W 1+inf-algebra

The (universal) one-dimensional central extension of the Lie algebra consisting of the differential operators on the unit circle $S ^ { 1 }$ which have a finite Fourier expansion (cf. also Fourier series). As such it appeared for the first time in [a4] (there the notation $W _ { 1 + \infty}$ was not used). After passing to the complex plane, it can equivalently be described as the central extension of the Lie algebra consisting of the holomorphic differential operators on the punctured disc $\mathbf{C} ^ { * } = \mathbf{C} \backslash \{ 0 , \infty \}$ which have algebraic poles at $0$ and $\infty$. A basis of the algebra is given by the set

\begin{equation*} \left\{ z ^ { n } \left( \frac { d } { d z } \right) ^ { m } : n \in {\bf Z} , m \in {\bf N} _ { 0 } \right\} \end{equation*}

and a central element $C$. Introducing $D = z d / d z$, the elements of $W _ { 1 + \infty}$ can be given as linear combinations of elements of the type $z ^ { k } f ( D )$ (with $k \in \mathbf{Z}$ and $f$ arbitrary polynomials) and the central element $C$. The Lie bracket calculates as

\begin{equation*} [ z ^ { n } f ( D ) , z ^ { m } g ( D ) ] = \end{equation*}

\begin{equation*} = z ^ { n + m } ( f ( D + m ) g ( D ) - f ( D ) g ( D + n ) ) + \end{equation*}

\begin{equation*} + \psi ( z ^ { n } f ( D ) , z ^ { m } g ( D ) ) \cdot C, \end{equation*}

with the Lie algebra two-cocycle

\begin{equation*} \psi ( z ^ { n } f ( D ) , z ^ { m } g ( D ) ) = \end{equation*}

\begin{equation*} = \left\{ \begin{array} { l l } { \sum _ { - n \leq i \leq - 1 } f ( i ) g ( i + n ) , } & { n = - m > 0, } \\ { - \sum _ { n \leq i \leq - 1 } f ( i - n ) g ( i ) , } & { n = - m > 0, } \\ { 0 , } & { \left\{ \begin{array} { l } { n + m \neq 0, } \\ { n = m = 0. } \end{array} \right.} \end{array} \right. \end{equation*}

The cocycle can also be represented as

\begin{equation*} \psi \left( a ( z ) \left( \frac { d } { d z } \right) ^ { n } , b ( z ) \left( \frac { d } { d z } \right) ^ { m } \right) = \end{equation*}

\begin{equation*} = \frac { m ! n ! } { ( m + n + 1 ) ! } \frac { 1 } { 2 \pi i } \oint _ { z = 0 } a ^ { ( m + 1 ) } ( z ) b ^ { ( n ) } ( z ) d z. \end{equation*}

This cocycle defines, up to isomorphy, the unique non-trivial one-dimensional central extension.

The Lie algebra $W _ { 1 + \infty}$ becomes a ($\bf Z$-) graded algebra (with infinite-dimensional homogeneous subspaces) by defining the degree as $\operatorname { deg } ( z ^ { n } f ( D ) ) = n$ and $\operatorname { deg } ( C ) = 0$.

The space of differential operators of (differential) degree one is a subalgebra of the Lie algebra of all differential operators. Taking also the central element into account one obtains the Virasoro algebra.

By setting $L _ { n } = - z ^ { n } D$, $n \in \mathbf{Z}$, one obtains its well-known structure equations

\begin{equation*} [ L _ { n } , L _ { m } ] = ( n - m ) L _ { n + m } + \frac { 1 } { 12 } ( n ^ { 3 } - n ) \delta _ { n , - m } . C ^ { \prime }, \end{equation*}

with $C ^ { \prime } = - 2 C$. The Virasoro algebra plays an important role as symmetry algebra in conformal field theory. E.g., in the context of Wess–Zumino–Novikov–Witten models, for the admissible representations of affine Lie algebras (Kac–Moody algebras of affine types; cf. also Kac–Moody algebra), the (affine) Sugawara construction (i.e. the modes of the energy-momentum tensor) gives a representation of the Virasoro algebra. In the context of these models, the $W _ { 1 + \infty}$-algebra appears as a higher symmetry of conformal field theory.

Besides the Virasoro algebra, $W _ { 1 + \infty}$ has also other important Lie subalgebras.

1) The subspace of functions (considered as differential operators of degree zero) together with the central element is a subalgebra with $\{ A _ { n } = z ^ { n } : n \in \mathbf{Z} \}$ and the central element $C$ as basis. It is called the infinite-dimensional Heisenberg algebra (cf. also Weyl algebra).

2) The Cartan subalgebra of $W _ { 1 + \infty}$ is given by the linear combinations of $D ^ { k }$, $k \in \mathbf{N} _ { 0 }$, and the central element $C$.

3) The $W _ { \infty }$-algebra is obtained as the subalgebra (linearly) generated by $z ^ { n } f ( D )$ with $n \in \mathbf{Z}$ and $f$ arbitrary polynomials without constant terms, and the central element. For this subalgebra the above-introduced elements $A _ { n }$ have to be removed from the basis.

The representation theory of $W _ { 1 + \infty}$ was developed by V.G. Kac, A. Radul, E. Frenkel, W. Wang [a5], [a3] and H. Awata, M. Fukuma, Y. Matsuo, S. Odake [a2], [a1]. A complete classification of the "quasi-finite positive energy representations" exists. Some of the representations carry a canonical vertex algebra structure.

The usage of the symbol $W _ { 1 + \infty}$ to denote this algebra comes from the fact that they are also obtained as "limits" for $N \rightarrow \infty$ of the $W _ { N }$-algebras.. The $W _ { N }$-algebras appear, for example, in the context of the quantization of the second Gel'fand–Dickey structure for Lax equations. They are not Lie algebras. With a suitable rescaling for $N \rightarrow \infty$, the $W _ { \infty }$-algebra is obtained with its Lie structure. "Adding spin-one currents" (i.e. the elements $A _ { n }$, $n \in \mathbf{Z}$) yields the $W _ { 1 + \infty}$-algebra, see [a1], [a7]. Equivalently, in conformal field theory, $W _ { \infty }$ can be obtained by adjoining "higher spin currents" to the Virasoro algebra. The latter is considered as the algebra of spin-two currents.

Further applications are the quantum Hall effect, two-dimensional quantum gravity, integrable systems, Toda field theories, etc. For references to these applications, see [a1].

There are a lot of generalizations, like matrix $W$-algebras, super $W$-algebras, etc. Another direction of generalization originates from the description of $W _ { 1 + \infty}$ as central extension of the algebra consisting of the meromorphic differential operators on a compact Riemann surface of genus zero that are holomorphic outside of $0$ and $\infty$. The Virasoro algebra was generalized by I.V. Krichever and S.P. Novikov [a6] to a central extension of the Lie algebra of meromorphic vector fields on a compact Riemann surface (of arbitrary genus) that are holomorphic outside of two fixed points. This has been further extended to an arbitrary number of points and to the Lie algebra of differential operators by M. Schlichenmaier [a8]. By semi-infinite wedge product representations, a central extension is constructed which is the higher-genus generalization of the $W _ { 1 + \infty}$-algebra.

References

[a1]	H. Awata, M. Fukuma, Y. Matsuo, S. Odake, "Representation theory of the $W _ { 1 + \infty}$ algebra" Prog. Theor. Phys. Proc. Suppl. , 118 (1995) pp. 343–373
[a2]	H. Awata, M. Fukuma, Y. Matsuo, S. Odake, "Character and determinant formulae of quasifinite representation of the $W _ { 1 + \infty}$ algebra" Comm. Math. Phys. , 172 (1995) pp. 377–400
[a3]	E. Frenkel, V. Kac, A. Radul, W. Wang, "$W _ { 1 + \infty}$ and $W(g l _ { N } )$ with central charge $N$" Comm. Math. Phys. , 170 (1995) pp. 337–357
[a4]	V.G. Kac, D.H. Peterson, "Spin and wedge representations of infinite-dimensional Lie algebras and groups" Proc. Nat. Acad. Sci. USA , 78 (1981) pp. 3308–3312
[a5]	V. Kac, A. Radul, "Quasifinite highest weight modules over the Lie algebra of differential operators on the circle" Comm. Math. Phys. , 157 (1993) pp. 429–457
[a6]	I.M. Krichever, S.P. Novikov, "Algebras of Virasoro type, Riemann surfaces and structures of the theory of solitons" Funkts. Anal. Appl. , 21 : 2 (1987) pp. 46–63
[a7]	A.O. Radul, "Lie algebras of differential operators, their central extensions and $W$ algebras" Funkts. Anal. Appl. , 25 : 1 (1991) pp. 33–49
[a8]	M. Schlichenmaier, "Differential operator algebras on compact Riemann surfaces" H.-D. Doebner (ed.) V.K. Dobrev (ed.) A.G Ushveridze (ed.) , Generalized Symmetries in Physics, Clausthal 1993 , World Sci. (1994) pp. 425–435