Download On the q-exponential of matrix q-Lie algebras

Spec. Matrices 2017; 5:36–50
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 su q (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 SO q (2) and SU q (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 definition 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 N⊕q , which
appear in su q (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: [email protected]
© 2017 Thomas Ernst, published by De Gruyter Open.
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.
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On the q-exponential of matrix q-Lie algebras |
37
In section 3 we come to the q-Lie algebras, which are very similar to matrix Lie algebras. The only difference 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 general 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 multiplications. 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
n
{ a }q ≡
Y
1 − qa
; { n }q ! ≡
{k}q , {0}q ! ≡ 1, q ∈ C\{1},
1−q
(1)
k=1
The Gauss q-binomial coefficient is given by
!
{ n }q !
n
, k = 0, 1, . . . , n.
≡
{ n − k } q !{ k } q !
k
(2)
q
The q-derivative is defined by
Dq φ(x) ≡
φ(x) − φ(qx)
.
(1 − q)x
(3)
The q-exponential functions are defined by
Eq (z) ≡
∞
X
k=0
k
∞
X
1 k
q ( 2) k
z ; E 1 (z) ≡
z .
q
{ k }q !
{ k }q !
(4)
k=0
The q-trigonometric functions are defined by
Cosq (x) ≡
1
(Eq (ix) + Eq (−ix)).
2
(5)
Sinq (x) ≡
1
(Eq (ix) − Eq (−ix)).
2i
(6)
Definition 2. Let a and b belong to a commutative monoid. The Nalli–Ward–AlSalam q-addition (NWA) is
given by
!
!
n
n
X
X
n
n
n
k n−k
n
(a ⊕q b) ≡
a b , (a q b) ≡
a k (−b)n−k
(7)
k
k
k=0
k=0
q
q
The Jackson–Hahn–Cigler q-addition (JHC) is given by
!
!
n
n
X
X
k
k
n
n
n
n−k k
n
(
)
2
(a q b) ≡
q a b , (a q b) ≡
q(2) a n−k (−b)k .
k
k
k=0
k=0
q
(8)
q
Definition 3. We use the alphabet [3, p. 98], with zero denoted by θ. Let M be a subset of this alphabet. Then
hM i denotes the set generated by M together with the four operations ⊕q , q , q , q .
Definition 4. The basic construction which replaces the real numbers as function arguments in qtrigonometric functions etc. are the q-real numbers Rq , which are defined as follows:
Rq ≡ hRi.
(9)
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38 | Thomas Ernst
In [3, p. 167] we introduced the following concept:
Definition 5. There is a Ward number n q
n q ∼ 1 ⊕q 1 ⊕q . . . ⊕q 1,
(10)
where the number of 1 in the RHS is n. Let (N⊕q , ⊕q , q ) denote the semiring of Ward numbers k q , k ≥ 0
together with two binary operations: ⊕q is the usual Ward q-addition. The multiplication q is defined as
follows:
n q q m q ∼ nm q ,
(11)
where ∼ denotes the equivalence in the alphabet.
We will now extend this semiring to a biring. Therefore we first define our general biring. The following definition prepares for the biring in theorem 1.1.
Definition 6. Assume that R ≡ R1 ∪ −(R1 ), a gradation. A graded commutative biring is a set (R, ⊕, , , 0),
with two binary operations ⊕ and on R, a dual addition , a zero 0, (and a unit 1), which satisfy the following axioms. For each elements a, b, c ∈ R:
1. Additive associativity: (a ⊕ b) ⊕ c = a ⊕ (b ⊕ c).
2. Additive commutativity: a ⊕ b = b ⊕ a.
3. Additive identity: There exists an element 0∈ R such that 0 ⊕ a = a ⊕ 0 = a.
4. Additive inverse: There exists an element -a∈ R such that a ⊕ (−a) = (−a) ⊕ a = 0.
5. Multiplicative identity: There exists an element 1∈ R such that a 1 = 1 a = a.
6. Multiplicative associativity: (a b) c = a (b c).
7. distributivity: a (b ⊕ c) = a b ⊕ a c.
8. Multiplicative commutativity: a b = b a.
We assume that for b or c equal to −d, d ∈ R1 , we may replace ⊕ by .
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 ≡ N⊕q ∪ −N⊕q , where there are
two inverse q-additions ⊕q and q . 0q denotes the zero θ, and 1q denotes the multiplicative identity. The
dual addition is defined by
n q q −m q ∼ n − m q , n ≥ m.
(12)
Furthermore, the multiplication q is defined by (11) and
n q q −m q ∼ −nm q .
(13)
−m q ≡ −m q .
(14)
Finally, we define
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 n q , m q , k q ∈ N⊕q by [3, p. 167]. Instead, choose three elements
n q , −m q , k q ∈ Z⊕q . The associativity for (Z⊕q , ⊕q , q ) is shown as follows:
(n q q −m q ) ⊕q k q
by(12)
n − m q ⊕q k q ∼(n − m) + k q , n ≥ m.
(15)
n q ⊕q (k q q −m q ) ∼ n q ⊕q k − m q ∼n + (k − m)q , k ≥ m.
(16)
∼
by(12)
The associativity for (Z⊕q , q ) is shown as follows:
(n q q −m q ) q k q
by(13)
∼
−nm q q k q
by(13)
∼
−nmk q .
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(17)
On the q-exponential of matrix q-Lie algebras |
by(13)
n q q (−m q q k q ) ∼ n q q −mk q
by(13)
∼
−nmk q .
39
(18)
The identity is 1q . The commutativity for (Z⊕q , q ) follows from (13).
The distributive law is proved as follows:
Assume that k ≥ m.
by(12)
n q q (k q q −m q )) ∼ n q q k − m q ∼ n(k − m)q .
by(13)
(n q q k q ) q (n q q −m q ) ∼ −nm q ⊕q nk q
by(12)
∼
(19)
(nk − nm)q .
(20)
Definition 8. [5] Let Q⊕q denote the set of objects of the following type:
mq
, n q 0q ,
nq
(21)
v, R[x] × Q⊕q → R,
(22)
together with a linear functional
called the evaluation. If v(x) =
P∞
k=0
a k x k , then
v
mq
nq
≡
∞
X
k=0
ak
(m q )k
.
(n q )k
(23)
Definition 9. Let Q⊕q [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⊕
denote the set of objects generated
q
by Q⊕q [N], together with the two operators ⊕q , q ,
*
Q⊕
≡ hQ⊕q [N]i.
q
(24)
The following equation can be used for solutions of q-linear systems of equations in section 6.
Lemma 1.2.
q q = q q = ⊕q .
(25)
Proof. Use the fact that q is the inverse operator to ⊕q .
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
α ⊕q β ∼ γ
(26)
or
α q β ∼ γ ,
(27)
we can compute α explicitly. By (25) the solutions are
α ∼ γ q β or α ∼ γ q β.
(28)
We can calculate β explicitly from (27) as
−β ∼ α q γ .
(29)
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40 | Thomas Ernst
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 + and ⊕ denote addition of functions in two different sets F and F 1 with real argument.
q
Let · and * denote multiplication of functions in two different sets F and F 1 with real argument. We define
q
two sets of functions:
(F, +, ·, 1) ≡ R[Sinq , Cosq , Sinhq , Coshq , Eq ],
(30)
(F 1 , ⊕, *, E) ≡ R[Sinq , Cosq , Sinhq , Coshq , Eq , Sin 1 , Cos 1 , Sinh 1 , Cosh 1 , E 1 ].
q
q
q
q
q
(31)
q
The letters in both F and F 1 are in R.
q
Theorem 2.1. (F, +, ·, 1) is a commutative ring.
Proof. This follows since the elements are linear combinations of real numbers.
Theorem 2.2. (F 1 , ⊕, *, E) is a commutative ring.
q
Proof. This is proved similarly.
Definition 11. Let Φ ⊂ R, |Φ| < ∞ denote a finite set of arbitrary letters. Let f ≡
is the function F 7→ F 1 , which operates on the functions f i in the following way:
Q
i fi
∈ F. The function τ Φ
q
(
q → 1q ,
if f i depends on Φ;
I,
if f i is independent of Φ,
(32)
where I denotes the identity operator.
Then we have
Theorem 2.3. The function τ Φ is a ring morphism.
τ Φ (f1 + f2 ) = τ Φ (f1 ) ⊕ τ Φ (f2 ),
(33)
τ Φ (f1 · f2 ) = τ Φ (f1 ) * τ Φ (f2 ),
(34)
τ Φ (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 τ Φ is a ring epimorphism.
Proof. Assume that Γ ∈ F 1 , and Ψ ⊃ Φ contains all letters in the alphabet. Then we can always find an
q
element ∆ ∈ F which maps to Γ by τ Φ because 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 ·q is defined as follows: for two n × n matrices A(q) and B(q), with entries a ij and b ij , we define
(a ·q b)ij ≡
n−1
X
a im τ(b mj ).
(36)
a im τ Φ (b mj ),
(37)
m=0
We can modify this product in the following way:
(a ·Φ ;q b)ij ≡
n−1
X
m=0
where τ Φ is defined by formula (32).
Definition 13. Let
(
ξ (x) ≡
τ(x) if n is even,
(38)
I, the identity if n is odd.
The q-determinant of an n × n matrix α = [a ij ]n−1
i,j=0 (the first index denotes the row) is defined by the
formula
X
detq α ≡
sign(π)a0π(0) τ(a1π(1) ) . . . ξ (a n−1π(n−1) ).
(39)
π∈S n
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:
X
|α|s,t;q ≡
sign(π)a0π(0) τ s (a1π(1) )τ t (a2π(2) ).
(40)
π∈S3
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
xs ≡ Dq,s x and xt ≡ Dq,t x.
(41)
xt = (Sinhq (s)Cosq (t), Sinhq (s)Sinq (t), 1),
(42)
xs = (−Coshq (s)Sinq (t), Coshq (s)Cosq (t), 0)
(43)
The q-deformed vector product of
and
is given by
xt ×q xs ≡ ||t,−;q
e~x
Sinh (s)Cos 1 (t)
≡
q
q
−Coshq (s)Sinq (t)
e~y
Sinhq (s)Sin 1 (t)
q
Coshq (s)Cosq (t)
=
0 e~z
1
(44)
Coshq (s)(−Cosq (t), −Sinq (t), Sinhq (s)).
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) |det1 A ≠ 0}.
(45)
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42 | Thomas Ernst
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. Let Mn (R′ q′ [Zq ](n,n) ) denote (a q-analogue of) {g |g ⊂ gl(n)}, i.e. the set of n × n matrices with
entries in a Lie algebra.
Then Mn (R′ q′ [Zq ](n,n) ) is a bi R-vector space with the operations of ⊕′ q′ , matrix addition and R-scalar multiplication. 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 ≡ {A ∈ Mn (R′ q′ [Zq ](n,n) ),
(46)
with properties similar to Lie algebras, which is a vector space in N2 with an antisymmetric bilinear bracket
operation [·, ·] : gq × gq 7→ gq defined by
[A, B] ≡ AB − BA.
(47)
Something about the notation: we denote direct sums of matrices by ⊕′ q′ or ⊕q . The notation ⊕′ q′ denotes
direct sum of two matrices in the context of Lie algebra, and the notation ⊕q 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 write ⊕q = {⊕q , ⊕′ q′ }. We can solve for the
compact part:
t = gq ′ q′ p.
(48)
Definition 17. A subspace hq of a q-Lie algebra gq is 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 A i , A j , A k be some basis vectors in a q-Lie
algebra. The structure constants c kij are defined by
[A i , A j ] =
r
X
c kij A k .
(49)
k=1
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+1
≡
q
[gnq , gnq ]. We call gq solvable if gnq = 0 for some n.
Definition 20. The lower central series of gq is defined as follows: let gq;0 ≡ gq , gq;1 ≡ [gq , gq ], . . . , gq;n+1 ≡
[gq , gq;n ]. We call gq nilpotent if gq;n = 0 for some n.
Example 2. The q-Lie algebra of upper-triangular matrices in gq is solvable.
Definition 21. The derived q-Lie algebra is the q-subalgebra of gq , denoted g′q , which consists of all Lie brackets 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:

0

I≡ 0
0
I≡
0
0
0


0
0


1 , J ≡  0
0
0
0
0
1
0
!
, J≡


1
0


0  , K ≡  −1
0
0
0
0
0
1
0
0
1
!
, K≡
0
0
0
1

0

0 .
0
0
0
0
(50)
!
In the third case the derived algebra is two-dimensional:





0 0 1
0 0 0
−1





M ≡  0 0 0 , N ≡  0 0 1 ,K ≡  0
0 0 0
0 0 0
0
.
−1
−1
0
(51)

0

0 .
0
(52)
There is also the following example of a solvable but not nilpotent affine q-Lie algebra from [1, p. 197]:
!
!
0 1
1 0
I≡
, J≡
.
(53)
0 1
0 0
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 (A ⊕q B) = 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 X ∈ B and Y ∈ L.
The following definition is very important as in the ordinary case.
Definition 23. A q-Lie algebra gq is called semisimple if its only solvable q-ideal is 0.
Definition 24. Consider
L = L 1 ⊕q L 2 ⊕q · · · L k ,
(55)
P
where all L i are simple q-ideals in L. The q-direct sum of the q-Lie algebras L i is the vector space
⊕q L i
P
P
P
with component-wise addition, scalar multiplication, and product ( ⊕q a i )( ⊕q b i ) =
a i b i . The q-tensor
product of the q-Lie algebras L i is the q-tensor product
L 1 ⊗q L 2 ⊗q · · · L k
(56)
of the vector spaces L1 , . . . , L k , together with the bilinear product defined by
(a1 ⊗q a2 ⊗q · · · a k )(b1 ⊗q b2 ⊗q . . . b k ) ≡ a1 b1 ⊗q a2 b2 ⊗q · · · a k b k .
(57)
Definition 25. The Lie bracket for the q-direct sum gq;1 ⊕q gq;2 is
[(X1 , X2 ), (Y1 , Y2 )]gq;1 ⊕q gq;2 ≡ ([X1 , Y1 ]gq;1 , [X2 , Y2 ]gq;2 ).
(58)
Definition 26. Let hq ⊂ gq be a q-ideal and π : gq 7→ gq /hq denote 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 C∞
q .
Example 4. We explain what it means to be q-differentiable in all letters. Put
x ∼ α ⊕q β q γ , f (x) ≡ Eq (x).
(59)
Dq,x f (x) = Dq,α f (x) = Dq,β f (x) = f (x).
(60)
Then we have
The function f (x) is q-differentiable in all letters.
Definition 28. A q-Lie group (G n,q,·,·q , Ig ) ⊃ Eq (gq ), is a possibly infinite set of matrices ∈ GLq (n, R), with
two associative multiplications: ·, and the twisted ·q . Each q-Lie group has a unit, denoted by Ig . Each element
Φ ∈ G has an inverse Φ−1 with the property Φ ·q Φ−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.
2
Proof. It is an open subset of Rn . The functions in question in (F, +, ·, 1) all belong to C∞ (and C∞
q ).
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 ·1 g12 , g21 ·2 g22 ),
(61)
(g11 , g21 ) ·q (g12 , g22 ) = (g11 ·1:q g12 , g21 ·2:q g22 ).
(62)
and
Definition 30. If (G n,q , ·, ·q ) is a q-Lie group and H n,q is a nonempty subset of G n,q , then (H n,q , ·, ·q ) is called
a q-Lie subgroup of (G n,q , ·, ·q ) if
1.
Φ · Ψ ∈ H n,q and Φ ·q Ψ ∈ H n,q for all Φ, Ψ ∈ H n,q .
(63)
2.
Φ−1 ∈ H n,q for all Φ ∈ H n,q .
3.
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H n,q is a submanifold of G n,q .
Definition 31. The q-Lie group (G n,q , ·, ·q ) acts on the set X if there are two functions
Ψ q : G n,q × X 7→ X
and
Φ q : G n,q × X 7→ X
such that, when we write g(x) instead of Ψ q (g, x), and g q (x) instead of Φ q (g, x) we have
1. (g1 · g2 )(x) = g1 ((g2 )(x)) for all g1 , g2 ∈ G n,q , x ∈ X,
2. (g1 ·q g2 )(x) = (g1 )q ((g2 )(x)) for all g1 , g2 ∈ G n,q , x ∈ X,
3. Ig (x) = x, x ∈ X.
The q-Lie group G n,q acts faithfully on X if the only element of G n,q , which fixes every element of X under the
two operations Ψ q (g, x) and Φ q (g, x), is the identity.
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Theorem 4.2. If G n,q acts on a set X and x ∈ X, then
Stab x = {g ∈ G n,q |(g(x) = x) ∧ (g q (x) = x)} is a q-Lie subgroup of G n,q , called the stabilizer of x.
Definition 32. A q-Lie subgroup H n,q of a q-Lie group G n,q is called a normal q-Lie subgroup of G n,q if
ϕ−1 ·q Ψ ·q ϕ ∈ H n,q for all ϕ ∈ G n,q and Ψ ∈ H n,q .
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Definition 33. An invertible mapping f : (G n,q , ·, ·q ) → (H n,q , ·, ·q ) is called a q-Lie group morphism between
(G n,q , ·, ·q ) and (H n,q , ·, ·q ) if
f (ϕ · ψ) = f (ϕ) · f (ψ), ϕ, ψ ∈ Rq , and f (ϕ ·q ψ) = f (ϕ) ·q f (ψ), ϕ, ψ ∈ Rq .
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A q-Lie group morphism is called q-Lie group isomorphism if it is one-to-one.
Theorem 4.3. If f : (G n,q , ·, ·q ) → (H n,q , ·, ·q ) is a q-Lie group morphism, then
f (Ig ) = Ih and f (ϕ−1 ) = f (ϕ)−1 for all ϕ ∈ G n,q .
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Proof. 1. f (Ig ) = f (Ig · Ig ) = f (Ig ) · f (Ig ) = Ih .
2. f (ϕ) ·q f (ϕ−1 ) = f (ϕ ·q ϕ−1 ) = f (Ig ) = Ih . We find the desired result.
Definition 34. If f : (G n,q , ·, ·q ) → (H n,q , ·, ·q ) is a q-Lie group morphism, the kernel of f , which we denote
by Kerf , is the set of elements of G n,q that are mapped by f to the identity of H n,q .
Theorem 4.4. Let f : (G n,q , ·, ·q ) → (H n,q , ·, ·q ) be a q-Lie group morphism. Then
1. Kerf is a normal q-Lie group of G n,q ,
2. f is injective if and only if Kerf = I G .
Proof. We begin by showing that Kerf is a q-Lie subgroup.
Assume that α and β ∈ Kerf so that f (α) = f (β) = Ih .
Then
f (α · β) = f (α) · f (β) = I H · Ih = Ih , and α · β ∈ Kerf = f (α ·q β)
Furthermore,
−1
f (α−1 ) = f (α)−1 = I−1
∈ Kerf .
h = Ih and α
Assume that α ∈ Kerf and g ∈ G q , then
f (g −1 ·q α ·q g) = f (g −1 ) ·q f (α) ·q f (g) = f (g −1 ) ·q Ih ·q f (g) = f (g −1 ) ·q f (g) = Ih .
This implies that g −1 ·q α ·q g ∈ Kerf and Kerf is a normal q-Lie subgroup of G n,q . The second statement is
proved in the same way as for groups.
Theorem 4.5. For any q-Lie group morphism f : (G n,q , ·, ·q ) → (H n,q , ·, ·q ), the image of f , Imf =
{f (ϕ)|ϕ ∈ G n,q } is a q-Lie subgroup of H n,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 ) ·q f (g2 ) = f (g1 ·q g2 ) ∈ Imf . Also we have f (g1 )−1 = f (g1−1 ) ∈ Imf .
The morphism
exp : R −→ R*
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Eq : (Rq , ⊕q ) −→ R* ,
(69)
corresponds to the q-Lie group morphisms
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Eq (x ⊕q y) = Eq (x)Eq (y)
(70)
Eq : (Rq ; ⊕q , q ) −→ R* ,
(71)
Eq (x ⊕q y) = Eq (x)Eq (y),
(72)
Eq (x q y) = Eq (x)E 1 (y).
(73)
and
q
When x and y are so-called q-Lie algebras, we call these mappings q-one-parameter subgroups, for which
see next subchapter.
Definition 35. The set GL′n;q ⊂ GLq (n, R) is defined as the image Eq (X) for X ∈ Mn (R′ q′ ). We assume that
there exist two matrix multiplications in GL′n;q : · and ·q , such that Eq (X) ·q Eq (−X) = In , the unit matrix. The
second multiplication ·q is given by formula (36).
We have the following commutative diagram:
U×V
Eq
⊕′ q′
U ⊕′ q′ V
/ Eq (U) × Eq (V) .
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·
Eq
/ Eq (U ⊕′ q′ V)
With the following notation
T ≡ Eq (t), G ≡ Eq (gq ), P ≡ Eq (p),
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we have the following expression for a so-called q-symmetric space:
T = G/P,
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where g is a real form of a simple complex q-Lie algebra, g = t ⊕′ q′ p, with [p, p] ≤ p, [p, t] ≤ t, and [t, t] ≤ p.
5 Examples: su q (n), SU q (4), SU q (2) and SO q (4)
5.1 The q-Lie algebra su q (n)
We now turn to the q-Lie algebra su q (n). We note that our definition is different from the physicist notation
[6]. First we define the q-trace.
Definition 36. The q-trace of an n × n matrix A is defined by
tr q A ≡ A00 ⊕q A11 ⊕q . . . ⊕q A n−1,n−1 , where n q ⊕q −n q ∼ θ,
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the zero of the alphabet.
Definition 37. Let A * denote conjugated transpose. Then the q-Lie algebra su q (n) is defined by
su q (n) ≡ {A ∈ C[Zq ](n,n) |tr q (A) ∼ θ, A * = −A T }.
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Definition 38. The basis for su q (4) consists of the following 15 matrices:




0 1 0 0
0 i 0 0
 −1 0 0 0 
 i 0 0 0 




λ1 ≡ 
 , λ2 ≡ 
,
 0 0 0 0 
 0 0 0 0 
0 0 0 0
0 0 0 0




1 0 0 0
0 0 1 0
 0 −1 0 0 
 0 0 0 0 




λ3 ≡ 
 , λ4 ≡ 
,
 0 0 0 0 
 −1 0 0 0 
0 0 0 0
0 0 0 0




0 0 i 0
0 0 0 0
 0 0 0 0 
 0 0 1 0 




λ5 ≡ 
 , λ6 ≡ 
,
 i 0 0 0 
 0 −1 0 0 
0 0 0 0
0 0 0 0




0 0 0 0
1 0
0
0
 0 0 i 0 
 0 1
0
0 




λ7 ≡ 
,
 , λ8 ≡ 
 0 i 0 0 
 0 0 −2q 0 
0 0
0
0
0 0 0 0




0 0 0 1
0 0 0 i
 0 0 0 0 
 0 0 0 0 




λ9 ≡ 
 , λ10 ≡ 
,
 0 0 0 0 
 0 0 0 0 
−1 0 0 0
i 0 0 0




0 0 0 0
0 0 0 0
 0 0 0 1 
 0 0 0 i 




λ11 ≡ 
 , λ12 ≡ 
,
 0 0 0 0 
 0 0 0 0 
0 −1 0 0
0 i 0 0




0 0 0 0
0 0 0 0
 0 0 0 0 
 0 0 0 0 




λ13 ≡ 
 , λ14 ≡ 
,
 0 0 0 1 
 0 0 0 i 
0 0 −1 0
0 0 i 0


1 0 0
0
 0 1 0
0 


λ15 ≡ 

 0 0 1
0 
0 0 0 −3q .
47
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5.2 SU q (4)
We now compute some of the matrices in SU q (4), all with q-determinant 1. The general formula is
U t;i;q ≡ Eq (λ i t).
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


U t;1;q = 




U t;3;q = 




U t;5;q = 




U t;15;q = 

Cosq t
−Sinq t
0
0
Eq (t)
0
0
0
Cosq t
0
iSinq t
0
Eq (t)
0
0
0
Sinq t
Cosq t
0
0
0
Eq (−t)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
iSinq t
0
Cosq t
0
0
Eq (t)
0
0
0
0
Eq (t)
0
0
0
0
0






 , U t;2;q = 








 , U t;4;q = 








 , U t;9;q = 


0
0
0
Eq (−t3q )
Cosq t
iSinq t
0
0
Cosq t
0
−Sinq t
0
Cosq t
0
0
−Sinq t
iSinq t
Cosq t
0
0
0
0
0
0
0
0
0
0
Sinq t
0
Cosq t
0
0
Cosq t
0
0
0
0
0
0
0
0
0
0
0
0
0
0



,




,

Sinq t
0
0
0

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

,




.

Note that there is a juxtaposition (multiplication) in −t3q .
Theorem 5.1. For the q-Lie algebra su q (n), the matrix elements −N − 1q in the basis do not contribute to the
commutator, because they will always be multiplied with zeros. Because of the bilinearity, this applies to all
elements in su q (n).
5.3 The q-Lie group SU q (2)
In an earlier paper [2] we introduced a q-Lie group SU q (2), together with its maximal torus. We showed that
SU q (2), ·, ·q in the form of a maximal torus is closed under the two operations. Here we consider a slightly
different form:
Definition 39. Let ξ (q) denote the first zero > 0 of the function Sinq (ψ), and let ξ (q, 2) denote the second
zero > 0 of the function Sinq (ψ). The general form for the q-Lie group SU q (2) is
!
Cosq (ψ)
−Sinq (ψ)Eq (iα)
U ψ,α ≡
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Sinq (ψ)Eq (−iα)
Cosq (ψ))
for 0 ≤ ψ < ξ (q), 0 ≤ α < ξ (q, 2), with q-determinant 1.
Theorem 5.2. Let the matrices A and B have matrix elements a ij and b ij respectively. Each element U ψ,α has
an inverse U−ψ,α under the following matrix multiplication:
A ·q B ≡
a00 τb00 + a01 τb10
a10 b00 + a11 b10
a00 b01 + a01 b11
a10 τb01 + a11 τb11
!
,
which is a mixture of ordinary matrix multiplication and formula (36).
5.4 The q-Lie group SO q (4)
We can now construct arbitrary q-Lie groups by the tensor product (61), we give one example:
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Definition 40. The q-Lie group SO q (4) is defined by
SO q (4) ≡ SU q (2) × SU q (2).
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6 q-linear system of equations
We will now describe a q-linear system of equations with Ward numbers by an unorthodox multiplication
of Ward numbers by q-additions and unknown q-rational numbers, where additions and subtractions are
replaced by the four q-additions.
Definition 41. Matrix elements will always be denoted (i, j). Here i denotes the row and j denotes the column.
The matrix elements range from 1 to n.
*
Assume that the letters {α i,j }ni,j=1 ∈ (N⊕q , ⊕q , q ), and x1 , . . . , x n ∈ Q⊕
. Furthermore, let the matrix of
q
n
q-additions be given by Y = {y ij }i,j=1 , where I denotes the unit operator. Let A be the n×2n-matrix with matrix
elements α ij × y ij and let X be the n-vector with matrix elements x j . Then we define the matrix multiplication
A ·q X i ≡
n
X
(α i,m × y i,m )X m .
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m=1
We assume that y i,1 = I, y i,j ≠ I, j ≠ 1. The matrix element (Y ·q X)i then denotes the vector
X1 , y i,2 X2 , . . . , y i,n X n ,
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where y i,m denotes the respective q-addition, which precedes X m .
Note that the sign of letters is included in y i,m , so that we do not need Z⊕q .
Theorem 6.1. Assume that the letters β1 , . . . , β n ∈ (N⊕q , ⊕q , q ), and let B be the n-vector with components
β j . Furthermore, assume that the coefficient matrix for q = 1 is nonsingular, det{A i,j }ni,j=1 |q = 1 ≠ 0. Then the
q-linear system of equations
A ·q X ∼ B
(87)
Qn
2
*
has k=2 k n-vector solutions in Q⊕q , counted with multiplicity.
Proof. These solutions correspond to the rational solution of the corresponding linear system of equations.
To find one solution, use Gaussian elimination by formulas (28) and (29), using the four q-additions between
the respective equations, and by (25) two JHC become an NWA. The number of possible solutions is then found
by the multiplication principle.
To explain this we give an example.
Example 5. Give one solution to the q-linear system of equations


 x ⊕q z q y ∼ 4q ,

2 x ⊕q z q 3q y ∼ 1q ,
 q

3 x ⊕ 2 y ∼ 7 .
q
q q
q
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The corresponding linear system of equations has solution x = 1, y = 2, z = 5. We solve for y by the third
equation using (29) to give
7q q 3q x
y∼
.
(89)
2q
By (89), again using (29), we obtain

 x ⊕q z q
2q x ⊕q z
7q q 3q ·x
∼ 4q
2q
7q q 3q ·x
q 3q ·
2q
∼ 1q .
(90)
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Now compute the first equation q the second using (28) and (25).
q x ⊕q 7q q 3q · x ∼ 3q ⇔ x ∼ 1q .
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Formula (89) then gives
y∼
4q
.
2q
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Finally, the first formula (90) gives
z ∼ 3q ⊕q
4q
.
2q
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7 Discussion
In this article we have combined two mathematical subjects, which are ubiquitous in theoretical physics,
compare [6], [3]. We note furthermore that these considerations can be extended to complex reflection groups
and hyperplanes, where similar matrices occur. These interesting subjects, as well as q-differential geometry
and q-symmetric spaces will be covered in forthcoming articles.
Acknowledgement: I thank Karl-Heinz Fieseler for his kind assistance.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
C.W.Conatser, Contractions of the lowdimensional real Lie algebras. J. Math. Physics 13, (1972), 196-203
T. Ernst, q-deformed matrix pseudo–groups. Royal Flemish Academy of Belgium (2010), 151–162
T. Ernst, A comprehensive treatment of q-calculus, Birkhäuser 2012.
T. Ernst, An umbral approach to find q-analogues of matrix formulas, Linear Algebra Appl. 439 (2013), 1167–1182.
T. Ernst, Multiplication formulas for q-Appell polynomials and the multiple q-power sums. Ann. Univ. Marie Curie (2016)
W. Pfeifer, The Lie algebras su(N). An introduction. Birkhäuser (2003)
J.D.Talman, Special functions. A group theoretic approach. The Mathematical Physics Monograph Series. New YorkAmsterdam: W.A. Benjamin, 1968.
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