Lie algebra extensions of current algebras on S3

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International Journal of Geometric Methods in Modern Physics Vol. 12 (2015) 1550087 (7 pages) c World Scientific Publishing Company
DOI: 10.1142/S0219887815500875

Lie algebra extensions of current algebras on S 3

Int. J. Geom. Methods Mod. Phys. Downloaded from www.worldscientific.com by Professor Tosiaki Kori on 07/20/15. For personal use only.

Tosiaki Kori∗ and Yuto Imai† Department of Mathematics Graduate School of Science and Engineering Waseda University, Tokyo 169-8555, Japan ∗[email protected] †[email protected] Received 28 March 2014 Accepted 27 April 2015 Published 5 June 2015 An affine Kac–Moody algebra is a central extension of the Lie algebra of smooth mappings from S 1 to the complexification of a Lie algebra. In this paper, we shall introduce a central extension of the Lie algebra of smooth mappings from S 3 to the quaternization of a Lie algebra and investigate its root space decomposition. We think this extension of current algebra might give a mathematical tool for four-dimensional conformal field theory as Kac–Moody algebras give it for two-dimensional conformal field theory. Keywords: Infinite-dimensional Lie algebras; current algebra; Lie algebra extensions; quaternion analysis. Mathematics Subject Classification 2010: 81R10, 17B65, 17B67, 22E67

1. Introduction The set of smooth mappings from a manifold to a Lie algebra has been a subject of investigation both from a purely mathematical standpoint and from quantum field theory. In quantum field theory they appear as a current algebra or an infinitesimal gauge transformation group. Loop algebras are the simplest example. Loop algebras appear in the simplified model of quantum field theory where the space is onedimensional and many important facts in the representation theory of loop algebra were first discovered by physicists. As is well known Belavin et al. [2] constructed two-dimensional conformal field theory based on the irreducible representations of Virasoro algebra. The purpose of this paper is to introduce a current algebra on the three-dimensional sphere so that it may give a four-dimensional apparatus of conformal field theory. It turned out that in many applications to field theory one must deal with certain extensions of the associated loop algebra rather than the loop algebra itself. The central extension of a loop algebra is called an affine Lie algebra or a Kac–Moody algebra, and the highest weight theory of finite-dimensional Lie algebra was extended to this case [3, 6, 7, 15]. The theory of Kac–Moody algebras 1550087-1

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Int. J. Geom. Methods Mod. Phys. Downloaded from www.worldscientific.com by Professor Tosiaki Kori on 07/20/15. For personal use only.

T. Kori & Y. Imai

is related to many areas of mathematics such as the theory of modular forms, the theory of integrable systems and the theory of invariant forms. In mathematical physics, it has applications to statistical physics, to string theory and especially to conformal field theory. As an affine Lie algebra is a central extension of the Lie algebra of smooth mappings from S 1 to the complexification of a Lie algebra, so our objective is a central extension of the Lie algebra of smooth mappings from S 3 to the quaternization of a Lie algebra. In [13, 14] it was shown that, for any smooth manifold X and a simple Lie algebra g, there is a universal central extension of the Lie algebra Map(M, g). The kernel of the extension is given by the space of complex valued 1-forms modulo exact 1-forms; Ω1 (X)/dΩ0 (X). It implies that any extension is a weighted linear combination of extensions obtained as a pullback of the universal extension of the loop algebra Lg by a smooth loop f : S 1 → X. While two-dimensional conformal field theory is based on this central extension, we would like to provide a mathematical tool that could help constructing a four-dimensional conformal field theory. This is why we are dealing with central extensions of the Lie algebra of smooth mappings from S 3 to the quaternization of a Lie algebra. Here is a brief explanation of our results. The detailed proof will appear in [12]. Let H be the quaternion algebra. Let g be a complex Lie algebra and let U (g) be the enveloping algebra of g. The quaternification gH = (H ⊗ U (g), [ , ]gH ) of g is defined by the bracket [z ⊗ X, w ⊗ Y ]gH = (z · w) ⊗ (XY ) − (w · z) ⊗ (Y X), for z, w ∈ H and X, Y ∈ U (g). Let S 3 H be the (non-commutative) algebra of H-valued smooth mappings over S 3 and let S 3 gH = S 3 H ⊗ U (g). The Lie algebra structure on S 3 gH is induced naturally from that of gH . We introduce a 2cocycle on S 3 gH with the aid of a tangential vector field on S 3 ⊂ C2 and have the corresponding central extension S 3 gH ⊕ (Ca). As a subalgebra of S 3 H we have the algebra of Laurent polynomial spinors C[φ± ] spanned by a complete orthogonal system of eigenspinors {φ±(m,l,k) }m,l,k of the tangential Dirac operator on S 3 . Then C[φ± ] ⊗ U (g) is a Lie subalgebra of S 3 gH , and we have the central extension
g(a) = (C[φ± ] ⊗ U (g)) ⊕ (Ca) as a Lie-subalgebra of S 3 gH ⊕ (Ca). Finally we have a Lie algebra
g which is obtained by adding to
g(a) a derivation d which acts on
g(a) by the Euler vector field d0 . That is the C-vector space
g = (C[φ± ] ⊗ U (g)) ⊕ (Ca) ⊕ (Cd) endowed with the bracket [φ1 ⊗ X1 + λ1 a + µ1 d, φ2 ⊗ X2 + λ2 a + µ2 d]bg = (φ1 φ2 ) ⊗ (X1 X2 ) − (φ2 φ1 ) ⊗ (X2 X1 ) + µ1 d0 φ2 ⊗ X2 − µ2 d0 φ1 ⊗ X1 + (X1 | X2 )c(φ1 , φ2 )a. Ca is the 1-dimensional center of the Lie algebra
g, and the centralizer of d in
g is the direct sum: C[φ± ]0 ⊗ U (g) ⊕ Ca ⊕ Cd, where C[φ± ]0 is the subspace of C[φ± ] of homogeneous order 0. When g is a simple Lie algebra with its Cartan 1550087-2

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subalgebra h we shall investigate the weight space decomposition of
g with respect +(0,0,1)
⊗ h) ⊕ (Ca) ⊕ (Cd). to the subalgebra h = (φ

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2. Quaternification of a Lie Algebra Let H be the quaternion numbers. In this paper, we shall denote a quaternion  a + jb ∈ H by ab . This comes from the identification of H with the matrix algebra    a −b mj(2, C) = : a, b ∈ C . b a     So the multiplication of quaternions a + jb = ab and c + jd = dc is given by       a c ac − bd . · = b bc + ad d H becomes an associative  algebra and the Lie algebra structure (H, [ , ]H ) is induced on it. The trace of a = ab ∈ H is defined by tr a = a + a. For u, v, w ∈ H we have tr([u, v]H · w) = tr(u · [v, w]H ). Let (g, [ , ]g ) be a complex Lie algebra. Let U (g) be its enveloping algebra. The quaternification of g is defined as the vector space gH = H ⊗ U (g) endowed with the bracket [z ⊗ X, w ⊗ Y ]gH = (z · w) ⊗ (XY ) − (w · z) ⊗ (Y X), for z, w ∈ H and X, Y ∈ U (g). It extends the Lie algebra structure (g, [ , ]g ) to (gH , [ , ]gH ). The quaternions H give also a half spinor representation of Spin(4). That is, ∆ = H ⊗ C = H ⊕ H gives an irreducible complex representation of the Clifford algebra Clif(R4 ): Clif(R4 ) ⊗ C  End(∆), and ∆ decomposes into irreducible representations ∆± = H of Spin(4). Let S ± = C2 × ∆± be the trivial even (respectively odd) spinor bundle. A section of spinor bundle is called a spinor. The space of even half spinors C ∞ (S 3 , S + ) is identified with the space S 3 H = Map(S 3 , H). Now the space S 3 gH = S 3 H ⊗ U (g) becomes a Lie algebra with respect to the Lie bracket: [φ ⊗ X, ψ ⊗ Y ]S 3 gH = (φψ) ⊗ (XY ) − (ψφ) ⊗ (Y X), for X, Y ∈ g and φ, ψ ∈ S 3 H, which is extended to S 3 gH . In the sequel, we shall abbreviate the Lie bracket [ , ]H simply to [ , ]. Such an abbreviation will be often adopted for other Lie algebras. 3. Analysis on H We shall review the theory of spinor analysis after [1, 5, 9–11]. Let D : S + → S − be ∂ −∂/) be the polar decomposition on the (half spinor) Dirac operator. Let D = γ+ ( ∂n 3 2 S ⊂ C of the Dirac operator, where ∂/ is the tangential Dirac operator on S 3 and γ+ is the Clifford multiplication of the unit normal derivative on S 3 . The eigenvalues m+3 of ∂/ are given by { m 2 , − 2 ; m = 0, 1, . . .} with multiplicity (m+1)(m+2). We have 1550087-3

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an explicitly written formula for eigenspinors {φ+(m,l,k) , φ−(m,l,k) }0≤l≤m,0≤k≤m+1 m+3 corresponding to the eigenvalue m respectively and they give rise to 2 and − 2 2 3 + a complete orthogonal system in L (S , S ). A spinor φ on a domain G ⊂ C2 is called harmonic spinor on G if Dφ = 0. Each φ+(m,l,k) is extended to a harmonic spinor on C2 , while each φ−(m,l,k) is extended to a harmonic spinor on C2 \ {0}. Every harmonic spinor ϕ on C2 \ {0} has a Laurent series expansion by the basis φ±(m,l,k) :   ϕ(z) = C+(m,l,k) φ+(m,l,k) (z) + C−(m,l,k) φ−(m,l,k) (z). Int. J. Geom. Methods Mod. Phys. Downloaded from www.worldscientific.com by Professor Tosiaki Kori on 07/20/15. For personal use only.

m,l,k

m,l,k

If only finitely many coefficients are nonzero, it is called a spinor of Laurent polynomial type. We denote by C[φ± ] the set of spinors of Laurent polynomial type. The boundary restriction of C[φ± ] to S 3 is found to have a multiplication structure 1 3 +(0,0,1) of S H algebraically generated by φ = so that C[φ± ] is a subalgebra 0 ,  z2  z2  0 +(0,0,0) +(1,0,1) −(0,0,0) = −1 , φ = −¯z1 and φ = z¯1 . φ 4. 2-cocycle on S 3 H Recall that the central extension of a loop algebra Lg = C[z, z −1] ⊗ g is the Lie algebra (Lg ⊕ Ca, [ , ]c ) given by the bracket [P ⊗ X, Q ⊗ Y ]c = P Q ⊗ [X, Y ] + (X | Y )c(P, Q)a, 1 dP −1 ]. Here (· | ·) with the aid of the 2-cocycle c(P, Q) = 2πi |z|=1 dz · Q dz on C[z, z is a non-degenerate invariant symmetric bilinear form on g [6, 7]. We shall give an analogous 2-cocycle on S 3 H. Let θ be the vector field on S 3 defined by θ = z1 For ϕ =

u v

∂ ∂ ∂ ∂ + z2 − z¯1 − z¯2 . ∂z1 ∂z2 ∂ z¯1 ∂ z¯2

∈ S 3 H, we put 1 Θϕ = √ 2 −1



θu θv

(1)

 .

Let c : S 3 H × S 3 H → C be the bilinear form given by 1 tr[Θ φ1 · φ2 ]dσ, φ1 , φ2 ∈ S 3 H. c(φ1 , φ2 ) = 2π 2 S 3

(2)

c defines a 2-cocycle on the algebra S 3 H. That is, c satisfies the following equations: c(φ1 , φ2 ) = −c(φ2 , φ1 ) and c(φ1 · φ2 , φ3 ) + c(φ2 · φ3 , φ1 ) + c(φ3 · φ1 , φ2 ) = 0. 1550087-4

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5. Extensions of the Lie Algebra C[φ±(m,l,k) ] ⊗ U (g) We extend the 2-cocycle c on S 3 H to the 2-cocycle on S 3 gH by c(φ1 ⊗ X, φ2 ⊗ Y ) = (X | Y ) c(φ1 , φ2 ),

(3)

where (· | ·) is the non-degenerate invariant symmetric bilinear form on g extended to U (g).
3 gH = S 3 gH ⊕ Ca we Let a be an indefinite number. On the C-vector space S define the following Lie bracket. For X, Y ∈ U (g) and φ, ψ ∈ S 3 H we put Int. J. Geom. Methods Mod. Phys. Downloaded from www.worldscientific.com by Professor Tosiaki Kori on 07/20/15. For personal use only.

[φ ⊗ X, ψ ⊗ Y ]∧ = (φψ) ⊗ (X Y ) − (ψφ) ⊗ (Y X) + c(φ, ψ)(X | Y ) · a, ∧

[a, φ ⊗ X] = 0.
3 gH , [ , ]∧ ) becomes an extension of the Lie algebra S 3 gH with oneThen ( S dimensional center Ca.
3 gH induces also the central extension of The above Lie algebra structure on S ± the Lie subalgebra C[φ ] ⊗ U (g) of the spinors of Laurent polynomial type, that will be denoted by (
g(a), [ , ]∧ );
g(a) = C[φ± ] ⊗ U (g) ⊕ Ca. Finally denote by
g the Lie algebra which is obtained by adding to
g(a) a derivation d which acts on
g(a) as radial derivation d0 on S 3 and which kills a. The radial derivation is defined by d0 = 12 (z1 ∂z∂ 1 + z2 ∂z∂ 2 + z 1 ∂z∂ 1 + z 2 ∂z∂ 2 ). We have the following fundamental property of the cocycle c. c(d0 φ1 , φ2 ) + c(φ1 , d0 φ2 ) = 0. g with the bracket Theorem 1. Let
g = (C[φ± ] ⊗ U (g)) ⊕ (Ca) ⊕ (Cd). We endow
defined by [φ ⊗ X, ψ ⊗ Y ]bg = [φ ⊗ X, ψ ⊗ Y ]∧ , [d, a]bg = 0,

[a, φ ⊗ X]bg = 0,

[d, φ ⊗ X]bg = d0 φ ⊗ X.

Then (
g, [ , ]bg ) is an extension of the Lie algebra
g(a) on which d acts as d0 .
6. Structure of g Suppose that g is a simple Lie algebra with its Cartan subalgebra h. We shall investigate the weight space decomposition of
g with respect to the subalgebra
h = (φ+(0,0,1) ⊗ h) ⊕ (Ca) ⊕ (Cd), the latter is a commutative subalgebra and ad(
h) acts on g diagonally. First we prepare preliminaries on the representation of

the adjoint action of h on the enveloping algebra U (g) [4]. Let g = α∈∆ gα be the root space decomposition of g. Let Π = {αi ; i = 1, . . . , r = rank g} ⊂ h∗ be the set of simple roots and {α∨ i ; i = 1, . . . , r } ⊂ h be the set of simple coroots. 1550087-5

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The Cartan matrix A = (aij )i,j=1,...,r is given by aij = α∨ i , αj . Fix a standard ∨ set of generators Hi = αi , Xi = Xαi ∈ gαi , Yi = X−αi ∈ g−αi , so that [Xi , Yj ] = Hj δij , [Hi , Xj ] = −aji Xj and [Hi , Yj ] = aji Yj . We see that the set of weights of the representation (U (g), ad(h)) becomes  r   ∗ Σ= ki αi ∈ h ; ki ∈ Z, i = 1, . . . , r (4) i=1

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The weight space of λ ∈ Σ is by definition gU λ = {ξ ∈ U (g); ad(h)ξ = λ(h)ξ, ∀ h ∈ h},

r  0. Then, given λ = i=1 ki αi , we have when gU λ =

(5)

q1 qr l1 lr p1 pr gU λ = C[Y1 · · · Yr H1 · · · Hr X1 · · · Xr ;

pi , qi , li ∈ N ∪ 0, ki = pi − qi , i = 1, . . . , r]. The weight space decomposition becomes  U (g) = gU λ,

gU 0 ⊃ U (h).

(6)

(7)

λ∈Σ

Now we proceed to the representation (
g, ad(
h)). The dual space h∗ of h can be ∗
h∗ . We regarded naturally as a subspace of h . So Σ ⊂ h∗ is seen to be a subset of
∗
define δ ∈ h by putting δ, hi = δ, a = 0, 1
i
r, and δ, d = 1. Then the set
of the representation (
of weights Σ g, ad(
h)) is m 
= Σ δ + λ; λ ∈ Σ, m ∈ Z , 2 (8)  m δ; m ∈ Z . ∪ 2 The weight space decomposition of
g is given by      


g= g m2 δ g m2 δ+λ . m∈Z

λ∈Σ,m∈Z

Each weight space is given as follows.
g m2 δ+λ = C[φ± ; m] ⊗ gU λ
g m2 δ = C[φ± ; m] ⊗ gU 0

for m = 0 and λ = 0, for m = 0,



g0δ = (C[φ± ; 0] ⊗ gU 0 ) ⊕ (Ca) ⊕ (Cd) ⊃ h, where

    z ± m C[φ ; m] = ϕ ∈ C[φ ]; |z| ϕ = ϕ(z) . |z| ±

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Lie algebra extensions of current algebras on S 3

Int. J. Geom. Methods Mod. Phys. Downloaded from www.worldscientific.com by Professor Tosiaki Kori on 07/20/15. For personal use only.

References [1] F. Brackx, R. Delanghe and F. Sommen, Clifford Analysis (Pitman advanced publishing program, Boston-London-Melbourne, 1982). [2] A. A. Belavin, A. M. Polyakov, A. B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B 241 (1984) 333–380. [3] R. W. Carter, Lie Algebras of Finite and Affine Type, Cambridge Studies in Advanced Mathematics, Vol. 96 (Cambridge University Press, Cambridge, 2005). [4] J. Dixmier, Alg`ebres Enveloppantes, Cahiers Scientifiques, Vol. 37 (Gauthier-Villars, Paris, 1974). [5] J. Gilbert and M. Murray, Clifford Algebras and Dirac Operators in Harmonic Analysis (Cambridge University Press, Cambridge, 1991). [6] V. Kac, Infinite Dimensional Lie Algebras (Cambridge University Press, Cambridge, 1983). [7] B. Khesin and R. Wendt, The Geometry of Infinite-dimensional Groups, A Series of Modern Surveys in Mathematics, Vol. 51, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 3 (Springer-Verlag, Berlin, 2009). [8] T. Kori, Lie algebra of the infinitesimal automorphisms on S 3 and its central extension, J. Math. Kyoto Univ. 36 (1996) 45–60. [9] T. Kori, Index of the Dirac operator on S 4 and the infinite-dimensional Grassmannian on S 3 , Japan. J. Math. (N.S.) 22 (1996) 1–36. [10] T. Kori, Spinor analysis on C2 and on conformally flat 4-manifolds, Japan. J. Math. 28(1) (2002) 1–30. [11] T. Kori, Extensions of current groups on S 3 and the adjoint representations, J. Math Soc. Japan 66(3) (2014) 819–838. [12] T. Kori and Y. Imai, Quaternifiations and extensions of current algebras on S 3 , preprint (2013), arXiv: 1306.5030 [math-ph]. [13] J. Mickelsson, Kac–Moody groups, topology of the dirac determinant bundle and fermionization, Commun. Math. Phys. 110 (1987) 175–183. [14] A. Pressley and G. Segal, Loop Groups, Oxford Mathematical Monographs (Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1986). [15] M. Wakimoto, Infinite-Dimensional Lie Algebras, Translations of Mathematical Monographs, Vol. 195 (American Mathematical Society, Providence, RI, 2001).

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