Polynomial quantiztion and overalgebra for hyperboloid of one sheet Текст научной статьи по специальности «Математика»

ISSN 1810-0198. Вестник Тамбовского университета. Серия: естественные и технические науки

Том 23, № 123

2018

DOI: 10.20310/1810-0198-2018-23-123-353-360

POLYNOMIAL QUANTIZTION AND OVERALGEBRA FOR HYPERBOLOID OF ONE SHEET

V. F. Molchanov

Tambov State University named after G.R. Derzhavin 33 Internatsionalnaya St., Tambov 392000, Russian Federation E-mail: [email protected]

Abstract. We show that the multiplication of symbols in polynomial quantization is exactly an action of an overalgebra on the space of these symbols Keywords: quantization; representations; hyperboloids; Poisson transforms

In |1| we constructed quantization in the spirit of Berezin on para-Hcrmitian symmetric spaccs G/H, see also [2]. In [3] we showed that this quantization, anyway polynomial quantization - the most algebraic variant of quantization, can be considered as a part of the representation theory. In present paper we continue our activity in this direction, namely, we show that the multiplication of symbols is exactly an action of an overalgebra on the space of symbols, see Theorem 2. Here we restrict ourselves to a hyperboloid of one sheet in M3 . Besides, we write explicit formulae of this action.

The study of actions of ovcralgebras is a new theme, opened by Yu. A. Neretin and the author [4-6].

In this paper the group G is the group SL(2,M) , the subgroup H consists of diagonal matrices, the space G/H is a hyperboloid of one sheet in ffi3 . The overgroup G = G * G contains three subgroups Gd, Gi h G2 isomorphic to G. Namely, they consist of pairs (y,g) , (g, E), (E,g), respectively. Here E is identity matrix, g /G.

Let g be the Lie algebra of G. Then the Lie algebras of G and Gd, G1, G2 are 9 = 0 + 0 and , 0!, g2 , respectively. In order to write an action of the overalgebra g , it is sufficient to take some subspace complementary to gd. Now we take the subalgebra 02 -It consists of pairs (0, X) , where X / $ .

The work is supported by the Ministry of Education and Science of the Russian Federation (Project № 3.8515.2017/8.9).

The group G consists of real matrices of the second order with unit determinant:

ff=(" f), aS Pi = 1- (1)

Changing in (1) a e 5 and ft e 7, we obtain an involution g g in G given by

9=U «

The Lie algebra 0 of the group G consists of real matrices of the second order with zero trace. A basis in g consists of matrices:

=(f v2). i+=1" --]■ «

/0 1 \o 0

The commutation relations are:

[L+fL_]= 2Lu [L+, Li] = L+, [L1;L_]= L (3)

Denote by Env (g) the universal enveloping algebra of the Lie algebra g .

Rccall some material on representations of G. We shall use the notation

tx>" = \f\AsgnX t/R*=R }0|, A/C, i/ = 0,l.

For a / C, 1/ = 0,1, let us denote by the space of functions / in C°°(M) such

that the function fit) = t2a'"f(ljt) belongs to C°°(M) too. The representation of the group G acts on "^„(R) by (we consider that G acts from the right):

M9)f) (t) = f(tyg) (ßt + 5)^, t yg = ^L .

The contragredient representation tiv,„ is defined by the involution g g, so that

faMf) (*) = / (* (7 t + a)2°r

Representations ira}ll and tt^,, are equivalent by means of the operator / f.

Any irreducible finite-dimensional representation pk of the group G is labelled by the number k (the highest weight) such that 2k / N = }0. 1,2,.. . | . It acts on the space of polynomials ip(t) in t of degree ^ 2k (so that dim = 2k + 1) by

(Pk(g)<f) (t) = <p(t yg) 0Bt + 5fk.

Operators corresponding to elements of g and Env (g) in representations 7ra iJ do not depend on v, so we do not write v in indexes. For basis elements (2) we have

TT(r(L_) = ^, 7ia(L1) = t^ a, Tra(L+) = t22at.

and Tra(L±) = ^(Lzp), TTa(Li) = n^iLi). Replacing here a by k, wc obtain formulas for pk.

On the product <pip of functions tp and i/j the differential operators ^(L), L / g, (they have the first order) act as follows:

71a(L)(ip^) = (Xcr(L)ip) yup + y ><7To(L)ip), (4)

and similarly to na .

An operator defined by

/00

(1 ts) 2" 2'" f(s)ds

-00

intertwines 7rCTji, and :

Tf-cr-i,v(g)A(r^ = A^ir^g) , and also 7f^ and tt^i^ . The composition ^^ and A-a-is a scalar operator:

A—,7—1 ij j4(j ij = -7 r xE,

c(cr, 1/)

where

, . 2a+ 1 ( I),/ + cos2(77r

c(iT,eJ = —--x-;—--

Z7T Sin 2(77T

Let us realize the space M4 of vectors x = (x0, x1,x2, x3) as the space of real 2*2 matrices:

1 ( X0 X3 X1 + x2

X = —

2 \ x1 + x2 x0 + x3 The overgroup G acts as follows:

x gi1xg2, (31,^2) / G.

Let T> be the cone det x = 0 , x t= 0 . For a / C , u = 0,1, let denote the space

of C°° functions / on the cone 'D homogeneous of degree 2a and parity u:

f(tx) = t2^f(x), t/R*.

Let R<jj. be the representation of G by translations on the space Urj- i'V) (in fact, it is a representation of the group SO0(2, 2) associated with a cone, G covers SO0(2,2) with multiplicity 2):

(R<r,v(gi,92)f)(x) = f(gi1xg2).

The section { of T) by plane (trx) = 1 can be identified with a hyperboloid of one sheet x\ + x\ + x\ = 1 in M3 . Restrictions of functions in 7i<rto { form a space 'hbjj ({ )

of functions on { .It is contained in CM({ ) and contains 7i({ ). In the realization on { the representation Ra v is:

(R,A9u92)f)(x) = f ( 9/*92 ) {tr (g^xg*)}2"*, x/{.

Vtr(5l xg2)J

The section { is invariant with respect to the action x ë» g_1xg of Gd = G, it is just the space G/H. The restriction of Ra v to Gd = G is the quasiregular representation U of G on { . It preserves the space ¿¡"(I ) of polynomials on { and decomposes in the direct sum: U = pQ+pi~\-P2 + ■ ■ ■ with the corresponding decomposition S^l ) = Xo+X1+X2 +...

Introduce on { horospherical coordinates £,77:

s = 4 ( f ? ]> N = N(Cv) = 1 fr,

N \ £ 1 so that

_ Z + V £_V _ 1 + (rj

Xl ~ N ' X2" N ' X3" N " In these coordinates, to basis elements (2) the following operators correspond:

Recall some material on polynomial quantization. As a supercomplete system we take the kernel of the intertwining operator , namely,

^Atv) = m.v)^ ■

This function has an invariance property

O^a(g) =

This formula can be rewritten as

(Ms-1) 01) = (1OM9)) *.A£,v)- (5)

For elements L of the Lie algebra g , formula (5) gives:

(7ra(L) 01) = (10ML)) (6)

Covariant symbols of operators ir(T(X), X / Env (9) , are functions F(x) on { defined as follows:

FiLv) = OD^v).

In particular, covariant symbols for basis elements (2) are multiplied by ( a) polynomials

2'i x-2 , X3 , a'i + a'2 , respectively.

The multiplication of operators gives rise to the multiplication of covariant symbols, denote it by Namely, let Fj and F2 be covariant symbols of operators D1 and D2 respectively. Then the covariant symbol Fj ®F2 of the product DiD2 is

(F, ®F2)(C,v) = ^ ? (A 01)(^(i,ij) m,v)) ■ CO

Let V be a covariant symbol of the first order (corresponding to an clement L of the Lie algebra g ). Let F be an arbitrary covariant symbol (corresponding to an element X of the universal enveloping algebra Env (g))

Theorem 1. We have (the point means pointwise multiplication)

V®F = VxF + (tt0(L) Ol)) F (8)

F®V = V xF (lCtfro(L))F (9)

Proof. To prove (8), we take in formula (7) = 7rCT(L) and F2 = F, then we differentiate by (4), as a result we get (8).

Now let D = 7ra(X). Since (Dv^L)) Ol = (D Ol)ML) Ol), we have

Fm = J_ {D oi)(ML) OD^,

then by (6) we can change here the latter operator by the operator } (1 0^r{£))l and then transpose it with D 01 since they act on different variables. We obtain

F®V =

(10)

then we differentiate by (4) and use (6) again. It gives (9). □

Theorem 2. The multiplication of covariant symbols F by first order symbols V is the action of the overalgebra g on the space of covariant symbols:

V®F= Ra( 0, L)F, F®V = 0 )F. (11)

Proof. Formula (7) with D\ = 7va(L) and F2 = F gives exactly the first formula in

(11). The second formula is just (10). □

For k / N, we define the Poisson kernel F\(x; t) as follows. Denote

B(x;t) = B&mt)={t Vt), (12)

then

Pk(x;t) = B(x;t)k. (13)

This kernel is a fixed vector in the tensor product U OPk

(U(g) O Pk(g)Pk) (x; t) = Pk(x; t), g /G.

Therefore, Pk(x; t) is a generating function for polynomials in Xk ■

Let us introduce the following differential operators Sk(X) , k /N,and E(X) invariable t, linearly depending on X / g, for basic elements (2) they are

E(D = 1, Sk(L ) = ^ , E(L1)=t, 5^)=^

E(L+) = t2, Sk(L+) = 12(2k + 1 + (2fc + l)(2fc + 2).

at£ at

The following commutation relations hold

([.X,Y1) = Pk(X) Sk(Y) Sk(Y) Pk+1(X), E([X,Y}) = Pk(X) E(Y) E(Y) pkl(X).

Then, let us introduce the following coefficients ak , f3k , :

2 a k

Oik =

ft-

Ik =

(2k + 2)(2k + 1) ' 1 2'

(2(7 + A; + l)k

2(2k + l)

Theorem 3. Let X / g . The operator Ra{0, X) acts on the Poisson kernel Pk(x] t) as follows:

Ra{0, X) Pk = ak xSfc(X) Pk+1 + ft ypk(X) Pk + 7fc xE(X) Pk, (14)

in the left hand side the operator acts on a function of r/, and in the left hand side the operators act on functions of t.

Proof. First we take X = . Keeping in mind (12), (13), we find:

| = kB^J^ + i 2a + k)Bk^. (15)

On the other hand, we compute (d/dt)Bk and (d2/dt2)Bk+1:

~dt ~ kB X N

= kBkl+2kBk~1 (16)

= (fc + ljjfcS*"1 2(2fc + l)Bfcx|F}. (17)

Expressing from (16) and (17) the second summands in right hand sides, substituting in (15), we obtain (14) for X = . Now for X = Ly and X = L+ , we use equality (14) with X = L_ already proved and commutation relations - successively the first and the second ones in (3), and corresponding commutation relations for operators Sk(X) and E(X). □

REFERENCES

1. Molchanov V.F. Quantization on para-Hermitian symmetric spaces. Amer. Math. Soc. Transl., Ser. 2, 1996, vol. 175, pp. 81-95.

2. Molchanov V.F., Volotova N.B. Polynomial quantization on rank one para-Hermitian symmetric spaces. Acta Appl. Math., 2004, vol. 81, no. 1-3, pp. 215-232.

3. Molchanov V.F. Berezin quantization as a part of the representation theory. Vestnik Tamhovs-kogo universiteta. Seriya Estestvennye i tekhnicheskie nauki - Tambov University Reports. Series: Natural and Technical Sciences, 2017, vol. 22, no. 6, pp. 1235-1246. DOI: 10.20310/1810-0198-201722-6-1235-1246.

4. Neretin Yu.A. Deystvie nadalgebry v plans her elevskom razlozhenii i operatory sdviga v mnimom napravlenii [The action of an overalgebra in the Plancherel decomposition and shift operators in an imaginary direction]. Izvestiya Rossiyskoy akademii nauk. Seriya matematicheskaya -Izvestiya: Mathematics, 2002, vol. 66, no. 5, pp. 171-182. (In Russian).

5. Molchanov V.F. Canonical representations and overgroups. Amer. Math. Soc. Transl., Ser. 2, 2003, vol. 210, pp. 213-224.

6. Molchanov V.F. Canonical representations for hyperboloids: an interaction with an overalgebra. Geometric Methods in Physics. Bialowieza, 2016, pp. 129-138.

Received 23 April 2018 Reviewed 25 May 2018 Accepted for press 19 June 2018

Molchanov Vladimir Fedorovich, Tambov State University named after G.R. Derzhavin, Tambov, the Russian Federation, Doctor of Physics and Mathematics, Professor of the Functional Analysis Department, e-mail: [email protected]

For citation: Molchanov V.F. Polynomial quantization and overalgebra for hyperboloid of one sheet. Vestnik Tambovskogo universiteta. Seriya: estestvennye i tekhnicheskie nauki - Tambov University Reports. Series: Natural, and Technical Sciences, 2018, vol. 23, no. 123, pp. 353-360. DOI: 10.20310/1810-0198-2018-23-123-353-360 (In Engl., Abstr. in Russian).

DOI: 10.20310/1810-0198-2018-23-123-353-360 УДК 517.98

ПОЛИНОМИАЛЬНОЕ КВАНТОВАНИЕ И НАДАЛГЕБРА ДЛЯ ОДНОПОЛОСТНОГО ГИПЕРБОЛОИДА

-е- В. Ф. Молчанов

ФГБОУ ВО «Тамбовский государственный университет им. Г.Р. Державина» 392000, Российская Федерация, г. Тамбов, ул. Интернациональная, 33 E-mail: [email protected]

Аннотация. Мы показываем, что умножение символов в полиномиальном квантовании есть в точности действие надалгебры на пространстве этих символов. Ключевые слова: квантование; представления; гиперболоиды; преобразования Пуассона

СПИСОК ЛИТЕРАТУРЫ

1. Molchanov V.F. Quantization on para-Hermitian symmetric spaces 11 Amer. Math. Soc. Transl. Ser. 2. 1996. Vol. 175. P. 81-95.

2. Molchanov V.F., Volotova N.B. Polynomial quantization on rank one para-Hermitian symmetric spaces // Acta Appl. Math. 2004. Vol. 81. № 1-3. P. 215-232.

3. Molchanov V.F. Berezin quantization as a part of the representation theory j I Вестник Тамбовского университета. Серия Естественные и технические науки. Тамбов, 2017. Т. 22. Вып. 6. С. 1235-1246. DOI: 10.20310/1810-0198-2017-22-6-1235-1246.

4. Неретин Ю.А. Действие надалгебры в планшерелевском разложении и операторы сдвига в мнимом направлении j j Известия РАН. Серия математическая. 2002. Т. 66. № 5. С. 171-182.

5. Molchanov V.F. Canonical representations and overgroups // Amer. Math. Soc. Transl.. Ser. 2. 2003. Vol. 210. P. 213-224.

6. Molchanov V.F. Canonical representations for hyperboloids: an interaction with an overalgebra // Geometric Methods in Physics. Bialowieza. 2016. P. 129-138.

Поступила в редакцию 23 апреля 2018 г. Прошла рецензирование 25 мая 2018 г. Принята в печать 19 июня 2018 г.

Молчанов Владимир Федорович, Тамбовский государственный университет им. Е Р. Державина. Тамбов, Российская Федерация, доктор физико-математических наук, профессор кафедры функционального анализа, e-mail: [email protected]

Для цитирования: Молчанов В.Ф. Полиномиальное квантование и над алгебра для однополостного гиперболоида j j Вестник Тамбовского университета. Серия: естественные и технические науки. Тамбов, 2018. Т. 23. № 123. С. 353-360. DOI: 10.20310/1810-0198-2018-23-123-353-360

Работа выполнена при поддержке Министерства образования и науки Российской Федерации (проект № 3.8515.2017/8.9).