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Research Article Open Access

Thomas Ernst*

On the q-exponential of matrix q-Lie algebras

DOI 10.1515/spma-2017-0003

Received July 4, 2016; accepted September 27, 2016

Abstract: In this paper, we define several new concepts in the borderline between linear algebra, Lie groups and q-calculus. We first introduce the ring epimorphism τ, the set of all inversions of the basis q, and then the important q-determinant and corresponding q-scalar products from an earlier paper. Then we discuss matrix q-Lie algebras with a modified q-addition, and compute the matrix q-exponential to form the corresponding n × n matrix, a so-called q-Lie group, or manifold, usually with q-determinant 1. The corresponding matrix multiplication is twisted under τ, which makes it possible to draw diagrams similar to Lie group theory for the q-exponential, or the so-called q-morphism. There is no definition of letter multiplication in a general alphabet, but in this article we introduce new q-number systems, the biring of q-integers, and the extended q-rational numbers. Furthermore, we provide examples of matrices in suq(4), and its corresponding q-Lie group. We conclude with an example of system of equations with Ward number coefficients.

Keywords: Ring morphism, q-determinant, Nova q-addition, q-exponential function, q-Lie algebra, q-trace, biring

MSC: Primary 17B99; Secondary 17B37, 33D15

1 Introduction

The purpose of this article is to introduce the new concept of a q-matrix Lie algebra, and the corresponding q-Lie group, defined as the set of q-exponentials of the q-Lie algebra. We find that our q-additions fit naturally in the new context, since they virtually replace all additions for the ordinary case, especially for the so-called q-morphisms, or the q-exponentials. In the first article [2], we introduced the inversion operator τ, together with a general n × n q-determinant, with the purpose that our q-Lie groups SOq(2) and SUq(2) should have q-determinant 1. In practise, however, many definitions of q-determinants and many definitions of the basis inversion are necessary, since the q-orthogonalities look differently from case to case. We postpone the defi- nition of the set GLq(n, R), which refers to all q-Lie groups, until after the definitions of q-determinants. The two definitions of matrix multiplications, which precede GLq(n, R), actually only refer to the latter set. For the q-Lie algebras, there are no matrix multiplications, but a slightly different q-addition. As in the ordinary case, for obvious reasons, we decided to define the q-Lie algebras before the q-Lie groups.

It goes without saying that all computations in this paper apply to the so-called q-real numbers from [3], which are also defined in the paper. The corresponding umbral calculus, with definitions of the alphabet, which would take too long to reproduce here, can also be found in [3]. The q-natural numbers Nq, which appear in suq(n), are also defined in [3]. This paper is organized as follows: In section 1 we give a general introduction and the first definitions. Furthermore, we define two important q-analogues of Z and Q, which extend previous q-numbers [3] and [5]. We show that the first object is an extension of a graded commutative biring, which will be used for the solution of a general q-analogue of a linear system of equations in section 6.

In section 2 we prove that τ is a ring epimorphism and give corresponding definitions of q-scalar and q-vector products and of q-determinants.

*Corresponding Author: Thomas Ernst: Department of Mathematics, Uppsala University, P.O. Box 480, SE-751 06 Uppsala, Sweden, E-mail: thomas@math.uu.se

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In section 3 we come to the q-Lie algebras, which are very similar to matrix Lie algebras. The only differ- ence is that the q-additions occur both as matrix elements in the q-Lie algebras and as q-analogues of direct sums of Lie algebras (or ideals). However, we will not always use the same symbol for q-addition between letters as for q-addition between matrix Lie algebras, see formula (74). The reason is that matrices in gen- eral have noncommutative multiplication. Although it is a kind of q-addition, we will strive to keep the same nomenclature for Lie algebras as in the ordinary case. In section 4 we come to the important q-Lie groups, which have very similar properties as groups, but with the important difference that there are two multipli- cations. All q-Lie groups are manifolds.

In section 5 we present the first q-analogue of SU(4) and the corresponding maximal torus; we note that there are many definitions of these objects in the literature. We also give some other examples of q-Lie groups.

In section 6 we show that a q-analogue of linear systems of equations has a set of solutions, similar to the ordinary case, which will have later applications for so-called q-symmetric spaces.

We now start with the definitions, compare with the book [3].

Definition 1. The q-analogue and q-factorial are given by {a}q 1 − qa

1 − q ; {n}q!

n

Y

k=1

{k}q, {0}q!1, qC\{1}, (1) The Gauss q-binomial coefficient is given by

n k

!

q

{n}q!

{n − k}q!{k}q!, k = 0, 1, . . . , n. (2) The q-derivative is defined by

Dqφ(x) φ(x) − φ(qx)

(1 − q)x . (3)

The q-exponential functions are defined by Eq(z)

X

k=0

1

{k}q!zk; E1 q(z)

X

k=0

q(k2)

{k}q!zk. (4) The q-trigonometric functions are defined by

Cosq(x)1

2(Eq(ix) + Eq(−ix)). (5)

Sinq(x) 1

2i(Eq(ix) − Eq(−ix)). (6)

Definition 2. Let a and b belong to a commutative monoid. The Nalli–Ward–AlSalam q-addition (NWA) is given by

(aqb)n

n

X

k=0

n k

!

q

akbn−k, (a qb)n

n

X

k=0

n k

!

q

ak(−b)n−k (7)

The Jackson–Hahn–Cigler q-addition (JHC) is given by (a qb)n

n

X

k=0

n k

!

q

q(2k) an−kbk, (a qb)n

n

X

k=0

n k

!

q

q(k2) an−k(−b)k. (8)

Definition 3. We use the alphabet [3, p. 98], with zero denoted by θ. Let M be a subset of this alphabet. Then hMidenotes the set generated by M together with the four operationsq, q, q, q.

Definition 4. The basic construction which replaces the real numbers as function arguments in q- trigonometric functions etc. are the q-real numbers Rq, which are defined as follows:

Rq≡ hRi. (9)

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In [3, p. 167] we introduced the following concept:

Definition 5. There is a Ward number nq

nq 1q1q. . .q1, (10)

where the number of 1 in the RHS is n. Let (Nq,q, q) denote the semiring of Ward numbers kq, k ≥ 0 together with two binary operations:q is the usual Ward q-addition. The multiplication q is defined as follows:

nq qmqnmq, (11)

wheredenotes the equivalence in the alphabet.

We will now extend this semiring to a biring. Therefore we first define our general biring. The following defi- nition prepares for the biring in theorem 1.1.

Definition 6. Assume that RR1−(R1), a gradation. A graded commutative biring is a set (R,, , , 0), with two binary operationsand on R, a dual addition , a zero 0, (and a unit 1), which satisfy the fol- lowing axioms. For each elements a, b, cR:

1. Additive associativity: (ab)c = a(bc).

2. Additive commutativity: ab = ba.

3. Additive identity: There exists an element 0R such that 0a = a0 = a.

4. Additive inverse: There exists an element -aR such that a(−a) = (−a)a = 0.

5. Multiplicative identity: There exists an element 1R such that a 1 = 1 a = a.

6. Multiplicative associativity: (a b) c = a (b c).

7. distributivity: a (bc) = a ba c.

8. Multiplicative commutativity: a b = b a.

We assume that for b or c equal to −d, dR1, we may replaceby .

We can now extend the q-addition with JHC to obtain a graded commutative biring.

Definition 7. Let (Zq,q, q, q, 0q) denote ± the Ward numbers, i.e. Zq Nq−Nq, where there are two inverse q-additionsqand q. 0q denotes the zero θ, and 1q denotes the multiplicative identity. The dual addition is defined by

nqq−mqn − mq, n ≥ m. (12)

Furthermore, the multiplication qis defined by (11) and

nq q−mq−nmq. (13)

Finally, we define

−mq−mq. (14)

Theorem 1.1. An extension of [3, p. 167]. Assume that Zq is defined by the previous definition. Then (Zq,q, q, q, 0q) is a graded commutative biring.

Proof. The proof is achieved for three elements nq, mq, kqNqby [3, p. 167]. Instead, choose three elements nq, −mq, kq Zq. The associativity for (Zq,q, q) is shown as follows:

(nqq−mq)qkqby(12)

n − mqqkq(n − m) + kq, n ≥ m. (15)

nqq(kqq−mq)by(12) nqqk − mqn + (k − m)q, k ≥ m. (16) The associativity for (Zq, q) is shown as follows:

(nq q−mq) qkqby(13)

−nmq qkq by(13)

−nmkq. (17)

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nq q(−mq qkq)by(13) nq q−mkq by(13)

−nmkq. (18)

The identity is 1q. The commutativity for (Zq, q) follows from (13).

The distributive law is proved as follows:

Assume that k ≥ m.

nq q(kqq−mq))by(12) nq qk − mqn(k − m)q. (19)

(nq qkq) q(nq q−mq)by(13) −nmqqnkqby(12)

(nk − nm)q. (20)

Definition 8. [5] Let Qqdenote the set of objects of the following type:

mq

nq, nq  0q, (21)

together with a linear functional

v, R[x] × QqR, (22)

called the evaluation. If v(x) =P

k=0akxk, then v mq

nq



X

k=0

ak(mq)k

(nq)k. (23)

Definition 9. Let Qq[N] denote q-rational numbers, where we can replace Ward numbers in the numerator by products of Ward numbers. These products are denoted by ·. Let Q*qdenote the set of objects generated by Qq[N], together with the two operatorsq, q,

Q*q ≡ hQq[N]i. (24)

The following equation can be used for solutions of q-linear systems of equations in section 6.

Lemma 1.2.

qq= q q =q. (25)

Proof. Use the fact that qis the inverse operator toq.

We can combine (25) with combinations of minus signs in an obvious way. We assume that the following equations from [3, p. 99-100] are known: If

αqβ∼ γ (26)

or

α qβ∼ γ, (27)

we can compute α explicitly. By (25) the solutions are

α∼ γ qβ or α∼ γ qβ. (28)

We can calculate β explicitly from (27) as

−βα qγ. (29)

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2 The ring morphism τ, scalar products and q-determinants

We will now generalize the theory from the two previous articles [2] and [4] and define τ as a ring morphism.

Definition 10. Let + anddenote addition of functions in two different sets F and F1

q

with real argument.

Let · and * denote multiplication of functions in two different sets F and F1 q

with real argument. We define two sets of functions:

(F, +, ·, 1)R[Sinq, Cosq, Sinhq, Coshq, Eq], (30)

(F1

q

,, *, E)R[Sinq, Cosq, Sinhq, Coshq, Eq, Sin1

q

, Cos1

q

, Sinh1

q

, Cosh1

q

, E1

q

]. (31)

The letters in both F and F1

q are in R.

Theorem 2.1. (F, +, ·, 1) is a commutative ring.

Proof. This follows since the elements are linear combinations of real numbers.

Theorem 2.2. (F1 q

,, *, E) is a commutative ring.

Proof. This is proved similarly.

Definition 11. Let ΦR,|Φ|< ∞ denote a finite set of arbitrary letters. Let f Q

ifi F. The function τΦ is the function F7→F1

q

, which operates on the functions fiin the following way:

(q1q, if fidepends on Φ;

I, if fiis independent of Φ, (32)

where I denotes the identity operator.

Then we have

Theorem 2.3. The function τΦis a ring morphism.

τΦ(f1+ f2) = τΦ(f1)τΦ(f2), (33)

τΦ(f1· f2) = τΦ(f1) * τΦ(f2), (34)

τΦ(1) = E. (35)

Proof. Formulas (33) and (34) follow at once, since we first add or multiply elements on F and then invert the basis. This is the same as first inverting the basis and then adding or multiplying. The unit 1 is independent of q, and formula (35) follows.

Theorem 2.4. The function τΦis a ring epimorphism.

Proof. Assume that Γ F1

q

, and Ψ Φ contains all letters in the alphabet. Then we can always find an element ∆F which maps to Γ by τΦbecause of the completeness of the ring F.

We now come to the first matrix definitions. As before, matrix elements are denoted (i, j) and range from 0 to n − 1.

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Definition 12. For q ≠ 1, we have two matrix multiplications · and ·q. · is the ordinary matrix multiplication and ·qis defined as follows: for two n × n matrices A(q) and B(q), with entries aijand bij, we define

(a ·qb)ij

n−1

X

m=0

aimτ(bmj). (36)

We can modify this product in the following way:

(a ·Φ;qb)ij

n−1

X

m=0

aimτΦ(bmj), (37)

where τΦis defined by formula (32).

Definition 13. Let

ξ (x)

(τ(x) if n is even,

I, the identity if n is odd. (38)

The q-determinant of an n × n matrix α = [aij]n−1i,j=0(the first index denotes the row) is defined by the formula

detqα X

π∈Sn

sign(π)a0π(0)τ(a1π(1)) . . . ξ (an−1π(n−1)). (39) In other words, τ is applied to the matrix elements with first index odd.

The following variant of 3 × 3 q-determinant will also be used:

|α|s,t;q X

π∈S3

sign(π)a0π(0)τs(a1π(1)t(a2π(2)). (40)

Remark 1. For complex matrices we change τ to its complex conjugate as in the definition of SU(N).

In particular this definition applies to vector products.

Example 1. Put

xsDq,sx and xtDq,tx. (41)

The q-deformed vector product of

xt = (Sinhq(s)Cosq(t), Sinhq(s)Sinq(t), 1), (42) and

xs = (−Coshq(s)Sinq(t), Coshq(s)Cosq(t), 0) (43) is given by

xt×qxs≡ ||t,−;q

e~x e~y e~z

Sinhq(s)Cos1

q(t) Sinhq(s)Sin1

q(t) 1

−Coshq(s)Sinq(t) Coshq(s)Cosq(t) 0

=

Coshq(s)(−Cosq(t), −Sinq(t), Sinhq(s)).

(44)

We can now finally, mainly for symbolic purposes, define a q-analogue of the general linear group, which will be used in section 4.

Definition 14. The set GLq(n, R) is defined by

GLq(n, R)≡ {A(F, +, ·, 1)(n,n)|det1A ≠ 0}. (45)

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3 Basic definitions for matrix q-Lie algebras

We start with some definitions. We only consider matrix q-Lie algebras, which we call q-Lie algebras. These are q-analogues of matrix Lie algebras denoted by gq.

Definition 15. LetMn(Rq[Zq](n,n)) denote (a q-analogue of){g|ggl(n)}, i.e. the set of n × n matrices with entries in a Lie algebra.

ThenMn(Rq[Zq](n,n)) is a bi R-vector space with the operations ofq, matrix addition and R-scalar multi- plication. The zero vector is the n × n zero matrix On,n;q, which will often be abbreviated as O, when we know the size of the matrix.

Definition 16. A q-Lie algebra is a set of matrices

gln,C;Zq ≡ {AMn(Rq[Zq](n,n)), (46)

with properties similar to Lie algebras, which is a vector space in N2with an antisymmetric bilinear bracket operation [·, ·] : gq× gq 7→gqdefined by

[A, B]AB − BA. (47)

Something about the notation: we denote direct sums of matrices byqorq. The notationqdenotes direct sum of two matrices in the context of Lie algebra, and the notationq denotes sums of commuting matrices. The fact that the matrices of the q-Lie algebras do not commute is unimportant since we always multiply the q-Lie groups in a certain order. In the following, we writeq ={⊕q,q}. We can solve for the compact part:

t= gq qp. (48)

Definition 17. A subspace hqof a q-Lie algebra gqis said to be a q-Lie subalgebra if it is closed under the Lie bracket.

Definition 18. Assume that a basis for a q-Lie algebra is fixed. Let Ai, Aj, Akbe some basis vectors in a q-Lie algebra. The structure constants ckijare defined by

[Ai, Aj] =

r

X

k=1

ckijAk. (49)

We will see one example of these structure constants in section 5.

Definition 19. The commutator series of gq is defined as follows: let g0q gq, g1q [gq, gq], . . . , gn+1q [gnq, gnq]. We call gqsolvable if gnq = 0 for some n.

Definition 20. The lower central series of gqis defined as follows: let gq;0gq, gq;1[gq, gq], . . . , gq;n+1 [gq, gq;n]. We call gqnilpotent if gq;n= 0 for some n.

Example 2. The q-Lie algebra of upper-triangular matrices in gqis solvable.

Definition 21. The derived q-Lie algebra is the q-subalgebra of gq, denoted gq, which consists of all Lie brack- ets of pairs of elements of gq.

The simplest matrix q-Lie algebras are the same as in [7, p. 45-49]. We list some of them here.

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In the first two cases the derived algebra is one-dimensional:

I

0 0 0

0 0 1

0 0 0

, J

0 0 1

0 0 0

0 0 0

, K

0 0 0

−1 0 0

0 0 0

. (50)

I 0 1

0 0

!

, J 1 0

0 1

!

, K 0 0

0 1

!

. (51)

In the third case the derived algebra is two-dimensional:

M

0 0 1

0 0 0

0 0 0

, N

0 0 0

0 0 1

0 0 0

, K

−1 −1 0

0 −1 0

0 0 0

. (52)

There is also the following example of a solvable but not nilpotent affine q-Lie algebra from [1, p. 197]:

I 0 1

0 1

!

, J 1 0

0 0

!

. (53)

We will use almost the same q-Lie algebras as for Lie algebras. Then we use the q-exponential to compute an element of the corresponding q-Lie group.

We are now going to describe the phenomenon direct sum of two matrices in the context of q-Lie algebras.

Example 3. When A and B are q-Lie algebras, we have a formula for a q-analogue of certain continuous-time Markov processes:

Eq(AqB) = Eq(A)Eq(B). (54)

Definition 22. A subset B of a q-Lie algebra L is said to be a q-ideal if it is a vector subspace of L under q-addition, and [X, Y]B for any XB and YL.

The following definition is very important as in the ordinary case.

Definition 23. A q-Lie algebra gqis called semisimple if its only solvable q-ideal is 0.

Definition 24. Consider

L = L1qL2q· · · Lk, (55)

where all Liare simple q-ideals in L. The q-direct sum of the q-Lie algebras Liis the vector spaceP

qLi with component-wise addition, scalar multiplication, and product (P

qai)(P

qbi) =P aibi. The q-tensor product of the q-Lie algebras Liis the q-tensor product

L1qL2q· · · Lk (56)

of the vector spaces L1, . . . , Lk, together with the bilinear product defined by

(a1qa2q· · · ak)(b1qb2q. . . bk)a1b1qa2b2q· · · akbk. (57) Definition 25. The Lie bracket for the q-direct sum gq;1qgq;2is

[(X1, X2), (Y1, Y2)]gq;1qgq;2([X1, Y1]gq;1, [X2, Y2]gq;2). (58) Definition 26. Let hqgqbe a q-ideal and π : gq 7→gq/hqdenote projection onto the vector space quotient.

Then the bracket [π(x), π(y)]π([x, y]) is well-defined and defines the quotient algebra gq/hq.

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4 q-Lie groups

Definition 27. Denote the functions which are infinitely q-differentiable in all letters (variables) by Cq. Example 4. We explain what it means to be q-differentiable in all letters. Put

xαqβ qγ, f (x)Eq(x). (59)

Then we have

Dq,xf (x) = Dq,αf (x) = Dq,βf (x) = f (x). (60) The function f (x) is q-differentiable in all letters.

Definition 28. A q-Lie group (Gn,q,·,·q, Ig) Eq(gq), is a possibly infinite set of matricesGLq(n, R), with two associative multiplications: ·, and the twisted ·q. Each q-Lie group has a unit, denoted by Ig. Each element ΦG has an inverse Φ−1with the property Φ ·qΦ−1= Ig.

Theorem 4.1. A q-Lie group is also a manifold (with boundary). The boundary comes from the limits of some variables, usually angles for q-trigonometric functions.

Proof. It is an open subset of Rn2. The functions in question in (F, +, ·, 1) all belong to C(and Cq).

Many of the following theorems have their origin in the corresponding q-Lie algebras, and should sometimes be interpreted in a formal sense.

Definition 29. If (G1, ·1, ·1:q) and (G2, ·2, ·2:q) are two q-Lie groups, then (G1×G2, ·, ·q) is a q-Lie group called the product q-Lie group. This has group operations defined by

(g11, g21) · (g12, g22) = (g11·1g12, g21·2g22), (61) and

(g11, g21) ·q(g12, g22) = (g11·1:qg12, g21·2:qg22). (62) Definition 30. If (Gn,q, ·, ·q) is a q-Lie group and Hn,qis a nonempty subset of Gn,q, then (Hn,q, ·, ·q) is called a q-Lie subgroup of (Gn,q, ·, ·q) if

1.

Φ · ΨHn,qand Φ ·qΨ Hn,qfor all Φ, ΨHn,q. (63) 2.

Φ−1Hn,qfor all ΦHn,q. (64)

3. Hn,qis a submanifold of Gn,q.

Definition 31. The q-Lie group (Gn,q, ·, ·q) acts on the set X if there are two functions Ψq: Gn,q× X7→X

and

Φq: Gn,q× X7→X

such that, when we write g(x) instead of Ψq(g, x), and gq(x) instead of Φq(g, x) we have 1. (g1· g2)(x) = g1((g2)(x)) for all g1, g2Gn,q, xX,

2. (g1·qg2)(x) = (g1)q((g2)(x)) for all g1, g2Gn,q, xX, 3. Ig(x) = x, xX.

The q-Lie group Gn,qacts faithfully on X if the only element of Gn,q, which fixes every element of X under the two operations Ψq(g, x) and Φq(g, x), is the identity.

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Theorem 4.2. If Gn,qacts on a set X and xX, then

Stab x ={gGn,q|(g(x) = x)(gq(x) = x)}is a q-Lie subgroup of Gn,q, called the stabilizer of x.

Definition 32. A q-Lie subgroup Hn,qof a q-Lie group Gn,qis called a normal q-Lie subgroup of Gn,qif ϕ−1·qΨ ·qϕHn,qfor all ϕGn,qand ΨHn,q. (65) Definition 33. An invertible mapping f : (Gn,q, ·, ·q)(Hn,q, ·, ·q) is called a q-Lie group morphism between (Gn,q, ·, ·q) and (Hn,q, ·, ·q) if

f (ϕ · ψ) = f (ϕ) · f (ψ), ϕ, ψRq, and f (ϕ ·qψ) = f (ϕ) ·qf (ψ), ϕ, ψRq. (66) A q-Lie group morphism is called q-Lie group isomorphism if it is one-to-one.

Theorem 4.3. If f : (Gn,q, ·, ·q)(Hn,q, ·, ·q) is a q-Lie group morphism, then

f (Ig) = Ihand f (ϕ−1) = f (ϕ)−1for all ϕGn,q. (67) Proof. 1. f (Ig) = f (Ig· Ig) = f (Ig) · f (Ig) = Ih.

2. f (ϕ) ·qf (ϕ−1) = f (ϕ ·qϕ−1) = f (Ig) = Ih. We find the desired result.

Definition 34. If f : (Gn,q, ·, ·q)(Hn,q, ·, ·q) is a q-Lie group morphism, the kernel of f , which we denote by Kerf , is the set of elements of Gn,qthat are mapped by f to the identity of Hn,q.

Theorem 4.4. Let f : (Gn,q, ·, ·q)(Hn,q, ·, ·q) be a q-Lie group morphism. Then 1. Kerf is a normal q-Lie group of Gn,q,

2. f is injective if and only if Kerf = IG.

Proof. We begin by showing that Kerf is a q-Lie subgroup.

Assume that α and βKerf so that f (α) = f (β) = Ih. Then

f (α · β) = f (α) · f (β) = IH· Ih= Ih, and α · βKerf = f (α ·qβ) Furthermore,

f (α−1) = f (α)−1= I−1h = Ihand α−1Kerf . Assume that αKerf and gGq, then

f (g−1·qα ·qg) = f (g−1) ·qf (α) ·qf (g) = f (g−1) ·qIh·qf (g) = f (g−1) ·qf (g) = Ih.

This implies that g−1·q α ·qg Kerf and Kerf is a normal q-Lie subgroup of Gn,q. The second statement is proved in the same way as for groups.

Theorem 4.5. For any q-Lie group morphism f : (Gn,q, ·, ·q) (Hn,q, ·, ·q), the image of f , Imf = {f (ϕ)|ϕGn,q}is a q-Lie subgroup of Hn,q.

Proof. This is again similar to the proof for groups. Assume that f (g1), f (g2) Imf . It follows that f (g1) · f (g2) = f (g1· g2)Imf , and f (g1) ·qf (g2) = f (g1·qg2)Imf . Also we have f (g1)−1= f (g−11 )Imf .

The morphism

exp : R−→R* (68)

corresponds to the q-Lie group morphisms

Eq : (Rq,q)−→R*, (69)

References

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Historiskt sett har bolagets substansvärde per aktie (justerat för återinvesterad utdelning) under perioden 1 januari 1995 – 31 mars 2018 haft en effektiv årsavkastning på 7,3 % och

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Den 30 september 2017 uppgick Aktiv fondportfölj till ett värde om 168,1 MSEK (251,1 MSEK), motsvarande 70,0 % (97,1 %) av det totala substansvärdet.. Av det sammanlagda

Under fjärde kvartalet var det åter igen lägre volymer då Finland upplevde en tydlig andra våg av pandemin, vilket ledde till återhållsamhet bland konsumenter.. Den

✓ Rörelseresultatet minskar till 325 mnkr (328) beroende på minskade energileveranser av både el och fjärrvärme till följd av temperaturer långt ifrån ett normalår...