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Международная научная конференция "Гармонический анализ на однородных пространствах, представления групп Ли и квантование"
Международная научная конференция "Гармонический анализ на однородных пространствах, представления групп Ли и квантование" была проведена в Тамбове на базе Тамбовского государственного университета имени Г. Р. Державина с 25 по 29 апреля 2005 года. Она была организована кафедрой математического анализа университета. В работе конференции приняли участие 58 человек - как из-за рубежа (Франция, Голландия, США), так и различных городов России (Тамбов, Москва, С.-Петербург, Петрозаводск, Оренбург, Белгород, Ижевск, Сыктывкар, Волгоград). Заседания конференции посещали аспиранты и студенты университета.
Гармонический анализ на однородных пространствах - часть современной математики (функционального анализа), чрезвычайно актуальная и имеющая важные приложения в теоретической физике (в частности, квантовой механике). Он развивается многими математиками в России и за рубежом (Франция, США, Нидерланды, Германия, Япония, Дания, Швеция). Группа математиков Тамбовского университета занимает в этом ряду достойное место.
Цель конференции состояла в дальнейшем развитии этого перспективного и имеющего богатые приложения раздела математики, укреплении и расширении связей и сотрудничества между учеными России и других стран.
Тематика докладов, заслушанных на конференции, была достаточно обширной, она относилась к современным разделам математики и теоретической физики (программа Гельфапда-Тиндикииа, квантования, канонические и граничные представления групп Ли, представления алгебраических структур, квантовые группы, вполне интегрируемые системы, формулы Планшереля и теоремы Пэли-Винера, спецфункции и др.).
Конференция была посвящена 70-летию двух замечательных российских математиков: Д.П.Желобенко (Российский университет Дружбы Народов, Москва) и В.М.Тихомирова (Московский государственный университет имени М.В.Ломоносова). Мы были рады видеть их в числе участников. Они оба выступили с докладами.
Конференция была поддержана Российским фондом фундаментальных исследований (грант № 05-01-10038г), Министерством образования и науки РФ (Ведомственная научная программа "Развитие научного потенциала высшей школы", проект № 380), Федеральным агентством по науке и инновациям и Тамбовским государственным университетом имени Г.Р. Державина.
Труды конференции печатаются в настоящем и следующем выпусках Вестника Тамбовского университета. Серия: Естественные и технические науки.
Сопредседатель Оргкомитета
В.Ф. Молчанов
Conference “Harmonic analysis on homogeneous spaces, representations of Lie groups and quantization”
The conference “Harmonic analysis on homogeneous spaces, representations of Lie groups and quantization” was held in April 25 - 29, 2005, Tambov (Russia) at G.R. Derzhavin Tambov State University. It was organized by the chair of mathematical analysis of the university. Participants (58 people) came both from Russia (Tambov, Moscow, S.-Peterburg, Petrozavodsk, Orenburg, Belgorod, Izhevsk, Syktyvkar, Volgograd). Sessions of the conference were also attended by students and postgraduate students of the university.
Harmonic analysis on homogeneous spaces is a part of modern mathematics (of functional analysis), very actual and having important applications to theoretical physics (in particular, quantum mechanics). It is being developed by many researchers in Russian and abroad (France, USA, the Netherlands, Germany, Japan, Denmark, Sweden). A group of mathematicians from the Tambov University is known for its significant contribution in this research.
The work of the conference was aimed at enlarging future prospects of this part of mathematics with its most promising applications, strengthening and extending of connections between researchers from Russia and other countries.
The theme of the talks delivered at the conference were wide and various, they concerned modern parts of mathematics and theoretical physics (the Gelfand—Gindikin program, quantizations, canonical and boundary representations of Lie groups, representations of some algebraic structures, quantum groups, integrable systems, Plancherel formulas and Paley—Wiener type theorems, special functions etc.).
The conference was devoted to the 70-th anniversary of two remarkable Russian mathematicians: D.P.Zhelobenko (Russian University of People Friendship, Moscow) and V.M. Tikhomirov (M.V.Lomonosov Moscow State University). We were very glad to welcome them as the participants of the conference. They both gave talks.
The conference was supported by the Russian Foundation for Basic Research (grant № 05-01-0038g), the Ministry of Education and Science of Russian Federation (the Scientific Program “Development of scientific potential of higher school”, project № 380), the Federal Agency on Science and Innovations, and G.R.Derzhavin Tambov State University.
Proceedings of the conference are published in this and next issues of the journal “Vestnik of Tambov University. Series: Natural and Technical Sciences”.
Co-chairmen of the Organized Committee
V.F. Molchanov
GAUSS ALGEBRAS AND QUANTUM GROUPS 1
D. P. Zhelobenko Peoples Friendship Russian University, Moscow, Russia
The category GA of Gauss algebras was introduced in [10] to study its objects by unified methods. In particular, GA contains (up to an equivalence) the category SLA of symmetrizable Lie algebras [3] and the category DJA of Drinfeld-Jimbo algebras [2]. The embedding SLA —» GA is given by the universal functor U : g —> U(g). The language of GA is closely related to a study of famous quantum groups [1]. The aim of this paper is to focus on some aspects of this theory. A most of the text has a character of a review of relevant topics. Some new results are contained in Sections 10-12.
1 Main definitions
Let A be a unital associative algebra over a field F. We begin with the following preliminary definitions (GT), (CA).
(GT) A triple of subalgebras AL , H, N+ of the algebra A is called Gauss triple if they contain the unity of A and satisfy the following conditions:
(a) The algebra H normalizes N± : HN± = N±H\
(/?) The algebra A is equipped with Z-grading An (n G Z) meaning that N-,H,N+ is generated (respectively) by homogeneous subsets e_, eo, e+ consisting of elements x such that deg x < 0, deg x = 0, deg x > 0; (7) The algebra A admits a free (triangular) decomposition
A = N-HN+. (1)
(CA) The algebra (1) is called contragredient (CA) if A is equipped with an (anti) involution x i->- x1 such that h = h', A'n = A_n for any h 6 H, n € Z.
Setting e = e+, e' = e_, we define the canonical bilinear form tp : A x A —> H by the rule
(fi(x,y) = (x'y) 0, (2)
where x £0 is defined as a projection A H parallel to the subspace T = e'A + Ae.
Notice that X{) = x'Q for any x G A. Hence the form (2) is symmetric. Moreover, its kernel
ker ip contains the left ideal Ae. Hence ip is well defined on the quotient space
M = A/Ae. (3)
It is clear that M is a cyclic A-module generated by the vector 1+ = 1 + Ae, such that el+ = 0. Using (a) (resp., (18)), we obtain that M inherits a structure of A x iif-bimodule (resp., Z-grading) of A. Using (7), we obtain an isomorphism of graded spaces
M = Al+ = N-H1+ ~ B, (4)
where B = N^H. It is clear also that the grading An (resp., Mn) is orthogonal with respect to tp. The space M is called the universal Verma module of A.
'Supported by the Russian Foundation for Basic Research (grant No. 05-01-00001a).
The algebra A is called nondegenerate if the form ip is nondegenerate on M. Later we assume (for simplicity) carde < oo. Then we state the fundamental definition
(GA) The algebra (1) is called Gauss algebra (GA) if A is contragredient and nondegenerate.
It is easy to verify that the corresponding algebra H is commutative and without zero divisors. The algebra H is called Cartan subalgebra of A. The algebras B± = II N± are called Borel subalgebras of A.
2 Main examples
We consider the following list of examples:
Weyl algebras A = Wn,
the algebras A = U(g), where g is SLA,
the quantum envelopes (DJA) A = Ug(g), where g is symmetrizable Kac-Moody algebra (SKMA),
superversions of these algebras,
an asymptotic version A — Dq(g) of Uq(g) (Kashiwara algebras), some examples of translator algebras (Section 11), the Yangians Y = Y’(fi) (Section 12).
Details are given below.
(1) The classical Weyl algebra A — Wn is a unital algebra (over C) generated by 2n elements ei, fi {i = 1,..., n) equipping with genetics [eu e3) — [fu fj] = 0 and
[e;,/;]=<%, (5)
where Sij is the Kronecker symbol (i,j = l,,..,n). Setting e = (e,;), deg et = 1, e' = /,. we obtain that A is GA with M = C[a:], where x = (xi, ...,xn). The action of A on M is given by the rule ei = di = d/dxi, fi = x^ (i = 1 ,...,n).
(2) We start from the known triangular decomposition of SLA: 5 = n_ © 1) © n+, where f) is a Cartan subalerebra of a F71. Settinsr A = C/fa) we obtain the corresDondine decomposition fl).
V L J * ',«►’/ i U 4. \ / I
where H = U(()). N± = C/(n±). The algebra A is generated by elements h G t) and Chevalley generators ei G n+, fi € rt_ (i G I), where card! < 00. The obtained genetics is reduced to the known weight and Serre conditions [7] together with relations
[sji/j] = hi, (6)
where hi G f) (i G I). Setting e = (ei), dege* = 1, e\ = fi, we obtain that A is CA.
A study of the form ip is reduced to a study of a series of finite dimensional forms (pn (n G Z+) associated to <p [4]. Calculating the principal terms of these (polynomial) forms, we obtain [7] that A is GA.
(3) The algebras A = Uq{g) (where 0 is SKMA) are defined as quantum deformations of their classical versions U(g). It is known (see [7], for example) that A is GA.
(4) The algebras A = Dq(g) are introduced by M. Kashiwara [4] to a study of an asymptotic behaviour of Uq(g). In other words, Dq(g) can be defined as a ’’twisted differential algebra” of the quantum Serre algebra Sq(g) [8].
(5) A lot of superversions of the examples (l)-(4) can be easily defined. See [8] for example.
(6) A notion of translator algebras is discussed in Section 11. On occasion, these algebras are GA.
(7) The Yangian A = Y(g) was introduced by V. Drinfeld [1], [2] as a unique quantum
deformation of the algebra J7(0[i]), where 0 is simple (dimg < oc) and t is an independent
variable. Sometimes 0 can be reductive. If g is simple, then A is GA (Section 12).
Later we consider some questions of theory of representations of GA. As a preliminary, we consider some special extensions of GA.
3 Projective extensions
Setting e = (e,), i E I, we define en (n E Z+) as the set of monomials e^,..., ejn (e° = 1). Using
(7), we obtain that A is contained in the Cartesian product
00
Aext = I] Bg^ (7)
n=0
where B = B-. Elements / 6 Aext can be considered as formal series with partial sums fn = <5/o + ... + 5fn, where Sfn E Be11. In this sence, Aext coincides with the projective limit of spaces
An = A/Aen+l. (8)
The algebra A is embedded in Aex* as the direct sum of subspaces Ben (or the set of finite series in Aext)-
Proposition 1 [7]. Aext is an algebra with respect to the multiplication of formal series. Moreover, let Aint be the greatest Z-graded subalgebra of Aext- Then we have a chain of extensions
A C Aint C Aexf. (9)
Now, let C(e) be the category of A-modules X locally nilpotent with respect to e : en+lx = 0 for any x E X and any n ^ no(x). Setting
/x = fnx for n ^ n0(x), (10)
we define the action of Aext on X. In other words, the module X is equipped with the structure of a Aext-module.
Notice that C(e) is closed with respect to passing from a module to its submodule and its quotient module. If X 0 (in /3(e)), then Xe 0. Here the subspace
Xe = ker e = {x E X : ex = 0} (11)
is called the extremal subspace of X (with respect to e).
In particular, the module M (Section 1) belongs to C(e). The action of the algebra A (and Aext) 011 M is contained in the algebra
E(M) = End HM (12)
(the commutant of the right action of H on M).
4 Perfect algebras
(PA) A Gauss algebra A is called perfect (PA) if H a field (an extension of F).
Theorem 1 [7] If A is perfect, then the action of A on M determines an isomorphism of
algebras
Aext ~ E(M). (13)
Respectively, we obtain an isomorphism of graded algebras
Aint — Eint(M), (14)
where Eint(M) is the greatest h-graded subalgebra in E(M).
Corollary 1 The action of A (and Aext) on M is exact. Moreover, M is a simple A-module. Corollary 2 There exists a unique element p E Aext satisfying equations
ep = pe' = 0 (15)
and normalyzing condition poo = 1- Moreover, p E Aint and
P2 =P = P', degp = 0, (16)
with respect to the unique extension of x x' onto A{nt-
Proposition 2 The operator p projects any module X of category C(e) onto their extremal subspace Xe parallel to e'X. In particular, we have
X = Xe ® e'X. (17)
Example. Setting A = W\, we obtain that any / E EndCJVj can be written uniquely as a formal series:
oo
/= £ fmnXmdn, (18)
m,n=0
where d = d/dx, satisfying the finiteness condition (</?): for any n, only a finite set of coefficients fmn £ C can differ from zero.
Remark. Assume that A has no zero divisors. Then the equation p2 = p, i.e. p(l — p) = 0 has only trivial solutions p = 0,1. Alternatively, the algebra Aext admits a lot of projectors. The projector p given in Corollary 2 is called the extremal projector of algebra A.
Consider an application of the projector p in the theory of A-modules.
Proposition 3 If A is PA, then the category C(e) is monoidal, with unique simple object M. In other words, any object X of C(e) has a form X = nM (n is a cardinal number).
Proof. For any module X in the category £(e), we set Xq = AXe. Assuming X / Xq, we obtain (X/Xo)e 0, i.e. there exists a vector iSl such that x ^ Y, ex C Y. Setting y = px, we obtain y C Y. Notice also that x — px EY. Hence x E Y (contradiction). As a result, we have X = Y.
Moreover, Y is the vector sum of a family of cyclic modules Axo, where e Xe. Notice that the map al+ axo is correctly defined and determines an isomorphism Axo ~ M for xq 7^ 0 (because M is a simple A-module). Consequently, Y is the direct sum of submodules Axo, where xq runs a basis of Xe. Hence X = Y = nM (n = diva.Xe). □
Examples:
1. M = C[x] for A = Wn.
2. M = Sq(g) for A = Dq{g).
5 Local extensions
(SA) A Gauss algebra A is called standard (SA) if A is invertible as //-bimodule, i.e. the left
and right actions of an element 0 ^ h E H in A is invertible. Hence A admits a structure of
.ff'-bimodule, where H' = FractH (the quotient field of H). Setting
A! = A ®H H', (19)
we obtain a natural structure of an algebra in (19). Namely, we set af = a ® f in (19) and define fa by means of the normalyzing conditions (a), see Section 1, in N±. The algebra (19) is called the local extension of A (with respect to H).
Notice that H' is the Cartan subalgebra of A'. Hence A! is perfect. Consequently, any SA has a perfect extension.
Notice that the extensions (7) and (19) are permutable. Respectively, we can define the locally projective extension
Kxt = (Aext)' = (A')ext.
In particular, any standard algebra A admits the extremal projector p E A'ext.
6 Cartan type algebras
(СТА) A Gauss algebra A is called the Cartan type algebra (СТА) if A admits a ’’fine” Q-grading Afl (ц E Q), where Q is an ordered abelian subgroup in Autif. The condition x £ AjL
moQno +v.Q+
lllV/UillU UllUlU
hx = xhfj, forany h E H, (20)
where fi : h >-> means the action of /i E AutH. Using the additive language for Q, we obtain
a usual rule
АХА^ с АХ+1Л (21)
for any \,ц E Q. The term ’’fine” means that A^ С Anwhere the function ц п(ц)
(Q —>• Z) is additive, n(0) = 0. Setting
Q+ = {m G Q : ju ^ 0}, (22)
we assume that the sets eo, e± are homogeneous, Q—degx = 0 (resp., < 0, > 0) for x E eо (resp., x E e_. x E e+). The Q-grading can be used (instead of Z-grading) in the definition (GT), see Section 1.
For the case (СТА), the chain (9) can be completed as follows:
A С Afin С Ain% С Aexti (23)
where Afin is the greatest Q-graded subalgebra in Aext. It is also clear that any СТА is SA. Hence we have the corresponding chain in A1:
A' с A'fin с A'int с A'ext. (24)
The indices in (23), (24) can be interpreted (respectively) as ’’fine”, ’’integer”, ’’extensive” (and also as ’’final”, ’’internal”, ’’external”).
Notice that the universal Verma module M' for A' can be written as follows:
M' = A' / A' e = M®HH' (25)
(a localized Verma module of A). Recall that the action of A' on M' determines an isomorphism of algebras A'ext ~ E(M').
As usual, the algebra H consists of functions defined on some set A. Respectively, elements / ^ A'eXt can be considered as ’’rational functions” on A, except of a subset cr(f) consisting of singularities of /.
7 Category 0(e)
Let A be СТА, Х(Н) be the set of characters of the algebra H. For any ц G X(H) and any Я-module X we define the usual weight subspace
Хц{х =£ X : hx = fi(h)x, forany h G H}. (26)
Notice that the family (26) is linearly independent [7], and we have
A£Xp с Xeit, (27)
where Efj,(h) = fj,(h£) in terms of (20). A module X is called H-diagonal if it is graded by the subspaces (26), i.e.
X = ©^ep^u, (28)
for some subset P С X(H). Meaning (27), we assume that P is invariant with respect to Q (QP С P). It is essential that any submodule Y С X inherits the grading (28), i.e.
Y = (B/iepY/» (29)
where Y^Y П Xfj.
Let us fix a Q-sub module P С X(H). Let 0(e) be the subcategory of C(e) consisting of P-graded (if-diagonal) A-modules (28). It is clear that O(e) is closed with respect to passing from a module to its submodules and quotient modules.
Example. For any Л G X(H), we define the Verma module
M(\) = M®BCX, (30)
where В = B+, Сд is an one-dimensional В-module defined by the character Л (extended trivially on N+). Notice that
M( X) = A/h = M/JX, (31)
where I\ is a left ideal of A generated by the set e and by the elements h\ = h — X(h), J\ is a submodule of M generated by the elements h\ (h G H). Notice also that
M( X) = Л1А = Nl\, (32)
where 1л = 1 ® 1 in (30), 1д — 1 -f-1\ in the first part of (31).
Assume that Q acts effectively in P (i.e. the equality = /i implies e = 1 or /j = 0). In
that case, the relation A ^ ц for Л G Q+l-i determines an ordering in P. Using (32), we find that
M(X) contains only weights ц ^ A, and M(X)\ = С1л-
Moreover, there exists a greatest submodule N(A) of M(A) not containing the weight A. Setting
V(X) = M(X)/N(\), (33)
we obtain a unique (up to an isomorphism) simple A-module with highest weight A.
The theory of Verma modules (over СТА) is quite similar to that in classical case A = U(q). We concern some questions of this theory in Section 10.
8 Simplest cases
We consider the following four algebras: (1) A = W\, (2) A = U(si2), (3) A = Uq(s^), (4) A = Dq(s[2), equipping (respectively) with Cartan subalgebra H = C, C[/i], C(q), C(q). All these algebras are equipped with two generators e, / and the following genetics:
(1)[e,/] = l,
(2) [e,/] = h, [h,e] = 2e, [h,f] = -2/,
(3) [e, /] = hg, te = q2et, tf = q~2ft for t = qh, hq = (q - g-1)-1^ “ i_1)>
(4) ef = q2fe+ 1.
More exactly, in case (3) we can consider A as an algebra with generators e,/, t, t~l. It is clear that all these algebras are CTA, and the algebras (1), (4) are PA. In all the four cases, the extremal projector p has the following form:
00 (_1 \n
<M>
71=0 V
where (n) = n for cases (1), (4), (n) = [n] for cases (2), (3), and the factor ipn G H has the form
n n
<pn = l, Yl(h + j+ z), Y[[h + j+ z], 1 (35)
j=1 j=1
in cases (1) (4) (respectively). Here we use the symbol [x] for x = h + n, where n £ Z. Moreover, the projector (34) can be written as an infinite product
OO
p=J^(l-u}/an), (36)
n=1
where u = fe, an = n, n(h + n + 1), [n][h + n + 1], [n]qn~l (respectively).
Recall that the category C(e) in cases (1), (4) is monoidal. The correspoding simple object
M coincides with C[x], Sq(sl2) (respectively).
9 The case A = U(g)
We consider the examples (l)-(7) of § 2. Notice that cases (1), (4) are simple (PA), cases (2), (3) are similar to each other (in view of quantization), and cases (5) correspond to (1) - (4). Cases
(6), (7) will be considered later (Sections 11, 12). Hence in detail it suffices to consider case (2) only.
Let us fix the standard notations for g [7]. In particular, let I) (resp., A+) be a fixed Cartan subalgebra of g (resp., a set of positive roots of g). For any A € I)* we set AQ = A(ha), where a G A_|_. A vector p € f)* is defined by the condition pa = 1 for any simple a E A+.
Let p be the extremal projector of g (i.e. of A — C^(g))- Notice that p E F(g)o, where
.F(g) = U(g)'fin (the localization of U(g) over the field i?(i)) = Fract U(t))). On the other hand, let z be (generalized) Casimir element of the algebra g [7]. Notice that z E U(g)fin (z E U(q) for dimg < 00) [7]. Then we have
z = ZQ + UJ, (37)
where x xq denotes the canonical projection onto H (Section 1), extended to the algebra U (g) fin. If is convenient to use the identification H ~ P(f)*) (the algebra of polynomial functions on I)*. Setting ze(A) = zo(A + e), we obtain a family of affine functions a£ = ze -zq (e E Q+):
^(-^O — 2(A + p, e) + (e, e). (38)
Theorem 2 [9], [7] The extremal projector p E F(g) can be written as infinite product
p = H(l^/a£), (39)
e^O
where lo = z — zq. The expression (39) is hold for any ’’constructive” g-module [9] (in particular, for M).
The expression (39) is called the Lagrange form of p. The following result means a ” control on singularities” for p.
Theorem 3 [9], [7] The function (38) defines an essential singularity of p only for e = ja (quasiroot), where a E A+, D ^ j E Z+.
In particular, let dimg < oo. In that case, Theorem 4 follows from the known ’’normal form”
of p [8]. Namely, for any normal ordering [8] A+ = {«i,..., am}, we have
P ~ Poll "'Potm 1 (40)
where pa is an analog of (34) corresponding to the root a E A+. It is essential that (40) does not depend on the choice of a normal ordering in A+. An analog of (40) is valid also for affine Lie algebras [12].
Theorem 4 [9] For any constructive g-module X and any x E X, the left denomerator of the rational vector-function px has the form
ma
TTx = J! U(ha+Pa+j), (41)
c*eA+ j=1
where ma = max{n : e™x ^ 0}.
The latter result is nontrivial as a control on singularities for the function px. For example, let eax = 0 for a fixed a E A+. Then the factor pa can be omitted in px, independently on its position in (40).
10 Verma modules
Let A be СТА. We select briefly a way of using the projector p in the theory of Л-modules in the category О — 0(e).
Proposition 4 The singular set a(p) (Section 6) coincides with the following subsets of A E fy* : (a) there exists a nontrivial homomorphism M(A) —> M(A + e), for some e E Q+,
(f3) the module V(X) coincides with a factor of M(A + e) for some £ E Q+,
('y) the module M(A) is not projective.
In particular, let the algebra N = ЛГ_ is without zero divisors. Then the action of N in M is free. In particular, any nontrivial homomorphism M(A) —> M(A + £ is an injection.
Let X be an object of the category O. We define P(X) as a set of ц E P such that xIL ф 0. A vector x E Xfj, is called primitive if x ф Y, ex С Y for some submodule У of X. In that case the correspoding weight ц is called primitive. A subset of P(X) consisting of primitive weights is denoted Pprim(X).
A module X (of category O) is called regular if any primitive vector x E Хц is extremal (ex = 0) or the corresponding weight fi is regular in the following sence: /j ^ cr(p). The subcategory of О consisting of regular modules is denoted by Oreg.
346
Proposition 5 Any module X of the category O is generated by its primitive vectors. Any module X of the category Oreg is generated by its extremal vectors. In the latter case, we have
X = AXe. (42)
Theorem 5 The category Oreg is semisimple, with simple objects V(A).
The proof is similar to that of Proposition 3. In particular, for A = U(g), the category Oreg contains the subcategory OlTlt of ’’integrable” A-modules [3]. In the case dimg < oo, the category Oint coincides with the category S' of finite dimensional g-modules. In that case, theorem 6 coincides with a classical theorem by H. Weyl [7].
11 Hypersymmetry
We draw the attention to a special way of constructing of GA. A pair of unital algebras (A, B) is called admissible if 1 £ B C A, where B is GA and 1 is the unity of A. For any A-module X, we set Xe = kere (with respect to B). The space
T = {a £ A : ea C Ae} (43)
coincides with Norm Ae (= the greatest subalgebra of A containing Ae as a two-sided ideal). It is clear that TXe C Xe (for any A-module X). Setting
O — T I An fAA\
kJ - -L I i }
we define an action of 5 in Xe (via eXe = 0). Hence we have SXe C Xe. The algebra (44) is called the hypersymmetry algebra of the space Xe.
Using (43), we obtain that S' is a part of M. Passing to the rational hull M' = A'/A'e (Secton 6), we obtain the algebra
z = S' = pM', (45)
where p is the extremal projector of B (p 6 B'ext). The following example is crucial (in the theory of Lie algebras).
The pair (g, 6) of finite dimensional Lie algebras is called reductive if the algebra t is re-ductively embedded in g (i.e. g is a reductive (adS)-module). In particular, t is reductive and g = I © p where p is a complementary (adt)-module.
It is clear that the pair A = U(q), B = U(E) is admissible. In that case, the localization (45) is given over the field i?(f)) = Fract 17(f)), where f) is a fixed Cartan subalgebra of 6.
Theorem 6 [8] Let ai (i = l,...,n) be a weight basis of the (adt)-module p. Then the algebra Z = Z(g,t) defined in (45) is generated (over R(t))) by the elements Z{ = pai (i = l,...,n) equipping with a quadratic genetics (over R(\))).
Example. Set AZn = Z(gn,gn-1), where gn = g[„ (over C). The elements e* = e^n, e_i = enj (i = 1, ...,n — 1) and eo = enn form a weight basis of p. Respectively, AZn is generated by the elements Zi = pei (i = 0, ±1,..., ±(n — 1)). The corresponding genetics has the following form:
ZiZj - EijZjZi, for i + j ^ 0, (46)
71 — 1
ZiZ-i = £ aijZ-jZj + 7i = 0 (47)
3=1
with coefficients in i?(f)) [8]. In that case, the algebra AZn is GA.
12 Yangians
The Yangian Y(g) was introduced by V. Drinfeld [1], [2] as an important example of a quantum group. Here g is a complex simple Lie algebra. The algebra A = Y(g) is a unital algebra over
C. Moreover, V. Drinfeld described a genetics of y(g), in terms of special elements Hiz, X^z [1]. Actually, the algebra A = Y(g) has a triangular decomposition (1), in terms of these generators. Moreover, A is CA.
It is not difficult to verify (by the analogy with U(g)) that the canonical form of Y(g) is nondegenerate (on M). In other words, we obtain the following
Theorem 7 The Yangian Y"(g) (g is simple) is GA.
The theory of finite dimensional representations of Y(g) [10] can be easily embedded in the general theory of representations of GA. On the other hand, the definition of Y(g) can be sometimes extended to the case when g is a complex reductive Lie algebra (g = g(n, for example).
The Yangian Yn = Y (gl„) is a unital algebra (over C) generated (over C) by the set of elements
OO
%(«) = £ 40) = 6i<48)
s=0
where u is an independent variable (in fact Yn is generated by the coefficients in (48)). The correspoding genetics of Yn has the following form:
[tij(u),tM(v)] = tkj(u)tu(v) - tkj(v)tu{u). (49)
Example (n = 2). In the case, the matrix t(u) = (tij(u)) can be rewritten as follows:
«“> - (“S )' (5o)
where
oo
x(u) = 'sy^iXjU~% i=0
for x = a, /3,7, S. Setting (xy)ij = [xi,yj], we can write the genetics (49) in the following form:
(xy)i+i,j - [xy)i,j+1 = {xy)ij, (51)
where the symbol xy is defined in terms of permutation of rows st-^iin (50). Namely, xy = xy (resp., xy) if the elements x,y belong to a common row (resp., to distinct rows) in the matrix (50).
Using the genetics (51), we obtain that Y = Yz has triangular decomposition (1), where H (resp., N^,N+) is generated by the elements a(u),S(u) (resp., 7(u),f3(u)). Moreover, H = AD = DA, where A (resp., D) is generated by the elements a(u) (resp., S(u)). The algebra H is not commutative. Hence this decomposition does not imply the structure of GA.
However, it is easy to verify that the algebra Y coincides with a projective limit of Gauss algebras 7r(y), where ir is a finite dimensional representation of Y.
In particular, the general theory of representations of GA (in the category 0(e)) can be used to study finite dimensional representations of Y.
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