Спектры подалгебр Бете в Янгианах тема диссертации и автореферата по ВАК РФ 00.00.00, кандидат наук Машанова-Голикова Инна Антоновна

• Машанова-Голикова Инна Антоновна
• кандидат наук
• 2022

Оглавление диссертации кандидат наук Машанова-Голикова Инна Антоновна

Contents

Introduction

Main results

Organization of the text

1. Yangians

1.1. Yangian for simple g

1.2. Yangian for gln

1.3. Some homomorphisms between Yangians

1.4. Centralizer construction

2. Mo,n+2 and its real forms

2.1. Deligne-Mumford compactification

2.2. Stratification of M0,n+2

2.3. Real forms

3. Bethe subalgebras

3.1. Bethe subalgebras, Y (g) case

3.2. Bethe subalgebras, Y(gin) case

3.3. Bethe subalgebras, Y(gl2) case

3.4. Proof of Theorem

4. Representations of Yangians

4.1. Representations of Y(g)

4.2. Representations of Y(gin)

4.3. Representations of Y(gl2)

5. Hermitian property

5.1. Hermitian property compatible with the compact form

5.2. Hermitian property compatible with the split form

6. Quantum shift of argument subalgebras

7. Limits of shift of argument subalgebras

8. Limits of Bethe subalgebras

9. The family of shift of argument subalgebras in centralizer

10. Centralizer construction and Bethe subalgebras

11. Spectra of quantum shift of argument subalgebras

12. Possible goals

13. Simplicity of spectra in Y (g) case for simple g

13.1. Cyclic vector

13.2. Semisimplicity

14. Simplicity of spectra in Y(gln) case

14.1. Cyclic vector for Bethe subalgebras of Y(gin)

14.2. Real forms of M0 n+2 and simplicity of spectra

15. Simplicity of spectra in Y(gi2) case

15.1. Cyclic vector

15.2. Proof of Theorem

15.3. Simplicity of spectra

15.4. Proof of Theorem 15.13 38 Conclusion 42 References

Введение диссертации (часть автореферата) на тему «Спектры подалгебр Бете в Янгианах»

Introduction

Let g be a simple Lie algebra. The Yangian Y (g) is a Hopf algebra, historically one of the first examples of quantum groups. It has been defined by V. Drinfeld in [D85].

The simplest case is g = sln. The Yangian Y(sln) can be realized as a factor of the extended Yangian Y(gln). Thus most of this work concerns the case of Y(gln). Y(gln) is in certain sense the unique Hopf algebra deforming the enveloping algebra U(gln[t]), where gln[t] is the Lie algebra of gln-valued polynomials.

There is a flat family of maximal commutative subalgebras B(C) C Y(gln), called Bethe subalgebras, parameterized by invertible diagonal matrices C G GLn with pairwise different eigenvalues, which are stable under the C-action by shift automorphisms of Y(gln). For g = sln this algebra appears in the works of L. Faddeev and St.-Petersburg school in relation to the inverse scattering method, see e.g. [T84, TF]. In full generality this algebra firstly appears in the paper of V. Drinfeld [D85]. The maximality of Bethe subalgebras has been studied in [NO]. This family originates from the integrable models in statistical mechanics and algebraic Bethe ansatz. More precisely, the image of B(C) in a tensor product of evaluation representations of Y (gln) form a complete set of Hamiltonians of the XXX Heisenberg magnet chain, cf. [B, KBI].

The main problem in the XXX integrable system is the diagonalization of the subalgebras B(C) in the corresponding representation of the Yangian. The standard approach is the algebraic Bethe ansatz which gives an explicit formula the eigenvectors depending on auxiliary parameters satisfying some system of algebraic equations called Bethe ansatz equations, see for example [KR86].

The questions we address in the present work are closely related to the completeness of the algebraic Bethe ansatz, i.e. to the problem whether the eigenvectors obtained by Bethe ansatz form a basis in V. This problem is extensively studied for many years, see e.g. [MV03, MTV07, MTV09, MTV14, T18, CLV, RV]. As the first step towards the solution of this problem, it is necessary that the joint eigenvalues have no multiplicities. The latter is satisfied if and only if the following two conditions hold: first, there is a cyclic vector for the Bethe subalgebra in V (i.e. v G V such that B(C)v = V) and, second, the algebra B(C) acts on V semisimply.

In this work we prove the simplicity of spectra in several new cases including tame representations of the Yangian in type A with generic values of the parameters and some Kirillov-Reshetikhin modules in other types.

Main results. We have said that as the first step towards the solution of the Bethe Ansatz problem, it is necessary to establish that the joint eigenvalues of a Bethe subalgebra have no multiplicities. As noted, this condition is satisfied if and only if the following two conditions hold: first, there is a cyclic vector for the Bethe subalgebra in V (i.e., v G V such that B(C)v = V) and, second, the algebra B(C) acts on V semisimply.

In this section we will discuss under which assumptions these two conditions are satisfied.

Conjecture 0.1. B(C) acts with a cyclic vector in any irreducible finite dimensional representation of Y(g) for all C G Treg.

The condition that can be verified and that guarantees semisimplicity of operators from the Bethe subalgebra B(C) is that they act with normal operators with respect to a Hermitian form, i.e. B(C)+ = B(C).

Conjecture 0.2. For all tensor products of Kirillov-Reshetikhin modules QQN=1 W^. ,rj (uj) where Uj G iR, B(C) acts with norma! operators with respect to the Hermitian form defined in section 5.1 for C G TZ0ZP.

In this thesis, we obtain several results supporting these conjectures and these are the main results of our work. In the following subsections we will discuss them.

Y(g) case for simple g. For the cyclic vector condition we consider representations for which Wk,r |0 = VkUr. According to [KR87], for all classical Lie algebras such representations exist. In particular, in type A all Kirillov-Reshetikhin modules are of this form, and in type C all

Kirillov-Reshetikhin modules with r =1 are of this form. Also in the orthogonal case the spin representation has such a form.

Theorem 0.3 ([Ma22]). If Wkr is a Kirillov-Reshetikhin module satisfying the condition above, then for all C G Treg for generic Uj, i.e. a Zariski open subset in C®N, Bethe algebra B(C) acts in representation V = W^,rj (uj) with a cyclic vector.

Now we state the result on the semismplicity.

Consider Tcroemp, the fixed points of the Cartan involution on Treg. I.e. Tcroemp = Treg n Gcomp.

Theorem 0.4 ([Ma22]). Let V be a representations of the Yangian Y(g) which is a tensor product of representations Wk,r (u) where u G iR with the condition that Wk,r is irreducible as g-module. Then V is Hermitian.

The main result of our work in this general case is the following:

Corollary 0.5. If C G Tl^p, the spectrum of Bethe subalgebra B(C) is simple in representations that satisfy the conditions of Theorems 13.1 and 13.4.

Cyclic vector and simplicity of spectrum, Y(gln) case. Let X G Mo<n+2 and consider the Bethe subagebra B(X). Our conjecture in this setting is as follows:

Conjecture 0.6. B(X) has a cyclic vector in any tame representation of Y(gln) for all X G

Mo,n+2.

In fact, it is easy to see that the Conjecture is true for generic X, z1,..., zN. Indeed, consider the parameter space M0,n+2 x CN. The condition that B(X) acts with a cyclic vector on Vi(zi) determines a Zariski open subset of M0,n+2 x CN, therefore once we have a single point (X, z1,..., zN) G M0,n+2 x CN such that B(X) acts with a cyclic vector on N=1 Vi(z:) we automatically have the same property for generic (X, z1,..., zN). On the other hand, according to [NT] the Gelfand-Tsetlin subalgebra of Y(gln) (which is a particular case of B(X)) acts without multiplicities on any tame representation, so has a cyclic vector. Hence this Zariski-open subset is non-empty. The problem with this argument is that it does not give any representation such that B (X) acts cyclicly for all X G M0,n+2.

Theorem 0.7 ([IMR]). There is a Zariski open dense subset of I C CN such that B(X) has a cyclic vector in V for all X G M0,n+2 and (zi,. .., zn) G I.

Particularly, in a generic tame representation in the sense of [NT] every Bethe subalgebra B(X) with X G M0 n+2 acts with a cyclic vector. This allows to study the joint spectrum of B(X) in a given tame representation as a finite covering of M0 n+2 and reformulate some properties of this spectrum in terms of geometry of Deligne-Mumford compactifications.

Theorem 0.8 ([IMR]). Let Wk,r be a Kirillov-Reshetikhin module such that its weights have no multiplicities. Then for all C G Treg for generic Uj G iR Bethe subalgebra B(C) acts in the representation V = j Wjj. ,rj (uj) with a cyclic vector.

Another main result of this work is the following theorem. It was first proved in type A by Reshetikhin [Re] for g = s[3.

Theorem 0.9 ([Ma22]). Let V be a representation of the Yangian Y(g) such that V is a tensor product of representations Wk,r(u) where u G iR and Wk,r is irreducible as a g-module. Then V is Hermitian (in the sense of definition 5.1).

Theorem 14.2 implies that once B(C) acts semisimply, it has simple spectrum (i.e. the joint eigenvalues have no multiplicities). The usual sufficient condition for this is the existence of a Hermitian scalar product such that B(C)+ = B(C) i.e. all elements of B(C) act by normal operators. We give sufficient conditions on the representation of the Yangian guaranteeing that such scalar product exists provided that C belongs either to the closure of the set of regular elements of the compact real torus Tcomp C T or to that of the split real torus Tspnt C T. So we get

Corollary 0.10 ([IMR]). For C G TZ£gp the spectrum of Bethe subalgebra B(C) in representations satisfying the conditions of Theorems 14.3 and 14-4 is simple.

The case of the compact torus goes back to Kirillov and Reshetikhin [KR86]. Then Wk,r (u) is a different notation for VkUr (kf — + u). Note that the closure of the set of regular points of the compact torus Tcomp in M0,n+2 is the compact form MCnp.. Therefore for the gln case we obtain:

Theorem 0.11 ([IMR]). Suppose that all V's are Kirillov-Reshetikhin modules. Let hi x Ti be

ki_rt

2 2

w™ ™ ™ 0\N

= 1

the size of the corresponding Young diagram. Suppose that Zi = 2 — rt+ixi, where xi G R. Then, for (xi, .. . ,xn) G RN from Zariski open subset, B(X) has simple spectrum on QQN=i Vi(zi) for

ail x g mc0nr+p2.

The closure of the set of regular points of the split real torus Tspiit in M0,n+2 is the real form

MoPn+2. Our next main result is

Theorem 0.12 ([IMR]). Let Vi,i = 1,. .. ,N be a set of skew representations of Y (gln). Then, for (xi,. .. ,xn) from a non-empty Zariski open subset in RN, B(X) has simple spectrum on

®N=i Vi(xi) for all X G M^.

Y(gl2) case. In the case of Y(gl2) we were able to find explicitly the Zariski open subset we discussed before.

A string is a set S(a, b) = {a — 1, a — 2,... ,b + 1,6} for a,b G C, a > b, a — b G Z. It is known that the representation L(a,b) = L(a1, b1) (... (g> L(aN, bN) is irreducible if and only if, for any 1 ^ i < j ^ N, one of three possibilities hold: S(ai,bi) U S(aj,bj) is not a string, or S(ai,bi) C S(aj,bj), or S(ai,bi) D S(aj,bj). We have the following two results:

Theorem 0.13 ([Ma21]). The action of any algebra in the family B in L(ai, bi)(.. .(L(aN, 6n) has a cyclic vector, if, for any 1 ^ i < j ^ n, S(ai, bi) U S(aj, bj) is not a string.

Secondly, we restrict to the closure of the subfamily corresponding to real diagonal matrices parametrized by the points of RPi ~ Z' C Z (see Theorem 3.5 for the definition of Z).

Theorem 0.14 ([Ma21]). For any x G Z' and any ai, bi . .., aN, 6n G R such that (ai, bi) and (aj, bj) are disjoint and S(ai, bi) U S(aj, bj) is not a string for each pair i,j, the subalgebra B(x) acts on L(ai, bi) ( ... ( L(aN, 6n) with simple spectrum.

Organization of the text. This text is organized as follows. In Section 1 we discuss the definitions of Yangians. In Section 2 we discuss the definition and properties of Deligne-Mumford spaces M0 n+2. In Section 3 we discuss the definitions and properties of Bethe subalgebras of Yangians. In Section 4 we present the representations of Yangians which are discussed in our work. In Section 5 we discuss what it means to be Hermitian for a representation of a Yangian. In Section 6 we discuss limits of Bethe subalgebras, i.e. the compactification of the flat family of Bethe subalgebras that preserves the properties of being maximal commutative. In Sections 7-10 we discuss quantum shift of argument subalgebras and their limits and their connection to Bethe subalgebras. In Section 11 we discuss the spectra of quantum shift of argument subalgebras. This concludes the preliminaries necessary for stating and proving our results.

In Section 12 we state the Conjectures we had in mind while developing our work. In Sections 13-15 we state the results and give the full proofs.

1. Yangians

1.1. Yangian for simple g. Let g be a simple complex Lie algebra, G is the corresponding connected simply connected Lie group, T is the maximal torus and Treg is the regular elements of T, h is the corresponding Cartan subalgebra, n = rk g = dim h.

Let $ be the root system corresponding to the Lie algebra g, are the positive roots, {ai,..., an} are the simple roots, {^i,..., wn} are the fundamental weights, (,) is the invariant

scalar product such that (a, a) = 2 for short simple roots, ga are the corresponding root subspaces of g, xa G ga,x- G g-a are such that (xa,x-) = 1, tUi G h is the element corresponding to G h* by the invariant scalar product. In the same way h: is the element corresponding to a~i = (a2aa -). Also we define the Casimir elements corresponding to the invarian scalar product

^ = (x+ < x- + x- < x+) + tUi < hj G g < g

a£$+ i

and

w = (x+x- + x-x+) + ^t^ihi G U(g).

a£$+ i

Definition 1.1. Yangian Y(g) is an associative algebra with a unit over C generated by the elements {x, J(x) | x G g} with the following relations:

xy - yx = [x,y], J([x, y]) = [J(x),y] J (cx + dy) = cJ (x) + dJ (y), [J(x), [J(y), z]] - [x, [J(y),J(z)]] = ([x, xa], [[y,xM], [z,xv]]){xA,xM,xv},

[[J(x), J(y)], [z, J(w)]] + [[J(z), J(w)], [x, J(y)]] = = 53 (([x,xa], [[y,xM], [[z,w],xv]]) + ([z,xa], [[w,xM], [[x,y],xv]])) {xx,x^,J(xv)}

for all x,y,z,w G g and c,d G C where {xA}AeA is some orthonormal basis of g, {x1, x2, x3} =

24 2ne&3 xn(1)xn

(2)xn(3) for x1 ,x2,x3 G Y(g).

The Yangian Y(g) is a Hopf algebra with comultiplication A, counit e, and antipode S defined

by

A(x) = x < 1 + 1 < x, A( J(x)) = J(x) < 1 + 1 < J(x) + 1[x < 1, H],

S(x) = -x, S(J(x)) = —J(x) + 1 cgx Vx G g,

e(x) = 0, e(J (x)) = 0,

where cg is the eigenvalue of w in the adjoint representation.

We will also denote by Aop the opposite comultiplication of Y(g); that is, Aop = a o A where

a = aY (g ),Y (g).

There is the shift automorphism tc of Y(g) defined by

x ^ x, J(x) ^ J(x) + cx, Vx G g.

Then we denote Tcd = tc < Td.

We are now prepared to introduce the universal R-matrix of Y (g).

Theorem 1.1 ([D85]). There is a unique formal series

R(u) = 1 + ^Rku-k G (Y(g) <g> Y(g))[[u-1]]

k=1

satisfying

(id < A)R(u) = R12(u)R13(u), T0,„Aop(y) = R(u)-1 (T0,„A(y))R(u) Vy G Y(g).

The series R(u) is called the universal R-matrix of Y(g) and it also satisfies the quantum Yang-Baxter equation

R12(u - v)R13(u)R23(v) = R23(v)R13(u)R12(u - v).

Here R12(u) = R(u) ( 1 € Y(g) ( Y(g) ( Y(g)[[u-1]], and R13(u) and R23(u) are defined similarly.

We can take the image of R(-u) under pv ( 1 for some finite-dimensional representation (V, pv) of Y(g). We will denote Tv(u) = pv ( 1(R(-u)) and call it T-operator. We can apply pv ( pV (1 to the Yang-Baxter equation and obtain the relations on the T-operator coefficients. The Fourier coefficients of the T-operator can be taken as another set of generators of Y(g).

Now we will take an algebra X (g) with such generators and following [W] will obtain a surjective homomorphism X(g) ^ Y(g).

Let V be a fixed finite-dimensional Y(g)-module with corresponding homomorphism p such that V has a non-trivial (not necessarily proper) irreducible submodule. We let R(u) denote the image of the universal R-matrix R(-u) under p ( p:

R(u) = (p ( p)R(-u) € End(V ( V)[[u-1]].

We fix a basis {e1,... ,eN} of V and we let {Ej}1^i,j^N denote the usual elementary matrices with respect to this basis.

The extended Yangian X (g) is the unital associative C-algebra generated by elements {tj | 1 ^ i,j ^ N,r ^ 1} subject to the defining RTT-relation

R(u - v)T1(u)T2(v) = T2(v)T1(u)R(u - v) in (End V)®2 ( X(g)[[v-1, u-1]],

where T(u) = YNNjj=1 Eij ( tj(u) with tj(u) = Sj + YT>1 t(ji)u-T for all 1 < i,j < N, Ta(u) = J2 ij 1®(a-1) ( Eij (1®(n-a) (tij (u) and R(u - v) hasbeen identified with R(u - v) (1. The extended Yangian is a Hopf algebra, with the Hopf algebra structure given by

A(T (u)) = T[1] (u)T[2] (u), S(T (u)) = T (u)-1, e(T (u)) = Id,

where T[1] (u) = YNN,j=1 Eij(tij(u)(1 € End V((X(g))®2 and T[2](u) = YNN,j=1 Eij®1®Uj(u) € End V ( (X(g))®2., ,

The RTT-Yangian Y(g) is the quotient of X(g) by the two-sided ideal generated by the

(T)

elements zj , for 1 ^ i,j ^ N and r ^ 1, defined by

N

ij zij— s \u))± + ^cs) 1

i,j=1

Z(u) —Y, Eij ® Zij(u) — S2(T(u))T(u + -Cg)

where zj (u) = Sij + Y1 T>1 Zj^u T for each pair of indices 1 ^ i,j ^ N.

The equivalence of two definitions was stated by V. Drinfeld [D85] and proved by C. Wend-landt in [W].

1.2. Yangian for gln. The definition of the extended Yangian X(g) for g = gln with the standard representation V of gln gives us the Yangian of gln.

The algebra Y(gln) is generated by elements tj, 1 < i,j < n, r € Z>o and tj = Sij. (The

elements tj correspond to Ejzr G fll„[z] where Ej G QÏn is the standard matrix unit.) The relations are

[t(r+1) t(s)i [t(r) t(s+1)-, — t(r)t(s) t(s)t(r) [tij , bkl J [tij ,bkl J — bkj Lil bkj Lil .

Introduce the formal power series in u-1, where u is a formal variable,

tij(u) — t(r'u

(r) -r ij

>0

These formal power series can be combined into a matrix with values in formal series with coefficients in Y(flln)

T (u) — ^ eij <g> tij (u) G End(Cn) <g> Y (fl[„ )[[u-1JJ, i,j

where ej is the standard matrix unit. Hence the relations can be rewritten as

R(u — v)T1(u)T2(v) — T2 (v)T1(u)R(u — v)

where

R(u) = 1 <g) 1 — u-1 ejj <g) ejj

is the R-matrix.

The algebra Y (g) for g = sin is the subalgebra of Y (gin) which consists of all elements stable under the automorphisms T(u) ^ f (u)T(u) for all f (u) G 1 + u-1C[[u-1]].

Proposition 1.2. ([Mo, Theorem 1.8.2, Proposition 1.84.])

(1) Y(gij = Y(si„) Z(Y(gin)), where Z(Y(gij) is the center of Y(gij;

(2) Y (sin) is a Hopf subalgebra of Y (gin).

For the details and links on the gln case, we refer the reader to the book [Mo] by A. Molev.

1.3. Some homomorphisms between Yangians. Let us define two different embedding of Y(gln) to Y(gln+k):

ik : Y(gln) ^ Y(gln+k); t(j ^ tj ^k : Y(gln) ^ Y(gln+k); tj ^ tkr)i,k+j

By definition, put

Wn : Y(gln) ^ Y(gln); T(u) ^ (T(-u - n))-1. It is well-known that wn is an involutive automorphism of Y(gln). We define a homomorphism

^k = Wn+k o fk o Wn : Y(gln) ^ Y(gln+k).

Note that is injective.

Proposition 1.3. [IR18, Lemma 2.12] The homomorphisms ik and ^n define an embedding

ik < ^n : Y(gln) < Y(glk) ^ Y(gln+k).

1.4. Centralizer construction. Consider the map

: Y(gln) ^ U(gln+k) given by $k = ev ow„+fc o ik. From [Mo, Proposition 8.4.2] it follows that Im$k C U(gln+k)g[fc. Here we use an embedding

glk ^ gln+k, Eij ^ Ei+n,j+n.

Let A0 = C[E1, E2,...] be the polynomial algebra of infinite many variables. Define a grading on A0 by setting deg Ei = i. For any k we have a surjective homomorphism

zk : A0 ^ Z(U(gln+k)); Ej ^f(n+k).

where £(n+k), i = 1, 2, 3,... are the following generators of Z(U(gln+k)) of degree i, see [Mo, Section 8.2]:

n+ k

1 + ^ Eiu-i = ev(qdet T(u))

i=1

Consider the algebra Y (gln) < A0. This algebra has a well-defined ascending filtration given by n

deg a < b = deg a + deg b For any k ^ 0 we define homomorphisms of filtered algebras

nk : Y(gln) < A0 ^ U(gln+k)g[fc; a < b ^ $k(a) • zk(b)

These homomorphisms are known to be surjective. Denote by (Y(gln) < A0)N the N-th filtered component, i.e. the vector space of the elements of degree not greater than N. From [Mo, Theorem 8.4.3] we have:

Theorem 1.4. The sequence {nk} is an asymptotic isomorphism. This means that for any N there exists K such that for any k > K the restriction of nk to the N-th filtered component (Y(gln) < A0)n is an isomorphism of vector spaces (Y(gln) < A0)n — U(gln+k)Nrfc.

Список литературы диссертационного исследования кандидат наук Машанова-Голикова Инна Антоновна, 2022 год

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