof. We must show that the (i, j)th etry of each matrix is equal to α i, α j i its respective algebra, thereby satisfyig Defiitio 1.5. Give our selected bases i Propositios 1. ad 1.3 respectively, the majority of calclatios are the same. Sice α i, α i = (αi,αi) (α i,α i) =, the diagoal etries are justified. With two exceptios, we have the followig hold for both algebras: α i, α j = (α i, α j ) (α j, α j ) = (ε i ε i+1, ε j ε j+1 ) (ε j ε j+1, ε j ε j+1 = 1 if i j = 1, ad 0 otherwise. The first exceptio is for α l ad α l 3 B, as α l, α l 1 = ε l ε l 1, ε l 1 = (ε l ε l 1, ε l 1 ) (ε l 1, ε l 1 ) = 1 1 =, ad the secod exceptio is for α l 1, α j D, as α l 1, α l 3 = ε l + ε l 1, ε l 3 ε l = (ε l + ε l 1, ε l 3 ε l ) (ε l 3 ε l, ε l 3 ε l ) = 1 = 1 = α l 3, α l 1, showig that all the etries are as required by Defiitio 1.5. We have thus far cosidered Lie algebras, particularly the orthogoal Lie algebras, as vector spaces cosistig of matrices. Though this is a coveiet method to ituitively derive prelimiary results, a more comprehesive ad applicable way to describe them is through geerators ad relatios, as is itroduced i the followig theorem. 11

13 Theorem 1.7 (Serre s Theorem). Fix a root system Φ, with base = {α 1,..., α l 1 }. Let g be the Lie algebra geerated by the 3l elemets {e αi, f αi, h αi 0 i l 1} subject to the followig relatios, (S1) [h αi, h αj ] = 0 for all i, j; (S) [h αi, e αj ] = α j, α i e αj ad [h αi, f αj ] = α j, α i f αj for all i, j; (S3) [e αi, f αj ] = δ ij h αi ; (S4) [e αi, [e αi,... [e αi, e αj ]... ]] = 0; 1 α j, α i times (S5) [f αi, [f αi,... [f αi, f αj ]... ]] = 0; 1 α j, α i times where δ is the Kroecker delta. The g is a (fiite dimesioal) semisimple algebra with Carta subalgebra h spaed by the h αi, with correspodig root system Φ. Proof. This is proved i [4, 18.3]. Defiitio 1.8. Let {e αi, f αi, h αi α i } be the geeratig elemets of the Lie algebra g as defied i Theorem 1.7. We call these elemets Serre geerators. Modules I this sectio we defie ad costruct the spi modules for the simple Lie algebras B ad D. We begi by defiig modules ad represetatios ad itroducig stadard results associated with these structures. We the itroduce the cocept of a miuscule system, which is ot covetioal ad is itroduced i [1]. We show that a miuscule system imposes the structure of a module o a vector space. Next, we itroduce classificatio theorems of fiite dimesioal irreducible modules. Combiig these results, we are able to explicitly costruct the required spi modules, ad explicitly observe their iteractio with the Serre geerators, which allows further study of their structures i Sectio 3. Defiitio.1. Let g be a Lie algebra over k ad let V be a fiite-dimesioal k-vector space. A represetatio of g is a Lie algebra homomorphism ϕ g gl(v ). A g-module V is a fiite- Defiitio.. Let g be a Lie algebra over k. dimesioal k-vector space V with a map g V V (x, v) x v 1

14 satisfyig: for all x, y g, v, w V, ad λ, µ k. (λx + µy) v = λ(x v) + µ(y v), x (λv + µw) = λ(x v) + µ(x w), [xy] v = x (y v) y (x v), Give a represetatio, ϕ, we ca make a correspodig g-module by defiig x v = ϕ(x)(v) for x L, v V. Coversely, give a g-module V ad a basis of V, we ca use the same equatio to defie a represetatio of g. Defiitio.3. Let V be a g-module. A submodule of V is a subspace W of V such that x w W for all x g, w W. Defiitio.4. Let V be a g-module. We say that V is simple or irreducible if it is o-zero ad it has o submodules other tha 0 ad V. Defiitio.5. Let V ad W be g-modules. A module homomorphism from V to W is a k-liear map θ V W such that θ(x v) = x θ(v) for all v V, x g. bijective module homomorphism. A module isomorphism is a Let g be a simple Lie algebra with Carta subalgebra h, ad suppose that the k-vector space V is a fiite dimesioal g-module. Sice every elemet of h is semisimple by Defiitio 1.14, by Lemma 1.5 the map h V V iduces a decompositio of V ito simultaeous eigespaces for h as follows: V = V λ, λ Ψ where Ψ is the set of λ h such that λ 0 ad V λ 0, ad V λ = {v V h v = λ(h)v for all h h}. Defiitio.6. Maitai the otatio of the previous paragraph. The decompositio of V is kow as the weight space decompositio. The elemets λ Ψ are kow as weights with correspodig weight space V λ. The elemets of V λ are kow as weight vectors of weight λ. The cocept of a miuscule system i the followig defiitio is ot stadard, ad is itroduced i [1]. 13

15 Defiitio.7. Let be the set of simple roots of a simple Lie algebra g. Suppose that Ψ is the set of elemets dual to the Carta subalgebra h of g, such that for all α ad λ Ψ, (i) λ, α { 1, 0, 1}, ad (ii) λ, α = 1 if ad oly if λ + α Ψ, λ, α = 1 if ad oly if λ α Ψ, ad λ, α = 0 if ad oly if λ ± α Ψ. We say that Ψ is a miuscule system with respect to the simple system. Defiitio.8. Let Ψ be a miuscule system with respect to the simple system ad let λ Ψ. We defie b λ to be the vector idexed by λ, ad we defie V Ψ to be the k-vector space with basis b Ψ, where b Ψ = {b λ λ Ψ}. Defiitio.9. We defie the actio of the Serre geerators o V Ψ by specifyig a k-liear edomorphism o the elemets of b Ψ, such that for all λ Ψ, b λ+αi if λ + α i Ψ; e αi b λ = 0 otherwise. b λ αi if λ α i Ψ; f αi b λ = 0 otherwise. h αi b λ = λ, α i b λ. These maps impose the structure of a g-module o V Ψ, where g has base. This is because by costructio, V Ψ has a basis of weight vectors for the Carta subalgebra h of g. However, this is ot a sufficiet coditio, as we must also show that the axioms of Defiitio. are satisfied. That is, we must check that (x y y x) v = [x, y] v for all x, y g ad all v V Ψ. It is eough to show that this holds for all Serre geerators of g ad all basis elemets of V Ψ, because every elemet i g ca be expressed as iterated brackets ad liear combiatios of the Serre geerators, ad every elemet i V Ψ ca be expressed as liear combiatios of its basis elemets. It the follows by iductio, biliearity i [, ], ad liearity i V Ψ that this holds for all elemets of g ad V Ψ. The calculatios for this are doe i [1], ad we itroduce this result i the followig theorem. Theorem.10. Let Ψ be a miuscule system with respect to the simple system, ad let g be the correspodig simple Lie algebra. The C-vector space V Ψ has the structure of a g-module, where the Serre geerators act o V Ψ as i Defiitio.9. Proof. This is proved i [1, 3]. Defiitio.11. Let Ψ be a miuscule system with respect to the simple system. We defie a partial order o Ψ such that µ λ if ad oly if λ µ = a i α i, where a i Z ad a i 0. We say that Ψ has highest weight λ h α i if µ λ h for all µ Ψ. Similarly, we say that Ψ has lowest weight λ l if λ l µ for all µ Ψ. 14

16 Defiitio.1. Let h be the Carta subalgebra of the simple algebra g with base, spaed by elemets {h αi α i } as defied i Theorem 1.7. Let {ω i α i } be the basis of h dual to {h αi α i }. That is, ω i (h αj ) = δ ij. The weights ω i are kow as fudametal weights. We ca ow itroduce a well kow result which classifies all fiite dimesioal irreducible modules. Propositio.13. (i) Let g be a simple Lie algebra over C. If λ is a oegative Z-liear combiatio of the fudametal weights ω i the up to isomorphism there is a uique fiite dimesioal irreducible g-module L(λ) where v λ is of weight λ ad is the uique ozero highest weight vector of L(λ). The modules L(λ) are pairwise oisomorphic ad exhaust all fiite dimesioal irreducible modules of g. (ii) Suppose that V is a fiite dimesioal g-module cotaiig a ozero highest weight vector v λ of weight λ, ad that dim(v ) = dim(l(λ)). The V is isomorphic to L(λ). Proof. Part (i) is a special case of [5, Theorem 10.1]. Part (ii) is proved i [1, Propositio 1.4 (ii)]. We are ow able to defie the spi modules. Defiitio.14. Maitai the otatio of Propositio.13. The spi module for the simple Lie algebra of type B (for ), is the algebra L(ω 1 ). By [6, 60.E.], it has dimesio. The two spi modules for the simple Lie algebra of type D (for 4), are the modules L(ω ) ad L(ω 1 ). By [6, 60.E.], each of the modules has dimesio 1. Accordig to [5, Propositio 10.17], the fudametal weights ca be obtaied by the formula ω i = (A 1 ) ji α j, j where (A 1 ) ji is the (j, i)th etry of the iverse of the Carta matrix associated with a fixed base. This motivates the followig result. Lemma.15. Let h be the Carta subalgebra of the simple Lie algebra so(l + 1, C) with base B, as defied i 1.7. The fudumetal weights of h are give by the followig: i ω i = j=0 i 1 (j + 1)α j + k=1 (i + 1)α i+k = α 0 + α (i + 1)α i + (i + 1)α i (i + 1)α 1 = ε 0 + ε ε i 1 15

17 for 0 i < l 1, ad ω 1 = 1 1 i=0 (i + 1)α i = 1 α 0 + α α 1 = 1 ε ε ε 1. Proof. The iverse of the Carta matrix associated with B is give by , 1 3 l l l 1 which yields the idetity matrix whe multiplied by the Carta matrix i Lemma 1.6 (i). The fudumetal weight ω i is obtaied by summig all of the simple roots α i B with respect to the fixed order, with coefficiets from the ith row. This iduces the above equatios, completig the proof. Lemma.16. Let h be the Carta subalgebra of the simple Lie algebra so(l, C) with base D, as defied i 1.7. The fudumetal weights of h are give by the followig: i ω i = j=0 l i 3 (j + 1)α j + k=1 (i + 1)α i+k + 1 l 1 l 1 (i + 1)α j j=l = α 0 + α (i + 1)α i + (i + 1)α i (i + 1)α l 3 + i + 1 α l + i + 1 α l 1 = ε 0 + ε ε i 1 for 0 i < l, ad ω l = 1 ω l 1 = 1 l 3 i=0 (i + 1)α i + (l)α 1 + (l 1)α 1 = 1 α 0 + α l α l 3 + (l)α l + (l 1)α l 1 = 1 ε ε ε l ε l 1 ε l 1 l 3 i=0 (i + 1)α i + (l 1)α 1 + (l)α 1 = 1 α 0 + α l α l 3 + (l 1)α l + (l)α l 1 = 1 ε ε ε l ε l + 1 ε l 1. 16

18 Proof. The iverse of the Carta matrix with respect to D is , l l 1 3 l l l l l l 1 l which yields the idetity matrix whe multiplied by the Carta matrix i Lemma 1.6 (ii). The the rest of the proof follows the same argumet as i Lemma.15. Lemma.17. Let h be the dual space of the Carta subalgebra h, of the simple Lie algebra of type B. Defie Ψ B to be the subset of h cosistig of all vectors of the form 1 1 ± 1 ε 0 ± 1 ε 1 ± 1 ε ± 1 ε 1. The Ψ B is a miuscule system with respect to B. Proof. We must show that the axioms of Defiitio.7 are satisfied. Suppose first that α i B for some 0 i < 1, ad let λ Ψ B. Write λ = 1 c i ε i. We i=0 ote that c i c i+1 { 1, 0, 1}, as c i, c j {± 1 }. The proof is a case by case check accordig to the values of c i ad c i+1. There are three cases to check. The first possibility is that c i = c i+1. This implies that λ, α i = 0, as α i is orthogoal to λ. The coefficiets of ε i ad ε j i λ ± α i differ by 3 i absolute value, which meas that λ ± α Ψ B, satisfyig the coditios of Defiitio.7. The secod possibility is that c i = 1 = c i+1. This implies that λ, α i = 1. The coefficiets of ε i ad ε i+1 i λ + α i differ by 1, ad differ by i λ α i. This meas that λ + α i Ψ B ad λ α i Ψ B, satisfyig the coditios of Defiitio.7. The third possibility is that c i = 1 = c i+1. A aalysis like the previous paragraph shows that λ, α i = 1 ad λ α i Ψ B, λ + α i Ψ B. This satisfies the coditios of Defiitio.7. It remais to show that Defiitio.7 is satisfied i the case i = 1. There are two cases to check, accordig to the value of c 1. If c 1 = 1, the λ, α i = 1, ad λ α i has 1 as the coefficiet of ε 1, whereas λ + α 1 has coefficiet of 3 of ε 1, satisfyig the coditios of Defiitio.7. For the case of c 1, a similar argumet shows that Defiitio.7 is satisfied, completig the proof. Lemma.18. Let h be the dual space of the Carta subalgebra h of the simple Lie algebra of type D. Let Ψ + D ad Ψ D be the subsets of h cosistig of all vectors of the form ± 1 ε 0 ± 1 ε 1 ± 1 ε ± 1 ε 1, 17

19 with a eve ad odd umber of occureces of 1 ε i respectively. The Ψ + D ad Ψ D are miuscule systems with respect to the simple system D. Proof. We prove this for Ψ + D ad Ψ D respectively. For the proof of Ψ+ D, suppose first that α i D for some 0 i < 1. Write λ = 1 c i ε i. The proof follows a argumet similar to Lemma.17. It remais to show that Defiitio.7 is satisfied whe i = 1. There are three cases to check, accordig to the values c ad c 1. The first possibility is that c = c 1 This implies that λ, α 1 = 0. The coefficiets c ad c 1 of λ±α 1 differ by 3 i absolte value, satisfyig the coditios of Defiitio.7. The secod possibility is that c = c 1 = 1. This implies that λ, α i = 1. The coefficiets c, c 1 i λ + α 1 are both 1, ad the same coefficiets i λ α 1 are both 3, which satisfies the coditios of Defiitio.7. The third possibility is that c = c 1 = 1. This follows a similar argumet to the case c = c 1 = 1, satisfyig Defiitio.7 ad completig the proof for Ψ + D. We ow prove this for Ψ D. Suppose first that α i D for some 0 i 3 or i = 1. The proof follows a similar argumet to the first three cases of Lemma.17. Now, suppose that i =. The proof follows the same argumet as above for Ψ + D i the case i = 1, satisfyig the coditios of Defiitio.7 ad completig the proof. i=0 To costruct the spi modules, we desire the sets Ψ B, Ψ + D, ad Ψ D closed uder egatio, ad therefore itroduce the followig Lemma. to be Lemma.19. (i) The set Ψ B is closed uder egatio. (ii) The sets Ψ + D ad Ψ D are closed uder egatio if ad oly if they are subsets of dual spaces of dimesio. Proof. For part (i), it is clear that the egative of ay vector of Ψ B is also of the form ± 1 ε 1 ± ± 1 ε. For part (ii), suppose Ψ + D is a subset of a dual space of dimesio, ad λ Ψ + D. By Defiitio.18, λ has a eve umber of egative coefficiets, say x, ad x positive coefficiets. Therefore, if is odd, λ will cotai a odd umber of egative coefficiets. The same argumet applies to Ψ D, completig the proof. Corollary.0. The set Ψ B has highest weight λ = 1 1 ε i. i=0 Proof. To satisfy Defiitio.11, we must show that the differece betwee λ ad ad ay other weight i Ψ B is a oegative iteger liear combiatio of 18

20 simple roots. Sice every coefficiet i λ is positive, its differece with aother weight will be a liear combiatio of the vectors ε i, 0 i 1 with coefficiets 1 or 0. This is obviously a liear combiatio of the positive roots of the set b Φ B, defied i Propositio 1.0 ad derived i Propositio 1.. By Defiitio 1.19, these are i tur oegative iteger liear combiatios of simple roots, which satisfies Defiitio.11 ad completes the proof. Corollary.1. The sets Ψ + D has highest weight λ = 1 1 ε i. i=0 Proof. Suppose that µ Ψ + D. To satisfy Defiitio.11, we must show that λ µ is a oegative iteger liear combiatio of simple roots. Sice we require Ψ + D to be closed uder egatio, is eve by Lemma.19. Furthermore, sice λ cosists etirely of compoets with coefficiet 1, this implies that λ µ has a eve umber of compoets with positive coefficiet 1 ad a eve umber of compoets with coefficiet 0. This is clearly a oegative liear combiatio of the positive roots of the set b Φ D, defied i Propositio 1.1 ad derived i Propositio 1.3. Sice by Defiitio 1.19 the elemets of b are i tur a oegative liear combiatio of simple roots, this satisfies Defiitio.11 ad completes the proof. Corollary.. The set Ψ D has highest weight λ = 1 ε i 1 i=0 ε 1. Proof. Suppose that µ Ψ D. To satisfy Defiitio.11, we must show that λ µ is a oegative iteger liear combiatio of simple roots. Sice we require Ψ + D to be closed uder egatio, by Lemma.19 is eve. There are two cases to check, accordig to the coefficiet of ε 1 i µ. Suppose first that the coefficiet is 1. By Defiitio.18, this implies that there is a odd umber of compoets ε i with coefficiet 1 i µ, where 0 i. Sice the compoets ε i of λ have positive coefficiets 1 where 0 i, ad 1 whe i = 1, λ µ results i a odd umber of coefficiets of 1 for ε i where 0 i, ad oe coefficiet of 1 for ε 1, whereas the rest of the coefficiets are 0. This is clearly a oegative iteger liear combiatio of the positive roots of the sets b, m Φ D, defied i Propositio 1.1 ad derived i Propositio 1.3. Sice by Defiitio 1.19 the positive roots are i tur a oegative iteger liear combiatio of simple roots, this satisfies Defiitio.11. It remais to check the case where the coefficiet of ε 1 i µ is 1. I this case, µ cotais a eve umber of occureces of 1 ε i, such that 0 i. Therefore, the compoets of λ µ have a eve umber of coefficiets of 1, whereas all other coefficiets are 0. Thus, this is clearly a oegative iteger 19

21 liear combiatio of the positive roots i the set b Φ D, defied i Propositio 1.1. The rest of the proof follows the same argumet as the previous case, thereby satisfyig Defiitio.11 as required. We ow have all the tools to explicitly costruct the modules. Theorem.3. (i) The module V ΨB is the spi module of type of type B. (ii) The module V ΨB is irreducible. Proof. By Theorem.10, V ΨB is a module for the simple Lie algebra of type B. It has dimesio because it cosists of vector compoets each with a positive or egative coefficiet of 1. By Corollary.0, the highest weight vector of V ΨB is equal to the highest weight vector of the spi module defied i Defiitio.14 ad derived i Lemma.15. Therefore, by Propositio.13 (ii), the modules are isomorphic. Furthermore, by Propositio.13 (i), V ΨB is irreducible. Theorem.4. (i) The modules V Ψ + ad V Ψ are the two spi modules of type D. D D (ii) The modules V Ψ + ad V Ψ are irreducible. D D (iii) The modules V Ψ + ad V Ψ are oisomorphic. D D Proof. The dimesio of V Ψ + D ad V Ψ D is = 1 as they each have a odd ad eve umber of occurreces of 1 ε i, respectively. The proof for parts (i) ad (ii) follows the same argumet as Theorem.3. By Propositio.13 (i), the modules are oisomorphic. We ote that the spi modules V Ψ + D ad V Ψ D are ot defied for D whe is odd, as by Lemma.19, they are ot closed uder egatio. 3 Mai Result I this sectio we defie a biliear form o the spi modules costructed i the previous sectio. We show that this biliear form is g-ivariat, ad is the uique g-ivariat biliear form up to multiplicatio by a ozero scalar o these modules. We the itroduce our origial results. We defie equatios for the height of weights of the spi modules ad use them to prove that the biliear form alterates betwee orthogoal ad symplectic depedig o the dimesio of its uderlyig Lie algebra. Defiitio 3.1. A biliear form o a k-vector space V is a map that is liear i both compoets. B V V k 0

22 Defiitio 3.. Let B be a biliear form o a k-vector space V. The radical of B is the set rad(b) = {v V B(v, w) = 0 for all w V }. A biliear form B is odegeerate if rad(b) = 0. Example 3.3. Let V = R be the euclidea vector space of dimesio. The usual dot product is a biliear form o V. Furthermore, it is odegeerate as the oly elemet orthogoal to every vector is 0. Defiitio 3.4. Let g be a Lie algebra over a field k with characteristic, ad let B be a biliear form o the g-module V. (i) The biliear form B is said to be g-ivariat if B(g v, w) + B(v, g w) = 0 for all g g, v, w V. (ii) The biliear form B is said to be orthogoal if it is odegeerate ad B(v, w) = B(w, v) for all v, w V. (iii) The biliear form B is said to be symplectic if it is odegeerate ad B(v, w) = B(w, v) for all v, w V. Lemma 3.5. Suppose that the k-vector space V is a g-module, ad suppose B is a g-ivariat biliear form o V. The rad(b) is a submodule of V. Proof. It is clear that rad(b) is a subspace of V, because B is liear i the first etry. Therefore, the proof reduces to showig that the subspace is ivariat uder the actio of the elemets of g. Suppose that r rad(b), g g, ad w V. The B(g r, w) = B(r, g w) = 0, completig the proof. Lemma 3.6. Let b Ψ be a basis for the g-module V Ψ over k, ad let S be the set of geerators for g. Suppose B is a biliear form o V Ψ such that The B is g-ivariat. B(s b λ, b µ ) + B(b λ, s b µ ) = 0 for all s S, b λ, b µ b Ψ. Proof. We show that the above property holds o the bracket of geerators of g. Sice every elemet of g ca be expressed by iterated liear combiatios ad brackets of the elemets of S, ad every elemet of V Ψ ca be expressed as a liear combiatio of the elemets of b Ψ, it follows by iductio ad biliearity i B, ad [, ], that B is g-ivariat. Let s, t S, ad u, v b Ψ. We have: B([s, t] v, w) + B(v, [s, t] w) = B((s t t s) v, w) + B(v, (s t t s) w) = B(s t v, w) B(v, t s w) + B(v, s t w) B(t s v, w) = B(s t v, w) B(v, t s w) + B(v, s t w) B(t s v, w) = B((s t v, w) + B(t v, s w)) (B(t v, s w) + B(v, t s w)) + (B(v, s t w) + B(s v, t w)) (B(s v, t w) + B(t s v, w)) = 0, completig the proof. 1

23 Defiitio 3.7. Let µ be a elemet of oe of the sets Ψ B, Ψ + D, or Ψ D, with highest weight λ h. We defie the height of µ, deoted µ, such that µ = a i, where µ + λ h = a i α i. α i α i Defiitio 3.7 itroduces ucovetioal otatio for the height of µ for coveiece, which is well defied i this special case due to Lemma.19. Corollary 3.8. Maitai the otatio of Defiitio 3.7. The height of µ is give by Proof. By Defiitio 3.7, µ = implyig as required. µ = λ h µ α i a i α i = µ + λ h α i a i such that = (λ h + λ h ) (µ + λ h ), a i = λ h µ α i Defiitio 3.9. Let Ψ be a miuscule system with respect to the simple system. We defie a biliear form o V Ψ by specifyig its effect o the elemets of the basis b Ψ of V Ψ, such that for all b λ, b µ b Ψ, ( 1) λ if λ = µ; B g (b λ, b µ ) = 0 otherwise. Propositio The biliear form B g is g-ivariat. Proof. Let S be the set of Serre geerators of g. By Lemma 3.6, it is sufficiet to show that the property B g (s b λ, b µ )+B g (b λ, s b µ ) = 0 is satisfied for all s S ad b λ, b µ b Ψ. We prove this i three parts, by showig that the coditio is satisfied for all h αi, e αi, ad f αi S, respectively. The first part reduces to a cases by case check o the value of λ relative to µ. There are two cases to check. First, suppose that λ µ. We the have B g (h αi b λ, b µ ) + B g (b λ, h αi b µ ) = λ, α i B g (b λ, b µ ) + µ, α i B g (b λ, b µ ) = λ, α i (0) + µ, α i (0) = 0,

24 as required by Defiitio 3.9. For the secod case, assume that λ = λ. We have Thus, B g (h αi b λ, b λ ) = λ, α i B g (b λ, b λ ) = λ, α i B g (b λ, b λ ) = B g (b λ, h αi b λ ), B g (h αi b λ, b λ ) + B g (b λ, h αi b λ ) = 0 as required by Defiitio 3.9. We proceed to check that the coditio is satisfied for all e αi S. As above, this reduces ito two cases accordig to the values of λ i relatio to µ. First, suppose for all i that λ + α i µ. This is equivalet to λ µ + α i. We have B g (e αi b λ, b µ ) + B g (b λ, e αi b µ ) = B g (b λ+αi, b µ ) + B g (b λ, b µ+αi ) by Defiitio 3.9. Now, suppose that for all i, λ + α i = µ. We the have = 0 B g (e αi b λ, b (λ+αi)) = B g (b λ+αi, b (λ+αi)) = ( 1) λ+αi = ( 1) λ +1 = ( 1) λ = B g (b λ, b λ ) = B g (b λ, e αi b (λ+αi)), as required by Defiitio 3.9. The proof for the elemets f αi argumet as for e αi, which completes the proof. follows the same Propositio The biliear form B g is odegeerate o V ΨB, V Ψ +, ad D V Ψ. D Proof. Let V Ψ deote V ΨB, V Ψ + or V Ψ, ad suppose for a cotradictio that D D rad(b g ) 0. By Propositio 3.10, B g is g-ivariat. Therefore, by Lemma 3.5 rad(b g ) is a submodule of V Ψ. However, by Theorems.4 ad.3, V Ψ is irreducible, which by assumptio implies that rad(b g ) = V Ψ. By Defiitios.17 ad.18, V Ψ is ot zero, ad by Corollary.19 it is closed uder egatio. Therefore, there exists a elemet b λ V Ψ such that B g (b λ, b λ ) = ( 1) λ 0, which is a cotradictio that implies rad(b g ) = 0, as required by Defiitio Theorem 3.1. The biliear form B g is the uique g-ivariat biliear form o V Ψ up to multiplicatio by a ozero scalar. Proof. We prove this i two steps, first by showig g ivariace imposes the coditio B g (b λ, b µ ) = 0 uless λ = µ. We the show that B g alterates sig as 3

25 the weights icrease or decrease by addig or subtractig a simple root, which satisfies Defiitio 3.9. Suppose B is a g-ivariat biliear form o V Ψ. The we must have This implies B g (h αi b λ, b µ ) + (b λ, h αi b µ ) = 0 for all h αi S ad b λ, b µ b Ψ. 0 = λ, α i B g (b λ, b µ ) + µ, α i B g (b λ, b µ ) = λ + µ, α i B g (b λ, b µ ). Sice the dot product, is odegeerate ad the elemets α i form a basis, this implies or 0 = B g (b λ, b µ ), 0 = λ + µ λ = µ, as required. Now, suppose that λ α i Ψ where α i. The actio of e αi S gives the followig: B g (b λ, b λ ) = B g (e αi b λ αi, b λ ) = B g (b λ αi, e αi b λ ) = B g (b λ αi, b (λ αi)). By Lemmas.0,.1, ad., every elemet µ Ψ ca be writte as the highest weight λ mius a oegative iteger liear combiatio of positive roots, c i Z. Therefore, it follows by iductio that as required. B g (b µ, b µ ) = ( 1) ci B g (b λ, b λ ) = ( 1) µ B g (b λ, b λ ) for all µ Ψ, The followig results are origial. Defiitio Let Ψ be a miuscule system with respect to the simple system, ad let λ Ψ. We defie the followig: i if ε i i λ has coefficiet of positive 1 b λi =, 0 otherwise. i if ε i 1 i λ has coefficiet of positive 1 d λi =, 0 otherwise. 4

26 Propositio Let λ Ψ B. The height of λ is give by the fuctio f(λ) = b λi. Proof. Let λ Ψ B, ad let k equal the height of λ. We prove our claim by iductio o k. Let λ l Ψ deote the lowest weight. For the base case k = 0, we show that that f(λ l ) = 0. The weight λ l = 1 1 ε i. Sice every coefficiet i=0 of ε i i λ l is egative, by Defiitio 3.13 every umber b λi = 0, ad therefore f(λ) = 0 = k, as required. For the iductive step, suppose that for some k > 0, k equals the height of λ, ad that f(λ) = k. We wish to show that there is a α i such that λ α i Ψ B, ad such that f(λ α i ) = k 1. By Defiitio.17, λ = ± 1 ε i. This is a case by case check accordig to the i=0 sig of coefficiets of the compoet vectors of λ. There are two cases to check. First, suppose that the sig of all of the coefficiets is positive. The subtractig the root α 1 from λ yields ( 1 ε ε 1) ε 1 = 1 ε ε 1 ε 1 Ψ B. Sice f is defied i terms of the value of coefficiets of compoets, we ca restrict our aalysis oly to those coefficiets that chage. The oly coefficiets which chaged were those of ε 1. Therefore, by Defiitio 3.13, ad b λ 1 = 1 ad b (λ α 1) 1 = 0, f(λ α 1 ) = f(λ) 1 = k 1, as required. For the secod case, suppose that λ has coefficiets of compoets ot all positive. The there exist compoets i λ such that 1 ε i 1 ε i+1 for some 0 i <, or 1 ε 1. This is a case by case check o each compoet. There are two cases. The first case for 1 ε 1 follows a similar argumet as case oe above, satisfyig f. For the secod case, suppose λ has compoets 1 ε i 1 ε i+1. We claim that λ α i Ψ B. As above, we ca restric our aalysis oly to compoets that chage. A direct check shows ( 1 ε i 1 ε i+1) ( 1 ε i 1 ε i+1) = 1 ε i + 1 ε i+1 Ψ B, ad b λi = i, b λi+1 = 0, b (λ αi) i = 0, b (λ αi) i+1 = i 1 Thus, f(λ) = f(λ α i ) 1 = k 1 as required, exhaustig all possibilities ad completig the proof. 5

27 Propositio Let λ Ψ B. The height of λ is give by f( λ) = i b λi. Proof. By Corollary 3.8, the height of λ is give by λ = λ h λ, where λ h is the highest weight. Therefore, usig the equatio of Propositio 3.14, λ = λ h λ = i b λi i=0 = i b λi. Example Let λ Ψ B, where B is such that = 5. Suppose λ = 1 ε ε 1 ε 3 1 ε ε 5. The b λ1 = 0, b λ = 4, b λ1 = 0, b λ1 = 0, b λ1 = 1, ad the heights of λ ad λ are respectively. λ = f(λ) = b λi = = 5 λ = f( λ) = i b λi = 1 + ( 4) (5 1) = 10, Propositio Let λ Ψ + D or Ψ D. The height of λ is give by g(λ) = 1 d λi. i=0 Proof. The proof follows a similar argumet to Propositio 3.14, ad is therefore omitted. Propositio Let λ Ψ + D or Ψ D. The height of λ is give by g( λ) = 1 i d λi. i=0 Proof. The proof follows a argumet similar to Propositio 3.15, ad is therefore omitted. Lemma Let Z. If = 0 or 3 (mod 4), the ()(+1) = 0 (mod ). If = 1 or (mod 4), the ()(+1) = 1 (mod ). Proof. This is a case by case check o the value of. There are four cases to check. Let = 0 (mod 4). The = 4x for some x Z. We the have, (4x)(4x + 1) = (x)(4x + 1) = 0 (mod ) 6

28 . Let = 3 (mod 4). The = 4x + 3. Thus, (4x + 3)(4x + 4) Let = 1 (mod 4). The = 4x + 1. Thus, = (4x + 3)(x + ) = (4x + 3)(x + 1) = 0 (mod ). (4x + 1)(4x + ) = (4x + 1)(x + 1) = 1 (mod ) as each compoet of the product is odd. Fially, let = (mod 4). The = 4x +. Thus, (4x + )(4x + 3) as above, completig the proof. = (x + 1)(4x + 3) = 0 (mod ) Lemma 3.0. Let λ Ψ B, for the orthogoal Lie algebra of type B. (i) If = 0 or 3 (mod 4), the λ = λ (mod ). (ii) If = 1 or (mod 4), the λ = 1 + λ (mod ). Proof. (i) Let = 0 or 3 (mod 4). By Propositio 3.14 we have by Lemma 3.19, by Propositio 3.15, λ = p λi (mod ) = 0 p λi (mod ), = ( + 1)() p λi (mod ) = i p λi (mod ) = i p λi (mod ), = λ (mod ) which proves part (i). (ii) Now, let = 1 or (mod 4). By Propositio 3.14 we have λ = p λi (mod ) = 0 p λi (mod ), 7

29 by Lemma 3.19, = 1 + ( + 1)() p λi (mod ) = 1 + i p λi (mod ) = 1 + i p λi (mod ), by Propositio 3.15, = 1 + λ (mod ), which proves part (ii) ad completes the proof. Lemma 3.1. Let λ Ψ + D or Ψ D, for the orthogoal Lie algebra of type D. (i) If = 0 (mod 4), the λ = λ (mod ). (ii) If = (mod 4), the λ = 1 + λ (mod ). Proof. The proof follows a argumet similar to Lemma 3.0, ad is therefore omitted. We ow have all the tools to prove our mai results. Theorem 3.. Let g be the simple Lie algebra of type B, ad let V ΨB be its correspodig spi module. Let B g be the g-ivariat biliear form o V ΨB. (i) If = 0 or 3 (mod 4), B g is orthogoal. (ii) If = 1 or (mod 4), B g is symplectic. Proof. (i) Let = 0 or 3 (mod 4). The by Lemma 3.0 we have B g (b λ, b λ ) = ( 1) λ λ (mod ) = ( 1) λ (mod ) = ( 1) = ( 1) λ = B g (b λ, b λ ). By Propositio 3.11, B g is odegeerate o V ΨB. Therefore, B g is orthogoal. 8

30 (ii)let = 1 or (mod 4). The by Lemma 3.0 we have, B g (b λ, b λ ) = ( 1) λ λ (mod ) = ( 1) 1+ λ (mod ) = ( 1) = ( 1) 1+ λ = ( 1) λ = B g (b λ, b λ ). By Propositio 3.11, B g is odegeerate o V ΨB. Therefore, B g is symplectic. Theorem 3.3. Let g be the simple Lie algebra of type D, where is eve, ad let V Ψ + ad V Ψ be its correspodig spi modules. Let B g be the g-ivariat D D biliear form o V Ψ + ad V Ψ. D D (i) If = 0 (mod 4), B g is orthogoal. (ii) If = (mod 4), B g is symplectic. Proof. (i) Let = 0 (mod 4). The by Lemma 3.1 we have B g (b λ, b λ ) = ( 1) λ λ (mod ) = ( 1) λ (mod ) = ( 1) = ( 1) λ = B g (b λ, b λ ). By Propositio 3.11, B g is odegeerate o V Ψ + D orthogoal. (ii)let = (mod 4). The by Lemma 3.1 we have B g (b λ, b λ ) = ( 1) λ λ (mod ) = ( 1) 1+ λ (mod ) = ( 1) = ( 1) 1+ λ = ( 1) λ = B g (b λ, b λ ). By Propositio 3.11, B g is odegeerate o V Ψ + D symplectic. ad V Ψ D. Therefore, B g is ad V Ψ D. Therefore, B g is 9

31 Refereces [1] R. M. Gree, Represetatios of Lie algebras Arisig From Polytopes, Iteratioal Electroic Joural of Algebra, 008. [] R. M. Gree, Combiatorics of Miiscule Represetatios, Cambridge Uiversity Press, 013. [3] K. Erdma ad M. J. Wildo, Itroductio to Lie algebras, Spriger, 006. [4] J. E. Humphreys, Itroductio to Lie algebras ad Represetatio Theory, Spriger, 197. [5] R. Carter, Lie Algebras of Fiite ad Affie Type, Cambridge Uiversity Press, 005. [6] K. Ito, Ecyclopedic Dictioary of Mathematics, Volume 1, The MIT Press,