mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the complex Witt algebra, named after

Ernst Witt Ernst Witt (26 June 1911 – 3 July 1991) was a German mathematician, one of the leading algebraists of his time. Biography Witt was born on the island of Alsen, then a part of the German Empire. Shortly after his birth, his parents moved the ...

, is the

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...

of meromorphic vector fields defined on the

Riemann sphere In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...

that are holomorphic except at two fixed points. It is also the complexification of the Lie algebra of polynomial vector fields on a circle, and the Lie algebra of derivations of the ring C 'z'',''z''⁻¹ There are some related Lie algebras defined over finite fields, that are also called Witt algebras. The complex Witt algebra was first defined by Cartan (1909), and its analogues over finite fields were studied by Witt in the 1930s.

Basis

A basis for the Witt algebra is given by the vector fields

L_n=-z^ \frac

, for ''n'' in ''

\mathbb Z

''. The

Lie bracket In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identit ...

of two vector fields is given by :

_m,L_n (m-n)L_.

This algebra has a central extension called the

Virasoro algebra In mathematics, the Virasoro algebra (named after the physicist Miguel Ángel Virasoro) is a complex Lie algebra and the unique central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string t ...

that is important in

two-dimensional conformal field theory A two-dimensional conformal field theory is a quantum field theory on a Euclidean two-dimensional space, that is invariant under local conformal transformations. In contrast to other types of conformal field theories, two-dimensional conformal fi ...

and

string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and inte ...

. Note that by restricting ''n'' to 1,0,-1, one gets a subalgebra. Taken over the field of complex numbers, this is just the algebra

sl(2,\mathbb)

of the

Lorentz group In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch phys ...

SL(2,C). Over the reals, it is the algebra ''sl''(2,R) = ''su''(1,1). Conversely, ''su''(1,1) suffices to reconstruct the original algebra in a presentation. D Fairlie, J Nuyts, and C Zachos (1988). ''Phys Lett'' B202 320-324.

Over finite fields

Over a field ''k'' of characteristic ''p''>0, the Witt algebra is defined to be the Lie algebra of derivations of the ring :''k'' 'z''''z''^''p'' The Witt algebra is spanned by ''L''_''m'' for −1≤ ''m'' ≤ ''p''−2.

References

Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry ...

''Les groupes de transformations continus, infinis, simples.''
Ann. Sci. Ecole Norm. Sup. 26, 93-161 (1909). * {{springer, author= , title=Witt algebra, id=W/w098060 Conformal field theory Lie algebras

Basis

Over finite fields

See also

References