Further results on q-Lie groups, q-Lie algebras and q-homogeneous spaces


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                  </jats:inline-formula> with corresponding <jats:italic>q-</jats:italic>Lie groups and <jats:italic>q</jats:italic>-geodesics. By introducing a <jats:italic>q</jats:italic>-deformed semidirect product, we can define exact sequences of <jats:italic>q</jats:italic>-Lie groups and some other interesting <jats:italic>q</jats:italic>-homogeneous spaces. We give an example of the corresponding <jats:italic>q</jats:italic>-Iwasawa decomposition for SL<jats:italic>q</jats:italic>(2).</jats:p>

The basic construction which replace the real numbers as function arguments in trigonometric functions etc. are the q-real numbers Rq . The de nition given for Rq is not the most general, but it will do for the moment. A closer introduction to this q-umbral calculus is given in [4] and in [5]. We nd that our objects have properties similar to manifolds. This paper is organized as follows: In this section we give some fundamental de nitions and theorems. In section 2 we start with a comparison with the similar Lie and topological groups. Several theorems from Lie groups have analogues for q-Lie groups, which is illustrated for SLq( ). Therefore, in subsection 2.1 we study the q-Lie group SLq( ) and prove the corresponding q-Iwasawa decomposition. Then we introduce the q-torus and the q-determinant from previous papers. The centers in subsection 2.2 have similar properties as in the ordinary case.
In section 3 a complete theory for q-Lie algebras is presented, which is very similar to Lie algebras. We start with the universal covering q-group in subsection 3.1. In subsection 3.2 we de ne the classical q-Lie algebras and in subsection 3.3 the important concepts solvable and nilpotent q-Lie algebras are de ned. In subsection 3.4 some properties of q-Cartan Killing forms are discussed. The description of q-root systems in subsection 3.5 is very similar to the ordinary case. The Weyl group in subsection 3.6 is also very similar. Dynkin diagrams and a table of important q-Lie algebras are discussed in subsection 3.7. In subsection 3.8 we make some remarks on the relation between complex semisimple q-Lie groups and real compact q-Lie groups.
In section 4 we brie y describe q-homogeneous spaces. In subsection 4.1 we introduce q-di erentials to prepare for the proof that q-homogeneous spaces are manifolds. In subsection 4.2 we study the q-Lie groups SUq( ) and SUq ( , ) together with the corresponding q-homogeneous space SUq ( , ) SOq( ) . In subsection 4.3 we study the q-Lie groups SOq( ) and SOq( ) together with the corresponding q-homogeneous space SOq( ) SOq( ) . In subsection 4.4 we compute q-roots and q-Cartan subalgebras for the well-known q-Lie groups. We develop the irreducible representation of SOq( ) from [3].
In subsection 4.5 we present the important concept q-deformed semidirect product. In section 5 we make a short conclusion.
Something about the notation: Since real and complex q-Lie groups often can be treated simultaneously, we shall from now on use the letter K in order to denote either R or C and use the term K-q-Lie group in order to refer to real resp. complex q-Lie groups.
We denote direct sums of matrices by ⊕ q or ⊕q. The notation ⊕ q denotes direct sum of two matrices in the context of q-Lie algebra or for sums like (44), and the notation ⊕q denotes sums of commuting matrices.
Because of this, we only consider values of x with < x < ( − q) − such that this function Eq(x) converges. Other values of x, except the poles, would give other branches.

Introduction to q-Lie groups and q-Tori
The following introduction to Lie groups and topological groups is also applicable for q-Lie groups. where A k ∈ R n− ,n− denotes the matrix obtained from A ∈ R n,n by deleting the k-th row and the -th column. With other words, the entries γ ij of A − are rational functions in the entries of A (with a non-vanishing denominator).

Example 2.3. The general linear group
In the following we shall present a series of closed subgroups G ⊂ GLn(K). In order to see that they are even Lie groups we use We consider a non degenerate bilinear form σ : K n × K n −→ K, write σ(x, y) = x T Sy with a matrix S ∈ K n,n .
Then a matrix A ∈ GLn(K) preserves σ, i.e. σ(Ax, Ay) = σ(x, y) for all x, y ∈ K n i A T SA = S. Obviously the set of all such "σ-isometries" forms a closed subgroup of GLn(K). We look at the map Then the Jacobian of F at E is Assume now S is either symmetric: S T = S or antisymmetric: S T = −S. Then F(GLn(K)) ⊂ Sn(K) resp. F(GLn(K)) ⊂ An(K), where Sn(K) denotes the vector space of all symmetric matrices and An(K) the vector space of all anti-symmetric matrices. Thus we may replace the target of both F and DF(E) with Sn(K) resp. An(K). Since a given matrix A ∈ Sn(K) resp. A ∈ An(K) is of the form A = DF(E)(X) with X ≡ (S − A), the Jacobian map of F at E is onto and hence our isometry group even a Lie group. If we take S = E we obtain the K-orthogonal group while for even n = m and S = E −E the analogous group and dim Spn(K) = n − dim An(K) = n(n + ).
We remark that the real orthogonal group On(R) is compact, and that det(A) = ± for A ∈ On(K) as well as for A ∈ Spn(K).
3. Now let us consider the hermitian form σ : C n × C n −→ C, σ(z, w) = z T w. The corresponding isometry group is the unitary group where, say, −π < arg(.) < π, with SU(n) = F − ( ) and Jacobian is a K-Lie group, consisting of the invertible upper triangular matrices, indeed the underlying di erentiable or complex manifold is nothing but (K * ) n × K n(n− )/ and we may argue as in the case of GLn(K). There is a canonical q-homomorphism its kernel UUn(K) ⊂ UTn is K-Lie group, it consists of all upper triangular matrices with diagonal entries equal to ("unipotent" matrices).
All our above Lie groups are closed subgroups of GLn(K).
Recall the de nition of q-Lie group [6].
De nition 2.5. We de ne the following commutative ring [6]: Then the q-Lie group Uq(n) is de ned by where the q-multiplication ·q is twisted by using well-known formulas for q-trigonometric functions, and the function argument for Eq(x) is multiplied by q-Ward numbers.
De nition 2.8. A q-Lie group Gq is called compact if the matrix elements of all matrices in Gq are bounded functions and the set of matrices in Gq is closed under the two operations . and .q.
De nition 2.9. A q-torus in a compact q-Lie group is a q-Lie subgroup, which is a nite tensor product of matrices of the forms and  A maximal q-torus is a q-torus Hq such that if Tq is another q-torus with Hq ⊂ Tq, then Hq = Tq.
The notion of maximal q-torus plays a special role in the theory of q-Lie groups, since all maximal q-tori are conjugate.
Then BAB − sends e j to v j to λ j v j to λ j e j . This implies that we can compute the corresponding diagonal elements of Tq from the λ j .
In this section we will rely on the very similar results from Curtis [2].

De nition 2.12.
A q-one parameter subgroup of a q-Lie group Gq over the normed eld K is a q-homomorphism M : K → Gq such that M(x ⊕q y) = M(x)M(y).
Theorem 2.13. A q-analogue of [2, p.93]. If γ, σ are q-one parameter subgroups of an abelian q-Lie group Gq, then γ × σ is a q-one parameter subgroup of Gq.
Proof. We use de nitions 29 and 30 in [6]. For the convenience of the reader, we brie y repeat the rst one.
Because of the abelian property only the rst formula comes into consideration. We nd that where we used the abelian property * in the second step. Obviously, γ × σ is a submanifold of Gq.
Theorem 2.14. A q-analogue of [2, p.95]. Any compact connected abelian q-Lie group is a q-torus.
is a maximal q-torus in Uq(n).
Proof. The q-Lie subgroup Tq of all diagonal matrices in Uq(n) is clearly isomorphic to T n q . It is actually a maximal q-torus, for if there is a strictly larger one, then one can nd some element g in Uq(n) which commutes with all elements of Tq. But Tq contains diagonal matrices with n distinct eigenvalues, and any matrix which commutes with such matrices must be diagonal, so we get a contradiction.
is a maximal q-torus in SUq(n).
Proof. Obviously, the above matrix has q-determinant 1. This implies that it is just the intersection of SUq(n) with the maximal q-torus given for Uq(n).

De nition 2.18. A maximal q-torus of a q-Lie group is a k, q-torus and which is not contained in any larger q-torus.
De nition 2.19. [3] The q-determinant of an n × n matrix Mn ≡ [m ij ] n− i,j= is de ned by detq Mn ≡ π∈Sn signπ m π( ) τ(m π( ) )m π( ) τ(m π( ) ) . . . ξ (m π(n− )n− ), (13) where ξ is the identity if n is odd, and ξ = τ if n is even. In particular, the q-determinant of a × matrix is given by the formula detq α = a τ(a ) − a τ(a ). The q-determinant of a tensor product of q-tori is de ned as the product of the corresponding q-determinants.
Proof. Apply the two matrix multiplications to the q-tori of the corresponding q-Lie groups.
. The q-Lie group SL q ( , K) We are going to give two di erent examples of realizations of SLq( , K).

Example 2.22.
A q-analogue of [19, p. 57]. The q-Lie algebra of SLq( , K) has a basis which consists of the well-known matrices We can form three q-one-parameter subgroups by the mapping X i → Eq(tX i ).
Example 2.23. A q-analogue of [19, p. 363]. The q-Lie algebra of SLq( , K) has a basis which consists of the matrices Similarly, we can form three q-one-parameter subgroups.
In the same way as for Lie groups we can show that the Eq mapping is not surjective on the whole SLq( ). We can de ne a q-hyperbolic plane as (compare with H q ) According to Iwasawa a real semisimple, non-compact Lie algebra g regarded as a vector space is a q-direct sum of subalgebras: where a is abelian and n is nilpotent. We have the following q-analogue assuming that we have a matrix q-Lie algebra: Then we can form q-Lie groups with the function Eq(x). For the Lie group case this corresponds to

Theorem 2.24. For each A ∈ GL(n, R) there is a S ∈ O(n), a diagonal matrix D with real positive diagonal elements and an upper triangular matrix U, such that A = SDU. This decomposition is unique.
The Iwasawa decomposition plays a key role in the representation theory. There is a corresponding q-Iwasawa decomposition for SLq( ).
Theorem 2.25. The q-Iwasawa decomposition for matrices An,q  Proof. Assume that α, γ > . Otherwise use negative ψ. A multiplication of the three matrices gives This matrix has q-determinant 1. We can compute ψ, t and x explicitly from (22) as Equation (23) always has a solution, since γ α > . This proves the uniqueness.

. Centers, q-central subgroups and coverings by maximal q-tori
We rst conclude that most of the centers for maximal q-tori have almost equivalent q-analogues.

De nition 2.26. The center, Z(Gq), of a q-Lie group Gq is the set of elements of Gq that commute with all other elements.
The center must lie in each maximal q-torus. More speci cally, Z(Gq) = λI, where λ ∈ C.
Proof. By Schur's lemma, Z(Gq) is a multiple of the unit matrix.
Proof. If B ∈ Z(Uq(n)) we infer that B must be diagonal with diagonal elements Eq(a i ). Therefore Then AB = BA shows that a = a etc., so all a i are equal. Clearly any Eq(iϑ)I is in the center, which proves the theorem.

De nition 2.29. A q-central subgroup Hq of a q-Lie group is de ned by Hq
Theorem 2.30. A q-discrete subgroup Hq of a q-Lie group Gq has the property e ∈ Hq is an isolated point is equivalent to all h ∈ Hq are isolated points in Hq. If Gq is connected, then every q-discrete subgroup is q-central. [2, p. 110]. The q-Lie group Uq(n) is covered by the conjugates of its maximal q-tori.
Theorem 2.32. A q-analogue of [2, p. 110]. The q-Lie group SUq(n) is covered by the conjugates of its maximal q-tori.

A theory for q-Lie algebras
We assume that the reader is familiar with the theory of q-Lie algebras from our previous article [6].

. The universal covering q-group
A q-Lie group q-homomorphism φ : Gq −→ Hq induces a q-Lie algebra q-homomorphism φ * . In this section we ask when for given q-Lie groups Gq , Hq and a q-Lie algebra q-homomorphism ψ : gq −→ hq we can nd a q-Lie group q-homomorphism φ : Gq −→ Hq inducing ψ, i.e. such that ψ = φ * . The strategy is as follows: So given ψ : gq −→ hq we look at the connected q-Lie subgroup Γq ⊂ Gq × Hq with The inclusion followed by the projection onto the rst factor Since both q-Lie groups, Gq and Γq are connected, it is a surjective q-homomorphism with discrete kernel. And if it is even an isomorphism, we can take φ ≡ pr Hq • π − . Indeed, there are q-Lie groups, where π necessarily is an isomorphism. Now the basic idea in the study of q-di erentiable q-groups is to replace the commutator map with the "bilinear part" of its q-Taylor expansion at (e, e) -here e ∈ Gq denotes the neutral element of the q-group Gq. Let us explain that: Denote DK(e, e) ∈ R m, m the q-Jacobian matrix of K at (e, e) -the linear part of the q-Taylor expansion. Then we have for small ξ , η ∈ R m the expansion It turns out that that map determines the group law near (e, e) ∈ Gq × Gq completely, so one can replace the local study of q-di erentiable q-groups with the study of certain bilinear maps [.., Let us discuss the example Gq = GLn(R) ⊂ R n,n and compute the map [.., ..] : R n,n × R n,n −→ R n,n . For a matrix A = (α ij ) we de ne its norm by and note that it is even well behaved with respect to products: ||AB|| ≤ ||A|| · ||B||. Now denote E ∈ GLn(R) the unit matrix (replacing e ∈ Gq) and take X, Y ∈ R n,n of norm < (replacing ξ and η).
a convergent series. Consequently with the dots representing terms of total degree > in X and Y. Thus the linear term vanishes and is the commutator of the matrices X, Y ∈ R n,n . This implies the following equivalent de nition for q-Lie algebras.

De nition 3.1. The q-Lie bracket of two elements A, B in a q-Lie algebra is de ned by
A matrix q-Lie algebra over K satis es the Jacobi identity:

De nition 3.2.
A q-Lie algebra q-homomorphism is a K-linear map φq : gq → hq, that preserves the q-Lie brackets:

De nition 3.3. The kernel of a q-Lie algebra q-homomorphism φq
Let Eq denote q-exponentiation of matrices. We have the following commutative diagram, almost a qanalogue of [2, p.192]: The q-Lie algebras suq(n, C) and soq(n, C) are called skew-Hermitian and skew-symmetric, respectively. They have no q-center except in very low dimensions.
Theorem 3.18. The real q-Lie algebras slq(n, R) and sp q (n, R) are all semisimple.

Theorem 3.19. The ( nitedimensional) q-Lie algebras together with the q-Lie homomorphisms form a category.
Theorem 3.20. A q-analogue of [2, p. 173]. Denote the q-Lie algebra of the q-Lie group Gq by L(Gq). The mapping Gq −→ L(Gq) de nes a covariant functor from the q-Lie group category C to the q-Lie algebra category C , which sends objects to objects and homomorphisms to homomorphisms.
Proof. Assume that α and β are q-Lie group homomorphisms. This follows from the following diagram: Theorem 3.21. The composition of two q-homomorphisms is a q-homomorphism.
Proof. Use composition of mappings and morphism properties.

. solvable and nilpotent q-Lie algebras
Back to q-Lie algebras! Note rst that, given a q-ideal a ⊂ gq of a q-Lie algebra gq we can endow the factor vector space with a natural q-Lie bracket the resulting q-Lie algebra being called the q-factor algebra gq /a of gq mod(ulo) the q-ideal a. Furthermore, we have gq = a D if dimgq = dima + and D = ad(X) for some X ∈ gq \ a.

De nition 3.22. The derived series of a q-Lie algebra gq is the sequence of q-ideals
We have an equivalent de nition.
De nition 3.23. The q-Lie algebra gq is called solvable if there is a nite sequence of q-Lie subalgebras Example 3.24. 1. If gq is solvable, then any subspace hq ⊂ gq with g i,q ⊂ hq ⊂ g i+ ,q is a q-Lie subalgebra and even a q-ideal in g i+ ,q . Hence we may re ne a given strictly increasing sequence as in Def. 3.23 in such a way that nally r = dim gq and dim g i+ ,q = dim g i,q + . In particular we see, that a solvable q-Lie algebra can be constructed by a repeated application of the g D,q -construction for a q-Lie algebra gq together with a derivation D ∈ Der(gq).

Denote C(gq) ≡ [gq , gq] the commutator q-Lie subalgebra of gq. A q-Lie algebra is solvable i the decreasing sequence of successive commutator q-Lie subalgebras
terminates at the trivial q-Lie subalgebra. 3. Let aq ⊂ gq be a q-ideal. If aq is solvable as well as gq /aq, so is gq. In particular, if aq , bq ⊂ gq are solvable q-ideals, so is aq ⊕ q bq. Hence there is a unique maximal solvable q-ideal in a q-Lie algebra gq: De nition 3.25. The maximal solvable q-ideal rq ⊂ gq is called the radical of the q-Lie algebra gq.
For solvable (q)-Lie algebras we have: Let Bn,q(K) denote the upper triangular matrices with 1:s along the diagonal. Then un,q(K) is the q-Lie algebra of Bn,q(K). Bn,q is not semisimple since the subset with 0 everywhere except for the upper left hand corner is a q-ideal. There is a common pattern in the following exposition. Many of the functions, which exist in q-(Lie) group theory reappear with the same name in q-Lie algebra theory. The reason is of course that these functions mean the same thing in either category. As the following exposition shows these terms can also be used in the case of q-Lie algebras (and q-Lie groups). In a nilpotent q-Lie algebra, gq, the Baker-Campbell-Hausdor series is nite and hence de nes a polynomial map Indeed, it provides gq with a group law: Consider the q-exponential map Eq : gq −→ Gq for the simply connected q-Lie group with Lie(Gq) ∼ = gq. Then, Eq being di eomorphic near ∈ gq, the conditions for a group law are satis ed near the origin and hence everywhere by the identity theorem for polynomial maps (A map V ×V −→ K for a K-vector space V is called polynomial if it is a K-linear combination of products of linear forms in one of both variables. A map V ×V −→ V is polynomial, if the composition with any linear functional V −→ K is polynomial.). Indeed the q-exponential map Eq : gq −→ Gq turns out to be an isomorphism of q-Lie groups. So we can replace Gq with (gq , C(., .)), the expression for C in terms of Lie monomials being independent from the nilpotent q-Lie algebra gq. Note that the n-th power of X ∈ gq for n ∈ Z with respect to C(., .) is nX.
Example 3.30. The q-exponential map Eq : un,q(K) −→ UUn(K) is polynomial: In order to understand all q-Lie groups with nilpotent q-Lie algebra we have to consider factor groups of (gq , C(., .)) mod q-central discrete q-Lie subgroups Dq ⊂ (gq , C(., .)). We claim that the center of the q-Lie group (gq , C(., .)) is the q-center ker(ad) of the q-Lie algebra gq: Since C(X, ) = = C( , Y), any of the q-Lie bracket monomials in C(X, Y) contains both X and Y as factor, hence C(Z, X) = Z ⊕ q X = X ⊕ q Z = C(X, Z) for Z ∈ ker(ad) and any X ∈ gq. On the other hand, for a q-central element Z of the q-Lie group (gq , C(., .)), all its integral powers nZ are q-central as well; hence C(nZ, mX) = C(mX.nZ) for all n, m ∈ Z. Both expressions being polynomials in n, m ∈ Z, comparison of the bilinear term yields [Z, X] = [X, Z] resp. [Z, X] = . So normal discrete q-Lie subgroups are exactly the lattices in the subspace ker(ad) ⊂ gq. Note furthermore that the connected q-Lie subgroups of (gq , C(., .)) are exactly the q-Lie subalgebras hq ⊂ gq (the q-exponential map being the identity on gq and subspaces being maximal connected submanifolds).
Theorem 3.31. 1. A q-Lie algebra gq is solvable i its commutator q-Lie algebra C(gq) is nilpotent. 2. A q-Lie algebra is nilpotent i ad(X) ∈ gl q (gq) is nilpotent for every X ∈ gq. 3. Any nilpotent q-Lie algebra is isomorphic to a q-Lie subalgebra of un,q(K) for some n ∈ N.

−→ Hq ⊂ GLq(n, K) is an isomorphism of q-Lie groups. Then, gq being solvable we may assume hq ⊂ utn(K), see Theorem 3.26 resp. Hq ⊂ UTn(K), see Cor. 3.27. Consider the element Z ≡ − [X, Y]. Since C(utn(K)) ⊂ un,q(K) we nd φ * (Z) ∈ un,q(K), hence φ(Z) = Eq(φ * (Z)) ∈ UUn(K) (note that Gq is here identi ed with its q-Lie algebra), but UUn(K) contains no non-trivial elements of nite order!
Now let us consider semisimple algebras and modules: But note that for v ∈ V the subspace gq v ≡ {Xv; X ∈ gq} in general is not a (gq-)submodule, so the irreducible factors are not necessarily q-factor algebras of gq (e.g. X(Yv) need not belong to gq v). On the other hand V ≡ gq is a gq-module with the q-Lie bracket as "scalar multiplication" (corresponding to the adjoint representation gq −→ gl q (gq), X → ad(X)). Then the irreducible factors are q-ideals of gq. Calling a q-Lie algebra gq simple if it is semisimple and admits no nontrivial q-ideals we obtain:

Theorem 3.34. A semisimple q-Lie algebra gq is the q-direct sum of simple q-Lie algebras
the factors g ,q , ..., gs,q being unique up to isomorphy and order. q-Ideals of semisimple algebras are direct factors and thus semisimple as well. As a consequence, no nontrivial solvable algebra is semisimple, since otherwise we would nd that the one-dimensional q-Lie algebra K is semisimple. Furthermore, a semisimple algebra has trivial radical. Indeed the reverse implication holds as well:

Theorem 3.35 (Theorem of Weyl). A q-Lie algebra gq with trivial radical is semisimple.
Example 3.36. Complex semisimple q-Lie algebras: The rst point applies only to real q-Lie algebras, since there are no compact simply connected q-Lie groups except the trivial group. (For q = , the only connected compact complex Lie groups are the tori G = C m /Λ with a lattice Λ ∼ = Z m of maximal rank.) But we can weaken our assumption: It is su cient that gq = k ⊕ q ik with a real q-Lie subalgebra k ⊂ gq belonging to a simply connected compact real q-Lie group Kq (not to be confused with the base eld). Given now a gq-submodule U ⊂ V, its orthogonal complement with respect to a K-invariant inner product on V is a K-invariant complex vector subspace, hence also k and gq = k ⊕ q ik-invariant. As example take gq = sln,qC) = sun,q ⊕ q isun,q .

Again, any complex semisimple q-Lie algebra is obtained in that way.
In the remaining part of this section we explain the classi cation of complex simple q-Lie algebras.

De nition 3.37. A q-Lie subalgebra hq ⊂ gq of a q-Lie algebra gq is called a q-Cartan subalgebra (CSA), if it is nilpotent and satis es
This can also be expressed in the following two de nitions:

De nition 3.38.
Let h q be a q-Lie subalgebra of a q-Lie algebra g q . Then N g q (h q ) ≡ {a ∈ hq|[a; hq] ⊂ hq} is a q-Lie subalgebra of g q , called the q-normalizer of h q .

De nition 3.39.
A q-Cartan subalgebra of a q-Lie algebra g q is a q-Lie subalgebra hq, which satis es the following two conditions: 1. h q is a nilpotent q-Lie algebra 2. N g q (hq) = h q Example 3.40. For gq = sln,q(C) the q-Lie subalgebra hq ≡ sdq n(C) consisting of all diagonal matrices in slq( , C) is a q-Cartan subalgebra.
De nition 3.41. Let gq be a q-Lie algebra. The mapping σ : gq → gq that preserves the algebraic operations on gq is called a σ-automorphism of gq.
Theorem 3.42. In a complex q-Lie algebra gq any two CSA hq , h q ⊂ gq are conjugate under an automorphism f = Eq(ad(X)) for some X ∈ gq, i.e. h q = f (hq).
We rst give an alternative characterization of a CSA.
Theorem 3.43. hq is a CSA if and only if hq = g ,q ad(Z)), where g ,q ad(Z)) contains no proper subalgebra of the form g ,q ad(X)).
Proof. Suppose hq = g ,q ad(Z) which is minimal in the sense of the proposition. Then we know that hq is its own q-normalizer. Also, hq ⊂ g ,q , ad(X)∀X ∈ hq. Hence ad(X) acts nilpotently on hq for all X ∈ hq. Hence, by Engel's theorem, hq is nilpotent and hence is a CSA. Suppose that hq is a CSA. Since hq is nilpotent, we have hq ⊂ g ,q ad(X), ∀X ∈ hq. Choose a minimal Z. Then, g ,q ad(Z) ⊂ g ,q ad(X), ∀X ∈ hq .
Thus hq acts nilpotently on g ,q ad(Z)/hq. If this space were not zero, we could nd a non-zero common eigenvector with eigenvalue zero by Engel's theorem. This means that there is a Y ∈ hq with [y, hq] ⊂ hq contradicting the fact hq is its own q-normalizer. Lemma 3.44. If Φ : gq → g q is a surjective q-homomorphism and hq is a CSA of gq then Φ(hq) is a CSA of g q .
Proof. Clearly Φ(hq) is nilpotent. Let k = KerΦ and identify g = g/k so Φ(hq) = hq ⊕ q kq. If X ⊕ q kq normalizes hq ⊕ q kq then X normalizes hq ⊕ q kq. But hq = g ,q (adZ) for some minimal such Z, and as an algebra containing a g ,q (adz), hq ⊕ q kq is self-normalizing. So X ∈ hq ⊕ q kq. Lemma 3.45. Let Φ : gq → g q be surjective, as above, and hq a CSA of gq.
Proof. hq is nilpotent by assumption. We must show it is its own q-normalizer in gq. By the preceding lemma, Φ(hq) is a Cartan subalgebra of hq . But Φ(hq) is nilpotent and hence would have a common eigenvector with eigenvalue zero in h /Π(h), contradicting the selfnormalizing property of Φ(hq) unless Φ(hq) = hq. So Φ(hq) = hq. If x ∈ gq normalizes hq, then Φ(X) normalizes hq. Hence Φ(X) ∈ h q so X ∈ mq so X ∈ hq.
De nition 3.46. The q-Lie subalgebra h q of g q is a real form of g q if there exists a C-linear isomorphism ϕ : h C q → gq such that ϕ|h q = I, where h C q denotes the complexi cation of hq.

De nition 3.47. Let hq be a real q-Lie algebra and let +q denote a twisted q-addition. Its complexi cation h
The case αq = does not occur, hq being a maximal abelian q-Lie subalgebra.  .

q-Cartan Killing form
First of all on a q-Lie algebra gq one de nes:

De nition 3.50. Let gq be a q-Lie algebra. The q-Cartan
Killing form is the following bilinear symmetric form Note that we changed trace to q-trace in the de nition. Let us mention: This is Cartan's criterion. Moreover in the latter case its restriction to hq ×hq with a CSA hq ⊂ gq is nondegenerate as well, and its dual form, also denoted is real valued on h * R,q × h * R,q and even positive de nite.
The proof that the above conditions hold for Φq ⊂ h * q relies on the knowledge of the irreducible sl ,q (C)modules. Indeed, given nonproportional αq , βq, the subspace k∈Z, q g βq+kαq is a module over the q-Lie subalgebra CHα,q ⊕ q gα,q ⊕ q g−α,q ∼ = sl ,q (C).

. Weyl groups and Weyl chambers
Now let us concentrate on abstract q-root systems:

De nition 3.53. Let Φq ⊂ V be a q-root system, and denote by Oq(V) the group of linear q-isometries of the euclidean space V. The Weyl group W(Φq) ⊂ O(V) is de ned as the subgroup of O(V) generated by the re ections sα , αq ∈ Φq.
The symmetries of a q-root system Φq given by the action of the Weyl group W make it possible to compress the information contained in it in a basis B:

De nition 3.54. A subset B ⊂ Φq is called a basis of the q-root system Φq ⊂ V if B is a basis of the vector space V and every βq ∈ Φq is an integral linear combination
where the coe cients kα ∈ Z satisfy either kα ≥ for all αq ∈ B or kα ≤ for all αq ∈ B.
A basis of a q-root system Φq ⊂ V gives rise to a decomposition On the other hand, starting with certain decompositions we get all the bases of a q-root system Φq: As a consequence we see that |W| < ∞. Furthermore that given a basis B we can recover the Weyl group W as well as W B . A q-root system is called reduced if αq , λαq ∈ Φq implies that λ = ± . If Φq is a q-root system in V, the q-coroot α ∨ q of a q-root αq ∈ Φq is de ned by The set of q-coroots also forms a q-root-system Φ ∨ q in V, called the dual q-root system. An element of Φ + q is called a simple q-root if it cannot be written as the sum of two elements of Φ + q , and the corresponding sα de ned by (43) is called a simple re ection.
Let us explain a little bit more in detail the di erent bases a root system Φq admits: Writing a hyperplane as H = Pγ ≡ γ ⊥ for γ ∈ V we see that it is a separating hyperplane for Φq, i.e., Pγ ∩ Φq = ∅, i γ ∉ V \ αq∈Φq Pα.
De nition 3.57. Let Φq ⊂ V be a q-root system. The connected components of V \ αq∈Φq Pα are called Weyl chambers. An element γ ∈ V is called regular if γ ∉ V \ αq∈Φq Pα. For such an element γ denote Ch(γ) the Weyl chamber containing γ.

. Dynkin diagram, table of all simply connected complex simple q-Lie groups
Now the information contained in a basis B ⊂ Φq can be encoded in a so called Dynkin diagram, a graph whose vertices are the base roots αq ∈ B. Two vertices αq , βq are connected by χ(βq , αq)χ(αq , βq) = cos (ϑ) edges, i.e. by one edge, if the angle ϑ ∈ [ , π) equals π , by two edges, if ϑ = π and by three edges if ϑ = π . In the last two cases the two or three edges are even oriented, the arrow pointing from the longer root to the smaller one. Note that the diagram does not depend on the choice of the basis B.
Let us come back to q-Lie algebras: A q-Lie algebra can be reconstructed -up to isomorphy -from its q-root system, and a q-root system from one of its bases resp. -again up to isomorphy -from its Dynkin diagram. First of all, it is connected if and only if the corresponding algebra is simple, and there is a complete classi cation of the connected Dynkin diagrams. Here it is, the index counting the number of vertices: being trivial. So there is not only a maximal q-Lie group -the simply connected one -but also a minimal q-Lie group with q-Lie algebra gq, since Ad(Gq) only depends on gq: It is the connected q-Lie subgroup of Aut(gq) with q-Lie algebra ad(gq). Indeed, for a semisimple algebra we have ad(gq) = Der(gq) and hence Ad(Gq) = Aut (gq), the component of the identity of Aut(gq).
If Gq is simply connected, any other connected q-Lie group with q-Lie algebra gq is of the form Gq /Dq with a discrete q-Lie subgroup Dq ⊂ Z(Gq). Indeed the center Z(Gq) ⊂ Gq is nite, it can even be read o from the q-root system Φq associated to a CSA hq ⊂ gq = Lie(Gq): If we denote Γ ⊂ V ≡ h * R,q the lattice generated by Φq (in fact Γ = αq∈B Zαq with any basis B ⊂ Φq) and Γ ≡ {γ ∈ V; χ(γ, Φq) ⊂ Z}, then, obviously Γ ⊂ Γ and (less obviously) The connected q-Lie subgroup Hq ⊂ Gq is a maximal (complexi ed) q-torus (C * ) , a closed q-Lie subgroup of Gq containing the center Z(Gq). Furthermore semisimple q-Lie groups can be realized as algebraic (in particular closed) q-Lie subgroups of GL(V) for some complex vector space V; and even better, any q-homomorphism Gq −→ GL(W) for an arbitrary vector space W is algebraic. Indeed this follows from the knowledge of all irreducible gq-modules W resp. all irreducible representations gq −→ gl q (W).
Here is the table of all simply connected complex simple q-Lie groups: Dynkin diagram simply connected q-Lie group Dimension Center Remark 4. 1. Here Spin n,q (C) denotes the universal covering group of SOn,q(C), and Oq(C) is the complexi ed algebra of q-Cayley numbers (q-octonions) [9]. The remaining exceptional q-groups do not have an immediate geometric realization. 2. Note that an arbitrary semisimple complex q-Lie group is of the form Gq /Dq, where Gq is a nite product of copies of the above q-Lie groups and Dq ⊂ Z(Gq), with Z(Gq) being the direct product of the centers of the simple factors. 3. The q-Lie algebra G has a base of 14 × matrices; it is also given by Der (Oq(C)). There are two types of (q)-octonions: The standard octonions and the split-octonions, an 8-dimensional nonassociative algebra over the real numbers.

. Complex semisimple and real compact q-Lie groups
Finally, let us comment on the relation between complex semisimple q-Lie groups and real compact q-Lie groups. Let us start with a semisimple q-Lie algebra gq and look for a "real compact form" k ⊂ gq, i.e., a real q-Lie subalgebra k ∼ = Lieq(Gq) for some real compact q-Lie group Gq satisfying gq = k ⊕ q ik. We start with a CSA hq ⊂ gq and write gq = hq ⊕ q α∈Φq gα,q , noting that, with respect to the q-Cartan Killing form ., . we have hq ⊥ gα,q for all αq ∈ Φq as well as gα,q ⊥ g β,q for αq + βq ≠ . We shall represent k = Fix(τ) as x algebra of a conjugation τ : gq −→ gq , i.e. an involutive automorphism of gq as real q-Lie algebra (τ = Ig q ) satisfying τ(iX) = −iX. Setting with h R,q ≡ span(Hα,q; αq ∈ Φq) (see Remark 3) we have gq = g ,q ⊕ q ig ,q and a conjugation σ ≡ Ig ,q ⊕ q −I ig ,q . We want to take τ = φ • σ with a complex automorphism φ : gq −→ gq: In order to de ne φ, we choose elements Zα,q ∈ gα,q , Z−α,q ∈ g−α,q with Zα,q , Z−α = − . Then the linear map φ : gq −→ gq with φ| hq = −I hq and φ(Zα,q) = Z−α is the desired q-Lie algebra automorphism.

Brief introduction to q-homogeneous spaces . On q-di erentials
De nition 4.1. Let R ⊕q denote the set generated by NWA of at most two letters in R.
Assume that x ⊕q y ∈ R ⊕q . Since R × R ∼ = C, this implies that the topological reasoning for C can be repeated. A connected open set is a domain. A domain, together with some of its boundary opints, is called a region. We can express the Nalli-Ward q-Taylor formula as follows: For n = this becomes Formula (47) (for formal power series) can obviously be expressed in the form with remainder term r(y) such that lim y→ r(y) |y| = . This means that the di erence between two function values can be approximated with a rst degree polynomial in y: dq F is called the q-di erential of F in x. Furthermore, the q-di erential of a multivariable function F(x , . . . , xn) is given by The following expositions are q-analogues of [16]. Proof. Let φ be the map Then dq φ ( , ) by( ) = X + Y. By the equivalent to the inverse function theorem, φ is a di eomorphism from an open neighborhood U × V of ( , ). In order to show that π • φ is one-to-one, we de ne some sets.
Suppose that X ∈ U , Y ∈ V and φ(X, Y) ∈ Hq then Since φ is a di eomorphism on U × V so X = and Y = Y and we conclude that In the next step we show the injectivity. Let U ⊆ U be a neighbourhood of 0 in mq such that Then π • Eq(U ) is injective, since if X , X ∈ U and π(Eq(X )) = π(Eq(X )) (57) The surjectivity on a neighborhood follows since φ is surjective from U × { } onto φ(U , ). By de nition π • Eq(mq) is continuous. If N is an open subset of U then π(Eq(N)) = π • φ (N, V ), which is open since φ is a di eomorphism here and we have the quotient topology. So the inverse is continuous. Thus π • Eq : mq −→ Gq /Hq is a local homomorphism at 0 in the quotient topology of Gq /Hq. Proof. Choose the quotient topology on Gq /Hq. Consider the map . The q-Lie groups SU q ( ) and SU q ( , ) In Audrey Terras' book [18] the rst symmetric space to be treated was the sphere (chapter 2), followed by the upper half plane H (chapter 3). We somehow follow the same plan and give formal de nitions of the most important q-analogues of homogeneous spaces in each section. In [3, (19) p.157] we de ned the following general form of a matrix in SUq( ) as: Every element of SUq( ) has an inverse for the multiplication · ψ ,q , namely U − ψ ,ϕ ,α = U ψ ,ϕ ,α , with the following three conditions on the umbrae: α ∼ α , ϕ ∼ −ϕ ψ ∼ −ψ . A straightforward calculation shows that formula (60) is equal to where This last expression can be written in the form Eq(T), where where the matrices don't commute. By the Weyl unitary trick, we nd the following expression for the noncompact q-Lie group SUq ( , ): where the matrices don't commute.
By the Weyl unitary trick, we nd the following expression for the noncompact q-Lie group SUq( , ), a q-analogue of [ 10, p. 202]: This corresponds to the q-Lie algebra decomposition Example 4.5. Put G * n,q = SUq ( , ) and K * q = SOq( ), Then the quotient space is the q-deformed hyperboloid (compare [10, p. 202]) e ·q e = , e = (x, y, z), where we use the following inde nite q-scalar product: We obtain the q-homogeneous space For this q-hyperbolic space we take For the q-Lie algebra the q-geodesic starting at p is This q-geodesic starting at p has direction A general q-geodesic is obtained by multiplication by an arbitrary element of SOq( ) : . The q-Lie groups SO q ( ) and SO q ( ) We can write O ψ as follows: The q-Lie algebra soq( ) of SOq( ) has the following basis in the space of skew-symmetric real ( × ) matrices: De nition 4.6. The q-deformed sphere, S q , a manifold, is de ned by q-spherical coordinates with unit radius: where ≤ γ < ξ (q); < φ < ξ (q, ).
Then we have SOq( ) For the q-sphere we take For the q-Lie algebra This q-geodesic has direction Again, a general q-geodesic is obtained by multiplication by an arbitrary element of SOq( ): We can already guess the structure of the q-Lie algebra for SOq(n). It is given by the algebra of n × n antisymmetric matrices A (n) according to the following mapping, a q-analogue of [10, (3.19) p.19]: Theorem 4.7. The mapping Eq is surjective for Gn,q = SUq( ).

. Sample q-roots and q-Cartan subalgebras
We will now study some special q-Lie groups. The results will be very similar to the classical Lie groups [2] and we only list formulas which di er from this classical case. In particular, we state q-roots ∈ Rq and q-Cartan subalgebras. For the computations of these q-roots see [2, p.197]. For SOq( ) we have the four q-roots α ⊕q α , −α q α , α q α , −α q α .
For SUq( ) we have the q-root α q α . For Sp q ( ) we have the q-root α. And for S0q( ) we have one q-root, which we denote −α. For SUq(n) the q-Cartan subalgebras are diagonal matrices with {d j } n j= ∈ IRq and ⊕ n q,j= d j ∼ θ.

. A q-deformed semidirect product
The following considerations from Simon [15] are very important in (q-Lie) group theory.
De nition 4.8. Let Nq be a q-Lie group. A mapping f : Nq → Nq is called a q-automorphism if f is bijective, and both f and f − are q-Lie group homomorphisms.

De nition 4.9.
A q-analogue of [15, p.6]. Let Nn,q and Hn,q be two q-Lie groups. Let α h be a q-automorphism Nq → Nq indexed by an element h ∈ Hn,q. The operations in Nn,q are and q. The operations in Hn,q are · and ·q. The q-semidirect product Nq α,q Hn,q of Nn,q and Hn,q is the q-Lie group of all ordered pairs (n, h) with the multiplication · de ned by (n , h ) · (n , h ) ≡ (n (n ), h · h )},

De nition 4.11. A sequence of q-Lie groups is de ned by
where f i are q-Lie group homomorphisms. The sequence (83) is said to be exact at G i,q if If the sequence is exact at every G i,q , it is called an exact sequence.  Proof. The proof goes along the same lines as for groups and is omitted. (88)

Conclusion
Various forms of q-symmetric spaces in quantum group form have appeared in the literature. Our approach to q-homogeneous spaces is completely di erent and much more promising, since it gives concrete formulas to be applied in physics. The Wigner functions D j m,n characterize the states in the Coulomb problem, because the three indices, j, m, n are, respectively, associated with the eigenstates of the Hamiltonean, the z-component of angular momentum, and the z-component of the Runge-Lenz vector. Similarly, the q-Wigner functions from [3] provide applicable q-analogues of these objects. The q-semidirect product will be used later to de ne the q-Euclidean group and the q-Poincaré group, which are both matrix q-Lie groups. Furthermore, we mention that quantum groups and complex motion groups are treated in the papers [13] and [14].