Parametrization of a Conjugacy Class of the Special Linear Group
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PARAMETRIZATION OF A CONJUGACY CLASS OF THE SPECIAL LINEAR GROUP Y. Palii∗
UDC 512.816.2
A local description of the foliation of the group SL(n) into conjugacy classes, and also the foliation of sl∗ (n) into coadjoint orbits, requires introducing parameters on a conjugacy class (a coadjoint orbit). Under the assumption that the parameters are rational functions of natural coordinates (the matrix elements) on SL(n), the problem is reduced to solving a system of linear equations. This system arises from the requirement that the parameters be invariant with respect to translations along vector fields normal to the conjugacy class. Similarly, the problem of parametrization of coadjoint orbits in sl∗ (n) can be solved using the Cartan–Weyl basis for sl(n). The adjoint action is the differential of the conjugation action. Consequently, the parameters on the conjugacy classes and the coadjoint orbits are related by a transformation determined by the mapping from the algebra sl(n) to the group SL(n). The groups SL(3), SL(4) are considered as examples. Bibliography: 13 titles.
1. Introduction The conjugation action of a Lie group G, Conjg (h) = g h g−1 ,
Conj : G × G → G,
(1.1)
where h is some fixed element of the group and g runs through the whole group, endows G with a structure of a singular Riemannian foliation F, see [1]. Foliation sheets are orbits OConj , i.e., conjugacy classes Ch of group elements. Global and local properties of this foliation are important both for the theory of Lie groups itself and for constructing dynamic models in theoretical and mathematical physics. The symplectic structure on the conjugacy classes and its generalization, the Dirac structure [2], are used, for example, in string models of the Wess– Zumino–Novikov–Witten type [3]. Recently, Darboux coordinates have been constructed on coadjoint orbits of the group GL(n, C), see [4]. By the theorem on maximal tori of linear algebraic groups [5], every semisimple element h ∈ G is conjugate to some element of a torus T , that is, T intersects every conjugacy class Ch .1 Moreover, T intersects Ch transversally and perpendicularly with respect to a bi-invariant metric on G. The number of intersections of an orbit Oh ≡ Ch with T is equal to the Euler characteristic χ(Oh ) of the orbit as a submanifold and the cardinality of the generalized Weyl group W of the section [6]. The space of conjugacy classes G/Conj is isomorphic to the quotient T /W : T
T /W
⊂
/G
∼ =
/ G/Conj,
where the vertical arrows stand for the projections onto the quotient spaces by the action of the corresponding groups. The traces of the powers of a group element, tk = Tr(gk ),
g ∈ G,
k = 1, . . . , rank G,
(1.2)
∗
Institute of Applied Physics, Chisinau, Republic of Moldova and Joint Institute for Nuclear Research, Dubna, Russia, e-mail: [email protected]. 1In this paper, we will consider only regular orbits corresponding to semisimple elements of the group.
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 485, 2019, pp. 155–175. Original article submitted Septe
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