Cayley-Hamilton theorem for quantum matrix algebras of GL(m|n) type

0.27 MB·

by  D. I. Gurevich

To L.D. Faddeev
with the respect and appreciation
5 Cayley-Hamilton theorem for quantum matrix
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2 algebras of GL(m|n) type

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D.I. Gurevich∗

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A ISTV, Universit´e de Valenciennes, 59304 Valenciennes, France
Q P.N. Pyatov†
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h Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Russia
t
a P.A. Saponov‡
m
Department of Theoretical Physics, IHEP, 142281 Protvino, Russia
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February 1, 2008
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1 Abstract
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0 The classical Cayley-Hamilton identities are generalized to quantum matrix alge-
/
h bras of the GL(m|n) type.
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1 Introduction
:
v
i
X
The well known Cayley-Hamilton theorem of classical matrix analysis claims that any square
r
a matrix M with entries from a field K satisfies the identity
∆(M) ≡ 0, (1.1)
where∆(x) = det(M−xI)isthecharacteristicpolynomialofM. ProvidedKbealgebraically
closed, the coefficients of this polynomial are the elementary symmetric functions in the
∗[email protected]
†[email protected]
‡[email protected]
1
eigenvalues of M. Below for the sake of simplicity we assume K to be the field C of complex
numbers.
In[NT,PS,GPS,IOPS,IOP1]theCayley-Hamiltontheoremwassuccessively generalized
to the case of the quantum matrix algebras of the GL(m) type. Any such an algebra is
constructed by means of R-matrix representations of Hecke algebras of the GL(n) type.
Intheworks [KT]theCayley-Hamilton identity was established formatrixsuperalgebras.
The aim of the present paper consists in a further generalization of the results of [KT] to
the case of quantum matrix superalgebras.
Letusformulateourmainresult—theCayley-Hamiltontheoremforthequantummatrix
algebras of the GL(m|n) type. All the definitions and notations are explained in the main
text below.
Theorem. Let M = kMjkN be the matrix of the generators of the quantum matrix
i i,j=1
algebra M(R,F) defined by a pair of compatible, strictly skew-invertible R-matrices R and
F, where R is a GL(m|n) type R-matrix (see sections 3.1 and 3.2). Then in M(R,F) the
following matrix identity holds
n+m
Mm+n−iC ≡ 0, (1.2)
i
i=0
X
where M¯i is the i-th power of the quantum matrix (see definition (3.27)) and the coefficients
C are linear combinations of the Schur functions (see definition (3.21)).
i
The Schur functions entering the expression of C are homogeneous polynomials of the
i
(mn+i) order in the generators Mj. Therefore, any matrix element in the left hand side of
i
(1.2) is a homogeneous polynomial of the (mn+m+n) order in Mj.
i
The expression of the coefficient C in terms of the Schur functions can be graphically
i
presented as follows
nnodes
...
C = min{i,m} (−1)kq(2k−i) mz.. }.|.. { ... )k (1.3)
i  .
k=maxX{0,(i−n)}
...

(i−k)nodes
where each Young diagram stands for the corresponding|S{czhu}r function.
If the matrices F and R are the transpositions in the superspace (m|n), then the cor-
responding algebra M(R,F) is a matrix superalgebra. In this case identity (1.2) coincides
with the invariant Cayley-Hamilton relations obtained in [KT]. Note that in the supersym-
metric case the linear size N of the matrix M is connected with the parameters m and n by
the equation N = m+n. As was shown in [G], in general this relation does not take place
in the quantum case. Our proof of identity (1.2) does not rely on such like connections.
2
The structure of the work is as follows. Section 2 contains definitions and necessary facts
on the Hecke algebras. In section 3 we define the quantum matrix algebra of the GL(m|n)
type. Ibidem the notions of the characteristic subalgebra and of the matrix power space are
introduced. The last section is devoted to the proof of the Cayley-Hamilton identity (1.2).
The authors express their appreciation to Alexei Davydov, Dimitry Leites, Alexander
Molev and Hovhannes Khudaverdian. Special gratitude to our permanent co-authors Alexei
Isaev and Oleg Ogievetsky for numerous fruitful discussions and remarks. The work of the
second author (P.N.P.) was partially supported by the RFBR grant No. 03-01-00781.
2 Some facts on Hecke algebras
In this section we give a summary of facts on the structure of the Hecke algebras of A
n−1
series. For their proofs the reader is referred to the works [DJ, M]. Detailed consideration of
the material and the list of references on the theme can also be found in the reviews [OP, R].
When considering the quantum matrix algebras we shall use this section as a base of our
technical tools.
2.1 Young diagrams and tableaux
First of all we recall some definitions from the partition theory. Mainly we shall use the
terminology and notations of the book [Mac].
A partition λ of a positive integer n (notation: λ ⊢ n) is a non-increasing sequence of
non-negative integers λ such that their sum is equal to n:
i
λ = (λ ,λ ,...), λ ≥ λ : |λ| ≡ λ = n.
1 2 i i+1 i
X
The number |λ| = n is called the weight of the partition λ. A convenient graphical image for
a partition is a Young diagram (see [Mac]) and below we shall identify any partition with
the corresponding Young diagram.
With each node of a Young diagram located at the intersection of the s-th column and
the r-th row we associate a number c = s−r which will be called the content of the node.
Define an inclusion relation ⊂ on the set of all the Young diagrams by stating that a
diagram µ includes a diagram λ (symbolically λ ⊂ µ) if after the superposition the diagram
λ is entirely contained inside of µ. In other words, λ ⊂ µ ⇔ λ ≤ µ i = 1,2,...
i i
Given a Young diagram λ ⊢ n one can construct n! Young tableaux by writing into the
nodes of the diagram all integers from 1 to n in an arbitrary order. The tableau will be
noted by the symbol
λ
, α = 1,2,...,n!, (2.1)
α
where the index α marks the diffnereont tableaux. The diagram λ will be called the form of a
λ
tableau .
α
n o
3
The set of all tableaux associated with a diagram λ ⊢ n can be naturally supplied with
an action of the n-th order symmetric group S . By definition, an element π ∈ S acts
n n
λ
on a tableaux by changing any number i for π(i) in all the nodes of the tableau. The
α
n o
λ
resulting tableau will be denoted as .
π(α)
In the set of all the Young tableaunx of tohe formλ we shall distinguish a subset of standard
tableaux which satisfy an additionalrequirement: the numbers written in a standard tableau
increases in any column from top to bottom and in any raw from left to right. From now
on we shall use only standard Young tableaux and retain notation (2.1) for them. However,
the index α will vary from 1 to an integer d which is equal to the total number of all the
λ
standard Young tableaux of the same form λ. Explicit expressions for d are given, for
λ
example, in [Mac].
Though the subset of the standard tableaux is not closed with respect to the action of
the symmetric group, it nevertheless is a cyclic set under such an action since an arbitrary
standard Young tableau of the form λ ⊢ n can be obtained from any other standard tableau
of the same form by the action of some element π from the group S that is
n
λ λ λ
∀α,β ∃π ∈ S : 7→ ≡ .
n α β π(α)
n o n o n o
The relation ⊂ defined above for the Young diagrams can be extended to the set of the
standard tableaux. Namely, we say that a tableau includes another one
λ µ
⊂ ,
α β
n o n o
µ
if λ ⊂ µ and the numbers from 1 to |λ| occupy the same nodes in the tableau as they
β
do in λ . n o
α
n o
2.2 Hecke algebras: definition and the basis of matrix units
The Hecke algebra H (q) of A series is a unital associative C-algebra generated by the set
n n−1
of elements {σ }n−1 subject to the relations
i i=1
σ σ σ = σ σ σ , k = 1,...,n−2 , (2.2)
k k+1 k k+1 k k+1
σ σ = σ σ , if |k −j| ≥ 2 , (2.3)
k j j k
σ 2 = 1+(q −q−1)σ , k = 1,...,n−1 , (2.4)
k k
where 1 is the unit element. The last of the above defining relations depends on a parameter
q ∈ C\{0} and is referred to as the Hecke condition.
H (1) is identical with the group algebra C[S ] of the symmetric group S . In case
n n n
qk −q−k
k := 6= 0, for all k = 2,3,...,n, (2.5)
q q −q−1
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the algebra H (q) is semisimple and is isomorphic to C[S ]. In what follows we shall suppose
n n
condition (2.5) to be fulfilled for all k ∈ N \ {0}. Under such an assumption the algebra
H (q) is isomorphic to the direct sum of matrix algebras
n
H (q) ∼= Mat (C). (2.6)
n dλ
λ⊢n
M
Here Mat (C) stands for an associative C-algebra generated by m2 matrix units E 1 ≤
m ab
a,b ≤ m which satisfy the following multiplication law
E E = δ E .
ab cd bc ad
Due to isomorphism (2.6) one can construct the set of elements Eλ ∈ H (q), ∀λ ⊢ n,
αβ n
1 ≤ α,β ≤ d , which are the images of matrix units generating the term Mat in the direct
λ dλ
sum (2.6)
Eλ Eµ = δλµδ Eλ . (2.7)
αβ γτ βγ ατ
The diagonal matrix units Eλ ≡ Eλ are the primitive idempotents of the algebra H (q)
αα α n
and the set of all Eλ forms a linear basis in H (q).
αβ n
Obviously, the matrix units Eλ canbe realized inmany different ways. The construction
αβ
presented in [M, R] (see also [OP]) satisfies a number of useful additional relations. It is this
type of the basis of matrix units which we shall use below. Now we shortly outline its main
properties.
Firstly, in this basis there are known remarkably simple expressions for the elements
which transform different diagonal matrix units into each other. In order to describe these
elements we need a few new definitions.
With each generator σ of the algebra H (q) we associate the element
i n
q−x
σ (x) := σ + 1, 1 ≤ i ≤ n−1. (2.8)
i i
x
q
Here x is an arbitrary non-negative integer, the symbol x being defined in (2.5). The
q
elements σ (x) obey the relations
i
σ (x)σ (x+y)σ (y) = σ (y)σ (x+y)σ (x), 1 ≤ i ≤ n−2, (2.9)
i i+1 i i+1 i i+1
(x+1) (x−1)
σ (x)σ (−x) = q q 1, 1 ≤ i ≤ n−1. (2.10)
i i
x2
q
λ
Consider an arbitrary standard Young tableau corresponding to a partition λ ⊢ n.
α
For each integer 1 ≤ k ≤ n−1 we define the numbneroℓ {λ} to be equal to the difference of
k α
the contents of nodes containing the numbers k and (k +1)
ℓ {λ} := c(k)−c(k +1). (2.11)
k α
5
λ λ
Let a tableau (not obligatory standard) differs from the initial tableau only in the
β α
transposition onf tohe nodes with the numbers k and (k +1) n o
β = π (α), 1 ≤ k ≤ n−1, (2.12)
k
λ
π ∈ S being the transposition of k and (k+1). If the tableau is standard then the
k n πk(α)
following relations hold n o
σ (ℓ {λ})Eλ = Eλ σ (−ℓ {λ}), (2.13)
k k α α πk(α) k k α
Eλσ (ℓ {λ}) = σ (−ℓ {λ})Eλ . (2.14)
α k k α k k α πk(α)
Now, take into account the cyclic property of the set of standard Young tableaux under
the action of the symmetric group and the fact that an arbitrary permutation can be (non-
uniquely) expanded into a product of transpositions. This allows one to connect any pair
Eλ and Eλ by a chain of transformations similar to (2.13), (2.14). Moreover, relations (2.9),
α β
(2.10) ensure the compatibility of all possible formulae expressing dependencies between Eλ
α
and Eλ.
β
λ
If the tableau is non-standard then
πk(α)
n o
σ (ℓ {λ})Eλ = Eλσ (ℓ {λ}) = 0. (2.15)
k k α α α k k α
The expressions for non-diagonal matrix units follow from (2.13), (2.14). Assuming both
λ λ
the tableaux and to be standard, one gets
α πk(α)
n o n o
Eλ := ω(ℓ {λ})Eλσ (ℓ {λ}), (2.16)
απk(α) k α α k k α
Eλ := ω(ℓ {λ})σ (ℓ {λ})Eλ, (2.17)
πk(α)α k α k k α α
where the normalizing coefficient ω(ℓ) satisfies the relation
ℓ2
ω(ℓ)ω(−ℓ) = q . (2.18)
(ℓ+1) (ℓ−1)
q q
λ
In particular, one can set ω(ℓ) := ℓ /(ℓ + 1). Note that in case the tableau is a
q q πk(α)
standard one, we have ℓ {λ} =6 ±1. Therefore, the above expression for the nnormaolizing
k α
coefficient is always correctly defined.
To formulate the second property of the matrix units which will be used below let us
consider the chain of Hecke algebra embeddings
H (q) ֒→ ... ֒→ H (q) ֒→ H (q) ֒→ ..., (2.19)
2 k k+1
which are defined on generators by the formula
H (q) ∋ σ 7→ σ ∈ H (q), i = 1,...,k −1. (2.20)
k i i+1 k+1
6
In each subalgebra H (q) one can choose the basis of the matrix units Eλ in such a way
k αβ
that the diagonal matrix units Eλ ∈ H (q) can be decomposed into the sum of diagonal
α k
matrix units belonging to any enveloping algebra H (q) ⊃ H (q)
m k
Eλ⊢k = Eµ⊢m, m ≥ k. (2.21)
α β
{λα}X⊂{µβ}
µ
Here the summation goes over all the standard Young tableaux , µ ⊢ m, containing the
β
standard tableau λ . n o
α
n o
3 Quantum matrix algebra M(R,F)
3.1 R-matrix representations of the Hecke algebra
Let V be a finite dimensional vector space over the field of complex numbers, dimV = N.
With any element X ∈ End(V⊗p), p = 1,2,..., we associate a sequence of endomorphisms
X ∈ End(V⊗k), k = p,p+1,..., according to the rule
i
X = Idi−1 ⊗X ⊗Idk−p−i+1, 1 ≤ i ≤ k −p+1, (3.1)
i V V
where Id is the identical automorphism of V.
V
An operator R ∈ Aut(V⊗2) satisfying the Yang-Baxter equation
R R R = R R R , (3.2)
1 2 1 2 1 2
will be called the R-matrix. One can easily see that any R-matrix obeying the condition
(R−qId )(R+q−1Id ) = 0, q ∈ C\{0} (3.3)
V⊗2 V⊗2
generates representations ρ for the series of the Hecke algebras H (q), k = 2,3,...
R k
ρ : H (q) → End(V⊗k) σ 7→ ρ (σ ) = R 1 ≤ i ≤ k −1, (3.4)
R k i R i i
where σ are the generators of H (q) (2.2) – (2.4). Such like R-matrices will be called the
i k
Hecke type (or simply Hecke) R-matrices and the corresponding representations ρ — the
R
R-matrix ones.
Let R be a Hecke R-matrix. Suppose that the corresponding representations ρ :
R
H (q) → End(V⊗k) are faithful for all 2 ≤ k < (m + 1)(n + 1) while the representation
k
ρ : H (q) → End(V⊗(m+1)(n+1)) possesses a kernel, generated by any matrix unit
R (m+1)(n+1)
corresponding to the rectangular Young diagram ((n+1)(m+1)) that is
ρ (E((n+1)(m+1))) = 0
R α
(3.5)
ρ (Eµ) 6= 0, for any µ ⊢ (m+1)(n+1), µ 6= ((n+1)(m+1)).
R α
Such an R-matrix will be referred to as an R-matrix of the GL(m|n) type.
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3.2 R-trace and pairs of compatible R-matrices
Let {v }N be a fixed basis in the space V. In the basis {v ⊗ v }N an operator X ∈
i i=1 i j i,j=1
End(V⊗2) is defined by its matrix Xkl: X(v ⊗v ) := N Xklv ⊗v .
ij i j k,l=1 ij k l
The operator X ∈ End(V⊗2) will be called skew-inPvertible if there exists a matrix ΨXkl
ij
such that
N
XkbΨXal = δlδk. (3.6)
ia bj i j
a,b=1
X
One can easily see that the property of skew-invertibility is invariant with respect to the
change of the basis {v } and, therefore, to the matrix ΨXkl there corresponds an operator
i ij
ΨX ∈ End(V⊗2). Introduce a notation
DX := Tr ΨX ∈ End(V), (3.7)
(2)
where the symbol Tr means taking the trace over the second space in the product V ⊗V.
(2)
A skew-invertible operator X will be called strictly skew-invertible in case the operator DX
is invertible.
Consider the set of N × N matrices Mat (W), their entries being vectors of a linear
N
space W over C. The set Mat (W) is endowed with the structure of a linear space over C
N
in the obvious way.
Let R be a skew-invertible R-matrix. The linear map
Tr : Mat (W) → W,
R N
defined by
N
Tr (M) = DRjMi M ∈ Mat (W), (3.8)
R i j N
i,j=1
X
will be called the calculation of R-trace.
Remark 1 If as an R-matrix one chooses the transposition operator in the superspace of
(m|n) type (see definition (3.32)), then the operation Tr coincides with the calculation of
R
the supertrace.
An ordered pair {R,F} of the two R-matrices R and F will be called a compatible pair
if the following relations are satisfied
R F F = F F R
1 2 1 2 1 2 (3.9)
R F F = F F R .
2 1 2 1 2 1
Now we list some properties of the compatible pairs of R-matrices.
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1. If R is a skew-invertible R-matrix and {R,F} is a compatible pair, then for all M ∈
Mat (W) the following relation holds
N
Tr (F±1M F∓1) = Id Tr M where M = M ⊗Id . (3.10)
R(2) 1 1 1 V R 1 V
2. In case R is skew-invertible, then
[R ,DRDR] = 0. (3.11)
1 1 2
A direct consequence of this formula is a cyclic property of the R-trace
Tr (R U) = Tr (UR ), U ∈ Mat (W)⊗2. (3.12)
R(12) 1 R(12) 1 N
3. If {R,F} is a compatible pair, then R := F−1R−1F is an R-matrix and the pair
f
{R ,F} is compatible.
f
4. Let {R,F} be a compatible pair, R-matrix R is skew-invertible and matrices R and
f
F are strictly skew-invertible. Then the map φ : Mat (W) → Mat (W) defined by
N N
φ(M) := Tr (F M F−1R ), M ∈ Mat (W), (3.13)
R(2) 1 1 1 1 N
is invertible. An explicit form of its inverse φ−1 is as follows (see [OP2])
φ−1(M) = Tr (F−1M R−1F ). (3.14)
Rf (2) 1 1 1 1
3.3 Quantum matrix algebras: definition
Consider a linear space Mat (W) and define a series of linear mappings Mat (W)⊗k →
N N
Mat (W)⊗(k+1), k ≥ 1, by the following recurrent relations
N
M := M, M 7→ M := F M F−1, M ∈ Mat (W)⊗k, (3.15)
1 k k+1 k k k k N
where M is an arbitrary matrix from Mat (W) and F is a fixed element of Aut(V ⊗V).
N
As the linear space W we choose an associative algebra A over C freely generated by the
unit element and N2 generators Mj
i
A = ChMji 1 ≤ i,j ≤ N.
i
Definition 2 Let {R,F} be a compatible pair of strictly skew-invertible R-matrices, the R-
matrix R = F−1R−1F being skew-invertible. By definition, the quantum matrix algebra
f
M(R,F) is a quotient of the algebra A over a two-sided ideal generated by entries of the
following matrix relation
R M M −M M R = 0, (3.16)
1 1 2 1 2 1
where M = kMjk ∈ Mat (A) and the matrices M are constructed according to (3.15) with
i N k
the R-matrix F.
9
Remark 3 The definition of the quantum matrix algebra presented here differs from that
given in [IOP1]. Nevertheless, under the additional assumptions on R and F made above,
bothdefinitionsareequivalent. Aswasshownin[IOP2], thematrixM = (DF)−1MDR obeys
relations imposed on the generators of the quantum matrix algebra in the work [IOP1].
f
Lemma 4 Thematrix M composedofthe generators ofthe quantum matrix algebraM(R,F)
satisfies the relations
R M M = M M R , (3.17)
k k k+1 k k+1 k
where the matrices M , M are defined by rules (3.15) with the R-matrix F.
k k+1
The proof of the lemma is presented in [IOP1].
3.4 Characteristic subalgebra and Schur functions
Further inthis sectionwe consider thequantum matrixalgebrasM(R,F)defined by aHecke
R-matrix R. We shall refer to them as the matrix algebras of the Hecke type.
Let a subspace Char(R,F) ⊂ M(R,F) be a linear span of the unit and of the following
elements
y(x(k)) = Tr (M ...M ρ (x(k))) k = 1,2,..., (3.18)
R(1...k) 1 k R
where x(k) runs over all elements of H (q). The symbol Tr stands for the calculation
k R(1...k)
of the R-trace over the spaces from the first to the k-th ones.
Proposition 5 Let M(R,F) be a quantum matrix algebra of the Hecke type. The subspace
Char(R,F) is a commutative subalgebra of M(R,F). The subalgebra Char(R,F) will be
called the characteristic subalgebra.
The detailed proof is given in [IOP1]. It is based, in particular, on the following technical
result.
Lemma 6 Consider an arbitrary element x(k) of the algebra H (q). Let x(k)↑i denotes the
k
image of x(k) in the algebra H (q) under the embedding H (q) ֒→ H (q) defined by (2.20).
k+i k k+i
Let {R,F} be a compatible pair of R-matrices, the Hecke R-matrix R being skew-invertible.
Then the following relation holds true
Tr (M ...M ρ (x(k)↑i)) = Id y(x(k)). (3.19)
R(i+1,...,i+k) i+1 i+k R V⊗i
Formula (3.19) will be repeatedly used in the sequel. It is immediate consequence of the
property (3.10) of the compatible pairs {R,F} and its proof is also presented in [IOP1].
Take into consideration two sets {p (M)} and {s (M)} of elements of the characteristic
k λ
subalgebra. The elements of these sets will be referred to as the power sums and the Schur
functions respectively. For an arbitrary integer k ≥ 1 the power sums are defined by
p (M) := Tr (M ...M R ...R ). (3.20)
k R(1...k) 1 k k−1 1
10