LECTURE 2: CRYSTAL BASES

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1 LECTURE 2: CRYSTAL BASES STEVEN SAM AND PETER TINGLEY Today I ll defne crystal bases, and dscuss ther basc propertes. Ths wll nclude the tensor product rule and the relatonshp between the crystals B(λ) of hghest weght modules and the nfnty crystal B( ). I ll then dscuss an ntrnsc characterzaton of crystal lattces and crystal bases n terms of a natural blnear form (sometmes called polarzaton ). Unless otherwse stated, the results here are due to Kashwara, and proofs can be found n [K]. For today, g s a symmetrzable Kac Moody algebra, U q (g) s ts quantzed unversal envelopng algebra, that V s a representaton whch s a (not necessarly fnte) drect sum of ntegrable hghest weght modules V (λ) (.e., an object n O nt ).. Defnton of crystal bases The goal s to fnd a bass of a representaton V of U q (g) such that the Kashwara operators Ẽ, F act by partal permutatons. Ths wll allow us to draw V (λ) as a colored drected graph. However, smple calculatons show that ths s mpossble, even for the adjont representaton of sl 3. But, n a sense that wll be made precse today, ths wll work at q = 0. Note: we have swtched conventons from last week, so that our crystals are at q = 0 rather then q =. The frst part of the theory works just as well at ether 0 or, but once we start dscussng tensor products our choce of coproduct wll determne ths choce. So, the reason we have changed ths conventon s that we are changng our conventon for coproduct. I dd ths to match Kashwara s conventons n [K, K2], whch are the man references for the next few lectures. Perhaps ths s a good tme to note that there are four choces of coproduct that work equally well n ths theory: () (E ) = E K + E ; (F ) = F + K F () (E ) = E K + E ; (F ) = F + K F () (E ) = E + K E ; (F ) = F K + F (v) (E ) = E + K E ; (F ) = F K + F. One must make a choce, and all four choces have been used n the lterature. We wll be usng the frst one from now on, whereas last week we used the second one. Let us also recall the defnton of Kashwara operators on a representaton V of U q (g). For each, E, F, K ± s a copy of U q (sl 2 ) nsde U q (g). The rreducble representatons of U q (sl 2 ) are root strngs [3] [2] [2] [3] where the arrows to the rght are the matrx coeffcents of F, and the ones to the left are the matrx coeffcents of E. Decompose V nto a drect sum of rreducble representatons of U q (sl 2 ). The Kashwara operator F s defned as the operator that moves step down n each root strng, wthout multplyng by any quantum nteger. Smlarly, Ẽ moves up the root strng. Ths s ndependent of the chosen decomposton. Let A 0 = {f(q) C(q) f regular at 0}. Ths s a local rng, and A/qA = C., Date: February 8, 20.

2 2 STEVEN SAM AND PETER TINGLEY Defnton.. Let V be an ntegrable module over C(q). A crystal lattce n V s an A 0 - submodule L = λ P L λ V compatble wth the weght decomposton of V, whch s closed under F and Ẽ. A crystal bass s a bass for L/qL such that the nduced actons of F and Ẽ on L/qL act by partal permutatons on B. Theorem.2. () Any ntegrable V has a crystal bass (L, B). (2) Gven (L (), B () ), and (L (2), B (2) ), there s an automorphsm Ψ of V such that Ψ(L () ) = L (2) and Ψ(B () ) = B (2). Note that (2) mples n partcular that, f V = V (λ) s rreducble, then the structure of B(λ) s unque. Remark.3. Gven an ntegrable module V, let {v,..., v k } be a weght bass for the space of hghest weght vectors of V. Set B = { F N F v j } and L to be the A 0 -span of B. Then L s a crystal lattce and the non-zero elements n the mage of B n L/qL form a crystal bass. Remark.4. If v s a hghest weght vector, then F nv = F (n) v where F (n) := F n/[n] q! s the nth dvded power of F. Ths s one explanaton for why dvded powers show up so often. Remark.5. U (g) s a trval deformaton of U(g) as an algebra. So the representaton theory of both algebras s the same (untl we start takng tensor product). U q (g) s not qute a deformaton, but ts representaton theory s stll very smlar (essentally there are some new non type representatons, but the remanng representatons look the same). So t s a bt suspcous that the crystal bass (whch s new) appears out of somethng whch s old. In fact, the crystal bass can be constructed wthout consderng the deformaton, see [BK]. However, some of the mportant propertes of crystals are less vsble n ths pcture. In partcular, as one would expect, the relatonshp wth tensor products descrbed below s more dffcult to see. 2. Tensor products of crystal bases Theorem 2.. If (L (), B () ) and (L (2), B (2) ) are crystal bases for V and W, then (L () L (2), B () B (2) ) s a crystal bass for V W. Remark 2.2. Untl now, the choce of coproduct was not mportant, and n fact we could have worked wth crystal bases at nstead of at 0 just as easly. Theorem 2. holds at q = 0 for coproducts () and (v), and at q = 0 for the other two. The tensor product rule as we gve t below only holds for coproduct (), but a smple modfcaton works for coproduct (v). Theorem 2. nduces a combnatoral tensor product, whch we now descrbe. When g = sl 2, crystals are just drected ntervals B(n) wth n + nodes, for each n Z 0. The tensor product rule for B(3) B(2) s expressed graphcally as follows: The frst tensor factor s placed at the top, and the second on the left sde. The vertces of the tensor product are all pars of vertces one n each factor, whch can be arranged n a grd as shown. The top row and rght column of the grd make one rreducble component. Then the nodes second from the top and second from the rght. Contnue untl all the nodes are used up.

3 LECTURE 2: CRYSTAL BASES 3 For other g, smply treat the arrows comng from each copy of U q (sl 2 ) separately. Decompose the crystal nto connected components for the arrows correspondng to that F, and take the tensor product of each par. Once ths has been done for all the dfferent F (.e., all dfferent colors of arrows n the crystal graph), the result s the tensor product of the g crystals. Example 2.3. Consder g = sl 3. Then the two fundamental crystals are B(ω ) = 2 and B(ω 2 ) =. Ther tensor product s whch llustrates that V (ω ) V (ω 2 ) = V (ω + ω 2 ) V (0). Theorem 2.4. B(λ) s connected. Ths mples that for any representaton V, ts rreducble components correspond to the connected components of ts crystal graph. Note that ths gves a combnatoral way to study the Clebsch Gordon rule (.e., to fnd multplctes of varous V (γ) n V (λ) V (µ)). The tensor product rule can also be expressed algebracally: set Then for b c B C, (2.5) ε (b) := max{m e m (b) 0}, ϕ (b) := max{m f m (b) 0}. { e (a) b e (a b) = a e (b) { f (a) b f (a b) = a f (b) 3. B( ) f ε (a) > ϕ (b), otherwse, f ε (a) ϕ (b), otherwse. For the moment I wll forget the relatonshp wth representatons, and thnk of crystals combnatorally. When I do ths, I wll denote the crystal operators by e, f nstead of Ẽ and F. By lookng at the top row of B(λ) B(µ), we see a copy of B(λ). Ths s not qute a subcrystal, but t wll be closed under the operators e. So the embeddng B(λ) B(λ) B(µ) s e -equvarant. It s also clear that the top-left element n the tensor product generates a copy of B(λ + µ). Thus we have a commutatve dagram B(λ) B(λ) B(µ) B(λ + µ). The nclusons B(λ) B(λ + µ) turn {B(λ) λ P + } nto a drected system. Defnton 3.. B( ) = lm B(λ). For each λ P +, we wll have a surjecton of crystals B( ) B(λ) {0} such that the elements not mappng to 0 map e -equvarantly onto B(λ).

4 4 STEVEN SAM AND PETER TINGLEY Fgure. The nfnty crystal B( ) for sl 3. The red arrow represent f, and always go to the leftmost avalable node (n the rght weght space). The green arrows represent f 2, and always go to the rghtmost avalable node. Any of the nodes drectly below the top vertex can be the lowest weght element of a B(λ). The thcker arrows show the copy of B(ω + 2ω 2 ). 3.. Fndng B( ) algebracally. Recall that, as a U q (g) module, V (λ) U q (g) /I λ, for some deal I λ. Let π λ denote the projecton from U q (g) to V (λ). If λ µ s domnant, then I λ I µ, so these projectons ft together ncely. Say that (L, B) s a local bass of U q (g) f L s an A 0 -lattce and B( ) s a bass for L/qL. Note that here we requre no compatblty wth the algebra structure of U q (g). Theorem 3.2. There exsts a unque local bass (L( ), B( )) of U q (g) such that the hghest weght space of L s spanned by U q (g), and, for all λ, π λ (L( ), B( )) s a crystal bass for V (λ). Queston 3.3. Can you characterze (L( ), B( )) usng Ẽ and F? Also, how should we defne Ẽ and F on U q (g)? We wll gve a postve answer to ths queston n Remark 4. below, but t s convenent to frst consder an alternatve characterzaton of both B(λ) and B( ). 4. Intrnsc characterzatons Ths secton s based on [K, Secton 5]. Let D be the dagonal matrx wth entres d such that DA s symmetrc, where A s the Cartan matrx. Set q = q d. Defnton 4. (Adjont map). Defne θ by θ(e ) = q F K θ(f ) = q K E θ(k ) = K. One can check that ths extends to an algebra ant-nvoluton whch s also a coalgebra somorphsm. Remark 4.2. There appears to be a typo n [K, Secton 5], n that the map used there does not actually defne an antautomorphsm. We have modfed θ(e ) to gve θ ths property. In most

5 LECTURE 2: CRYSTAL BASES 5 calculatons we wll only apply θ to elements of U, so these calculatons are unaffected by the change. Let v λ V (λ) be a fxed hghest weght vector. Defne a blnear form (, ) on V (λ) va (v λ, v λ ) = and (au, v) = (u, θ(a)v) for all u, v V (λ), a U q (g). Snce θ s a coalgebra somorphsm, If we defne (, ) V (λ) V (µ) = (, ) V (λ) (, ) V (µ), then θ stll acts as an adjont. Theorem 4.3. Notaton as above. () L(λ) = {u V (λ) (u, L(λ)) A 0 } = {u V (λ) (u, u) A 0 }. (2) B(λ) s an orthonormal bass of L(λ)/qL(λ) wth the nduced form. Remark 4.4. We are workng over C(q), but we really could work wth an approprate ntegral form. Then we could characterze B(λ) B(λ) as the set of vectors wth norm. 4.. Blnear forms on U q (g). If we want to defne as nner product as we dd for hghest weght representatons, we need to have an acton of U q (g) on U q (g). In fact, there s a whole famly of such actons, one correspondng to each hghest weght (.e. the Verma modules). These can be descrbed as follow. Recall the trangular decomposton of U q (g): As a vector space, For each λ, let defne (4.5) U q (g) U q (g) U 0 q (g) U + q (g). ψ λ : U 0 q (g) U + q (g) C(q) E 0 K q H,λ. We get an acton λ of U q (g) on U q (g) by left multplcton composed wth applyng ψ λ to the two rghtmost factors n the trangular decomposton (4.5). For each λ, we then get a blnear form on U q (g) defned by (4.6) (, ) = (a λ u, v) = (u, θ(a) λ v) for all u, v U q (g), a U q (g). If λ s domnant, the quotent of Uq (g) by the kernel of ths form s somorphc to V (λ). Thus t s natural that, to study B( ), we are really nterested n the lmt of the nner product as λ, and as q 0. K K Recall that E F j u = F j E u + δ,j q q u. As the hghest weght λ, K wll act on u by some large power of q and K acts by some large negatve power of q. Snce we are also takng the lmt as q 0, the K wll domnate. For ths reason, we modfy E to get an operator E by: Defnton 4.7. E acts on U q (g) by E = 0 and E F j = F j E + δ,j qk. Theorem 4.8. There s a unque blnear form (, ) on U q (g) such that () (, ) =, (2) (F u, v) = (u, q K E v). Remark 4.9. It s mportant that, one you commute the E to the front, all terms wth no E also have no K. One can check that ths does n fact happen. Kashwara handles ths ssue by defnng E = q K E. Then E F j = q H,α j F j E + δ,j, and t s clear that no K appear n ths commutaton relaton. Theorem 4.0. Notaton as above.

6 6 STEVEN SAM AND PETER TINGLEY () L( ) = {u V ( ) (u, L( )) A 0 } = {u V ( ) (u, u) A 0 }. (2) B( ) s an orthonormal bass of L( )/ql( ) wth the nduced form. Agan, by workng over an approprate ntegral form, we could characterze B(λ) B(λ) as the set of vectors wth norm. Remark 4.. We can also now see the correct defnton of Ẽ, F on Uq (g): Defne a Uq (g) to be -sngular f E (a) = 0. For all -sngular v U q (g), defne F n (n) (v) = F (v). These F play the role of the Kashwara operators on Uq (g), leadng to a crystal bass. See [K, Theorem 4]. References [BK] Arkady Berensten and Davd Kazhdan, Geometrc and unpotent crystals II: From unpotent bcrystals to crystal bases, Quantum groups, 3 88, Contemp. Math. 433, Amer. Math. Soc., Provdence, RI, 2007, arxv:math/06039v4. [CP] V. Char and A. Pressley, A gude to quantum groups, Cambrdge unversty press, Cambrdge (994). [K] Masak Kashwara. On crystal bases of the q-analogue of unversal envelopng algebras, Duke Math. J. 63 no. 2, , 99. [K2] Masak Kashwara. On crystal bases, Representatons of groups (Banff, AB, 994), 55-97, CMS Conf. Proc. 6, Amer. Math. Soc., Provdence, RI, 995.

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