A Cartan subalgebra of g is a maximal abelian subalgebra h Ă g consisting of semisimple (diagonalizeable) elements. (Analogous to Jucys-Murphy

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1 A Cartan subalgebra of g is a maximal abelian subalgebra h Ă g consisting of semisimple (diagonalizeable) elements. (Analogous to Jucys-Murphy elements!!)

2 A Cartan subalgebra of g is a maximal abelian subalgebra h Ă g consisting of semisimple (diagonalizeable) elements. (Analogous to Jucys-Murphy elements!!) Example: sl n has basis tx ij E ij, y ij E ji, h l E ll E l`1,l`1 1 ď i ă j ď n, 1 ď l ď n 1u. Our favorite Cartan subalgebra is h Cth l l 1,..., n 1u.

3 A Cartan subalgebra of g is a maximal abelian subalgebra h Ă g consisting of semisimple (diagonalizeable) elements. (Analogous to Jucys-Murphy elements!!) Example: sl n has basis tx ij E ij, y ij E ji, h l E ll E l`1,l`1 1 ď i ă j ď n, 1 ď l ď n 1u. Our favorite Cartan subalgebra is h Cth l l 1,..., n 1u. The weights of a Cartan h is the dual set h tµ : h Ñ Cu.

4 A Cartan subalgebra of g is a maximal abelian subalgebra h Ă g consisting of semisimple (diagonalizeable) elements. (Analogous to Jucys-Murphy elements!!) Example: sl n has basis tx ij E ij, y ij E ji, h l E ll E l`1,l`1 1 ď i ă j ď n, 1 ď l ď n 1u. Our favorite Cartan subalgebra is h Cth l l 1,..., n 1u. The weights of a Cartan h is the dual set h tµ : h Ñ Cu. Example: For i 1,..., n, define ε i : h Ñ C by h ÞÑ trpe ii hq, (i.e. it picks the ith diagonal element).

5 A Cartan subalgebra of g is a maximal abelian subalgebra h Ă g consisting of semisimple (diagonalizeable) elements. (Analogous to Jucys-Murphy elements!!) Example: sl n has basis tx ij E ij, y ij E ji, h l E ll E l`1,l`1 1 ď i ă j ď n, 1 ď l ď n 1u. Our favorite Cartan subalgebra is h Cth l l 1,..., n 1u. The weights of a Cartan h is the dual set h tµ : h Ñ Cu. Example: For i 1,..., n, define ε i : h Ñ C by h ÞÑ trpe ii hq, (i.e. it picks the ith diagonal element). So ε i ph l q δ i,l δ i,l`1.

6 A Cartan subalgebra of g is a maximal abelian subalgebra h Ă g consisting of semisimple (diagonalizeable) elements. (Analogous to Jucys-Murphy elements!!) Example: sl n has basis tx ij E ij, y ij E ji, h l E ll E l`1,l`1 1 ď i ă j ď n, 1 ď l ď n 1u. Our favorite Cartan subalgebra is h Cth l l 1,..., n 1u. The weights of a Cartan h is the dual set h tµ : h Ñ Cu. Example: For i 1,..., n, define ε i : h Ñ C by h ÞÑ trpe ii hq, (i.e. it picks the ith diagonal element). So Note ε 1 ` ` ε n 0. ε i ph l q δ i,l δ i,l`1.

7 A Cartan subalgebra of g is a maximal abelian subalgebra h Ă g consisting of semisimple (diagonalizeable) elements. (Analogous to Jucys-Murphy elements!!) Example: sl n has basis tx ij E ij, y ij E ji, h l E ll E l`1,l`1 1 ď i ă j ď n, 1 ď l ď n 1u. Our favorite Cartan subalgebra is h Cth l l 1,..., n 1u. The weights of a Cartan h is the dual set h tµ : h Ñ Cu. Example: For i 1,..., n, define ε i : h Ñ C by h ÞÑ trpe ii hq, (i.e. it picks the ith diagonal element). So ε i ph l q δ i,l δ i,l`1. Note ε 1 ` ` ε n 0. So we have h Ctε l ε l`1 l 1,..., n 1u.

8 Let x, y be a NIBS form on g. Then the map h ÝÑ h h ÞÑ xh, y h µ ÞÑ µ is an isomorphism, where h µ is the unique element of h such that xh µ, hy µphq for all h P h. Define x, y : h b h Ñ C by xµ, λy xh µ, h λ y.

9 Let g act on itself via ad, and define g α tg P g ad h pxq αphqxu. The set of weights R tα P h α 0, g α 0u is called the set of roots of g (i.e. these are the non-zero simultaneous eigenspaces inside the left-regular (adjoint) module).

10 Let g act on itself via ad, and define g α tg P g ad h pxq αphqxu. The set of weights R tα P h α 0, g α 0u is called the set of roots of g (i.e. these are the non-zero simultaneous eigenspaces inside the left-regular (adjoint) module). Example: sl 3 has basis x 1 E 1,2, x 2 E 2,3, x 3 E 1,3 h 1 E 1,1 E 2,2, h 2 E 2,2 E 3,3, y 1 E 2,1, y 2 E 3,2, y 3 E 3,1 You try: Compute rh l, x j s and rh l, y j s for j 1, 2, 3, and compute R in terms of ε i s (again, where ε i ph l q trpe ii h l q δ i,l δ i,l`1 q and g α for each α P R.

11 Let g act on itself via ad, and define g α tg P g ad h pxq αphqxu. The set of weights R tα P h α 0, g α 0u is called the set of roots of g (i.e. these are the non-zero simultaneous eigenspaces inside the left-regular (adjoint) module). Example: sl 3 has basis x 1 E 1,2, x 2 E 2,3, x 3 E 1,3 h 1 E 1,1 E 2,2, h 2 E 2,2 E 3,3, y 1 E 2,1, y 2 E 3,2, y 3 E 3,1 You try: Compute rh l, x j s and rh l, y j s for j 1, 2, 3, and compute R in terms of ε i s (again, where ε i ph l q trpe ii h l q δ i,l δ i,l`1 q and g α for each α P R. Note: As a vector space, Usl 3 pcq Ctpy λ 1 1 yλ 2 2 yλ 3 3 qphλ 4 1 hλ 5 2 qpxλ 6 1 xλ 7 2 xλ 8 3 qˇˇλ i P Z ě0 u U U 0 U `, where U is lower triangular, U 0 is diagonal, and U ` is upper triangular. This is called a triangular decomposition.

12 In general, for sl n pcq, letting α i,j ε i ε j, we have R t α i,j 1 ď i ă j ď nu and g αij CtE ij u for i j, and g 0 h.

13 In general, for sl n pcq, letting α i,j ε i ε j, we have R t α i,j 1 ď i ă j ď nu and g αij CtE ij u for i j, and g 0 h. In sl 3, we had R t pε i ε j q 1 ď i ă j ď 3u

14 In general, for sl n pcq, letting α i,j ε i ε j, we have R t α i,j 1 ď i ă j ď nu and g αij CtE ij u for i j, and g 0 h. In sl 3, we had R t pε i ε j q 1 ď i ă j ď 3u t β 1, β 2, pβ 1 ` β 2 qu where β 1 ε 1 ε 2 and β 2 ε 2 ε 3.

15 In general, for sl n pcq, letting α i,j ε i ε j, we have R t α i,j 1 ď i ă j ď nu and g αij CtE ij u for i j, and g 0 h. In sl 3, we had R t pε i ε j q 1 ď i ă j ď 3u t β 1, β 2, pβ 1 ` β 2 qu where β 1 ε 1 ε 2 and β 2 ε 2 ε 3. We call subset of B Ă R a base of R is every element of R can be written as (positive integral combination of elements of B).

16 In general, for sl n pcq, letting α i,j ε i ε j, we have R t α i,j 1 ď i ă j ď nu and g αij CtE ij u for i j, and g 0 h. In sl 3, we had R t pε i ε j q 1 ď i ă j ď 3u t β 1, β 2, pβ 1 ` β 2 qu where β 1 ε 1 ε 2 and β 2 ε 2 ε 3. We call subset of B Ă R a base of R is every element of R can be written as (positive integral combination of elements of B). In our example, B tβ 1, β 2 u is a base.

17 In general, for sl n pcq, letting α i,j ε i ε j, we have R t α i,j 1 ď i ă j ď nu and g αij CtE ij u for i j, and g 0 h. In sl 3, we had R t pε i ε j q 1 ď i ă j ď 3u t β 1, β 2, pβ 1 ` β 2 qu where β 1 ε 1 ε 2 and β 2 ε 2 ε 3. We call subset of B Ă R a base of R is every element of R can be written as (positive integral combination of elements of B). In our example, B tβ 1, β 2 u is a base. In particular, since R sits all within a 2-dimensional space, with xε 1 ε 2, ε 2 ε 3 y 1 β 1 =β 2 arccos a arccos xε1 ε 2, ε 1 ε 2 yxε 2 ε 3, ε 2 ε 3 y 2 2π 3

18 In sl 3, we had R t pε i ε j q 1 ď i ă j ď 3u t β 1, β 2, pβ 1 ` β 2 qu where β 1 ε 1 ε 2 and β 2 ε 2 ε 3. We call subset of B Ă R a base of R is every element of R can be written as (positive integral combination of elements of B). In our example, B tβ 1, β 2 u is a base. In particular, since R sits all within a 2-dimensional space, with xε 1 ε 2, ε 2 ε 3 y 1 β 1 =β 2 arccos a arccos xε1 ε 2, ε 1 ε 2 yxε 2 ε 3, ε 2 ε 3 y 2 2π 3, we can plot R as β 2 β 1 ` β 2 -β 1 β 1 pβ 1 ` β 2 q β 2

19 In sl 3, we had R t pε i ε j q 1 ď i ă j ď 3u t β 1, β 2, pβ 1 ` β 2 qu where β 1 ε 1 ε 2 and β 2 ε 2 ε 3. We call subset of B Ă R a base of R is every element of R can be written as (positive integral combination of elements of B). In our example, B tβ 1, β 2 u is a base. In particular, since R sits all within a 2-dimensional space, with xε 1 ε 2, ε 2 ε 3 y 1 β 1 =β 2 arccos a arccos xε1 ε 2, ε 1 ε 2 yxε 2 ε 3, ε 2 ε 3 y 2 2π 3, we can plot R as β 2 β 1 ` β 2 Negative roots, R (correspond to lower triangular root spaces) -β 1 pβ 1 ` β 2 q β 2 β 1 Positive roots, R` (correspond to upper triangular root spaces)

20 Notice, in sl n pcq, for all i ă j, the elements x i,j E i,j, y i,j E j,i, h i,j E i,i E j,j generate a Lie subalgebra isomorphic to sl 2 pcq.

21 Notice, in sl n pcq, for all i ă j, the elements x i,j E i,j, y i,j E j,i, h i,j E i,i E j,j generate a Lie subalgebra isomorphic to sl 2 pcq. Moreover, for all we have h c 1 E 1,1 ` ` c n E n,n P h (with c 1 ` ` c n 0) rh, x i,j s pc i c j qx i,j and rh, y i,j s pc i c j qy i,j.

22 Notice, in sl n pcq, for all i ă j, the elements x i,j E i,j, y i,j E j,i, h i,j E i,i E j,j generate a Lie subalgebra isomorphic to sl 2 pcq. Moreover, for all we have h c 1 E 1,1 ` ` c n E n,n P h (with c 1 ` ` c n 0) rh, x i,j s pc i c j qx i,j and rh, y i,j s pc i c j qy i,j. Let M be a finite-dimensional simple sl n pcq-module.

23 Notice, in sl n pcq, for all i ă j, the elements x i,j E i,j, y i,j E j,i, h i,j E i,i E j,j generate a Lie subalgebra isomorphic to sl 2 pcq. Moreover, for all we have h c 1 E 1,1 ` ` c n E n,n P h (with c 1 ` ` c n 0) rh, x i,j s pc i c j qx i,j and rh, y i,j s pc i c j qy i,j. Let M be a finite-dimensional simple sl n pcq-module. 1. h is simultaneously diagonalizable, so pick a simultaneous weight vector of all of h.

24 Notice, in sl n pcq, for all i ă j, the elements x i,j E i,j, y i,j E j,i, h i,j E i,i E j,j generate a Lie subalgebra isomorphic to sl 2 pcq. Moreover, for all we have h c 1 E 1,1 ` ` c n E n,n P h (with c 1 ` ` c n 0) rh, x i,j s pc i c j qx i,j and rh, y i,j s pc i c j qy i,j. Let M be a finite-dimensional simple sl n pcq-module. 1. h is simultaneously diagonalizable, so pick a simultaneous weight vector of all of h. 2. If v is an weight vector, then so are x i,j v and y i,j v.

25 Notice, in sl n pcq, for all i ă j, the elements x i,j E i,j, y i,j E j,i, h i,j E i,i E j,j generate a Lie subalgebra isomorphic to sl 2 pcq. Moreover, for all we have h c 1 E 1,1 ` ` c n E n,n P h (with c 1 ` ` c n 0) rh, x i,j s pc i c j qx i,j and rh, y i,j s pc i c j qy i,j. Let M be a finite-dimensional simple sl n pcq-module. 1. h is simultaneously diagonalizable, so pick a simultaneous weight vector of all of h. 2. If v is an weight vector, then so are x i,j v and y i,j v. 3. Thinking about Res sln sl 2 pmq for each i, j, each h i,j must act on v by an integer; and x i,j v is an weight vector of weight 2 more than v.

26 Notice, in sl n pcq, for all i ă j, the elements x i,j E i,j, y i,j E j,i, h i,j E i,i E j,j generate a Lie subalgebra isomorphic to sl 2 pcq. Moreover, for all we have h c 1 E 1,1 ` ` c n E n,n P h (with c 1 ` ` c n 0) rh, x i,j s pc i c j qx i,j and rh, y i,j s pc i c j qy i,j. Let M be a finite-dimensional simple sl n pcq-module. 1. h is simultaneously diagonalizable, so pick a simultaneous weight vector of all of h. 2. If v is an weight vector, then so are x i,j v and y i,j v. 3. Thinking about Res sln sl 2 pmq for each i, j, each h i,j must act on v by an integer; and x i,j v is an weight vector of weight 2 more than v. Specifically, if hv µphqv for µ P h, then hx i,j v pµ ` α i,j qphqx i,j v.

27 Notice, in sl n pcq, for all i ă j, the elements x i,j E i,j, y i,j E j,i, h i,j E i,i E j,j generate a Lie subalgebra isomorphic to sl 2 pcq. Moreover, for all we have h c 1 E 1,1 ` ` c n E n,n P h (with c 1 ` ` c n 0) rh, x i,j s pc i c j qx i,j and rh, y i,j s pc i c j qy i,j. Let M be a finite-dimensional simple sl n pcq-module. 1. h is simultaneously diagonalizable, so pick a simultaneous weight vector of all of h. 2. If v is an weight vector, then so are x i,j v and y i,j v. 3. Thinking about Res sln sl 2 pmq for each i, j, each h i,j must act on v by an integer; and x i,j v is an weight vector of weight 2 more than v. Specifically, if hv µphqv for µ P h, then hx i,j v pµ ` α i,j qphqx i,j v. 4. So there s a highest weight vector v` λ such that x i,jv` 0 for all i ă j and h i,j v` λ λ i,jv` λ. (Also, we know λ i,j P Z ě0.)

28 Let h α be the hyperplane (vector space of codimension 1) in h perpendicular to α. h β1 h β1`β 2 h β2 β 2 β 1 ` β 2 β 1

29 Let h α be the hyperplane (vector space of codimension 1) in h perpendicular to α. h β1 ` h β1`β 2 ` ` h β2

30 Let h α be the hyperplane (vector space of codimension 1) in h perpendicular to α. h β1 ` h β1`β 2 ` ` h β2

31 Let h α be the hyperplane (vector space of codimension 1) in h perpendicular to α. h β1 h β1`β 2 h β2

32 Let h α be the hyperplane (vector space of codimension 1) in h perpendicular to α. h β1 h β1`β 2 h β2

33 Let h α be the hyperplane (vector space of codimension 1) in h perpendicular to α. h β1 h β1`β 2 h β2

34 Let h α be the hyperplane (vector space of codimension 1) in h perpendicular to α. h β1 h β1`β 2 h β2 highest weight µ

35 Let h α be the hyperplane (vector space of codimension 1) in h perpendicular to α. h β1 h β1`β 2 h β2 highest weight µ

36 Let h α be the hyperplane (vector space of codimension 1) in h perpendicular to α. h β1 h β1`β 2 h β2 highest weight µ

37 Let h α be the hyperplane (vector space of codimension 1) in h perpendicular to α. h β1 h β1`β 2 h β2 highest weight µ

38 Let h α be the hyperplane (vector space of codimension 1) in h perpendicular to α. h β1 h β1`β 2 h β2 highest weight µ

39 Let h α be the hyperplane (vector space of codimension 1) in h perpendicular to α. h β1 h β1`β 2 h β2 highest weight µ Let W be the group generated by reflections across the hyperplanes h α.

40 Let h α be the hyperplane (vector space of codimension 1) in h perpendicular to α. h β1 h β1`β 2 h β2 highest weight µ Orbit W µ Let W be the group generated by reflections across the hyperplanes h α.

41 Let h α be the hyperplane (vector space of codimension 1) in h perpendicular to α. h β1 h β1`β 2 h β2 highest weight µ Orbit W µ Let W be the group generated by reflections across the hyperplanes h α. W is called the Weyl group of sl 3 pcq.

42 Let h α be the hyperplane (vector space of codimension 1) in h perpendicular to α. h β1 h β1`β 2 h β2 highest weight µ Orbit W µ Let W be the group generated by reflections across the hyperplanes h α. W is called the Weyl group of sl 3 pcq. Fact: The Weyl group of sl n pcq is S n.

43 Some stuff about simple complex Lie algebras g...

44 Some stuff about simple complex Lie algebras g Every simple g-module is a highest-weight module.

45 Some stuff about simple complex Lie algebras g Every simple g-module is a highest-weight module. 2. Every Lie algebra has a set of roots R, and every R has a base and an associated Weyl group W. [See Coxeter group.]

46 Some stuff about simple complex Lie algebras g Every simple g-module is a highest-weight module. 2. Every Lie algebra has a set of roots R, and every R has a base and an associated Weyl group W. [See Coxeter group.] For type A r, W S r`1 ; for B r &C r, W is the signed reflection group.

47 Some stuff about simple complex Lie algebras g Every simple g-module is a highest-weight module. 2. Every Lie algebra has a set of roots R, and every R has a base and an associated Weyl group W. [See Coxeter group.] For type A r, W S r`1 ; for B r &C r, W is the signed reflection group. 3. Every base B tβ 1,..., β r u is a basis of h. Its dual basis Ω tω 1,..., ω r u has the property that P ZΩ are the weights appearing in f.d. l modules, and P ` Z ě0 Ω are the weights appearing as highest weights. [See: fundamental weights, integral weights.]

48 Some stuff about simple complex Lie algebras g Every simple g-module is a highest-weight module. 2. Every Lie algebra has a set of roots R, and every R has a base and an associated Weyl group W. [See Coxeter group.] For type A r, W S r`1 ; for B r &C r, W is the signed reflection group. 3. Every base B tβ 1,..., β r u is a basis of h. Its dual basis Ω tω 1,..., ω r u has the property that P ZΩ are the weights appearing in f.d. l modules, and P ` Z ě0 Ω are the weights appearing as highest weights. [See: fundamental weights, integral weights.] 4. For type A r, B r, C r, D r, there s a bijection between P ` and integer partitions. Further, for λ P P `, the dimensions of the weight spaces appearing in Lpλq can be computed combinatorially. [See: semistandard tableaux; path model.]

49 Some stuff about simple complex Lie algebras g Every simple g-module is a highest-weight module. 2. Every Lie algebra has a set of roots R, and every R has a base and an associated Weyl group W. [See Coxeter group.] For type A r, W S r`1 ; for B r &C r, W is the signed reflection group. 3. Every base B tβ 1,..., β r u is a basis of h. Its dual basis Ω tω 1,..., ω r u has the property that P ZΩ are the weights appearing in f.d. l modules, and P ` Z ě0 Ω are the weights appearing as highest weights. [See: fundamental weights, integral weights.] 4. For type A r, B r, C r, D r, there s a bijection between P ` and integer partitions. Further, for λ P P `, the dimensions of the weight spaces appearing in Lpλq can be computed combinatorially. [See: semistandard tableaux; path model.] 5. The weights appearing with non-zero multiplicity in Lpλq are a subset of the convex hull of the orbit W λ. [See MV polytopes.]

Theorem The simple finite dimensional sl 2 modules Lpdq are indexed by

The Lie algebra sl 2 pcq has basis x, y, and h, with relations rh, xs 2x, rh, ys 2y, and rx, ys h. Theorem The simple finite dimensional sl 2 modules Lpdq are indexed by d P Z ě0 with basis tv`, yv`, y