A simple algorithm for extending the identities for quantum minors to the multiparametric case

0.1 MB·English

by  Zoran Škoda

A simple algorithm for extending the
identities for quantum minors to the
multiparametric case
8 Zoran Sˇkoda, [e-mail]
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0 Abstract. For any homogeneous identity between q-minors, we provide an
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identity between P,Q-minors.

n
a
1. ([1, 2, 8]) Suppose we are given n×n matrices P = (p ) and Q = (q )
J ij ij
1 with invertible entries in the ground field k, for which there exist q such that
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p q = q2,q = q−1, i < j, and q = p , for all i. (1)
] ij ij ij ji ii ii
A
Q Define an associative algebra M(P,Q;k,n) := khTi,i,j = 1,...,ni/I,
j
. where I is the ideal spanned by the relations
h
t
a TkTk = q TkTk, i < j
m i j ij j i
TkTl = p TlTk, k < l
[ i i kl i i
q TkTl = p TlTk, i < j,k < l
ij j i kl i j
1 TkTl −q q−1TlTk = (q −p−1)TkTl, i < j,k < l
v i j ij kl j i ij ij j i
5
6 M := M(P,Q;k,n) is a bialgebra with respect to the ”matrix” comulti-
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plication which is the unique algebra homomorphism ∆ : M → M ⊗ M
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. extending the formulas ∆Ti = Ti ⊗Tk with counit ǫTi = δi (Kronecker
1 j P k j j j
0 delta). The bialgebra M is called the multiparametric quantum linear
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group.
0
: Our conventions differ a bit from the cited references: we treatp-s and q-s
v
i symmetrically in the sense that if we interchange rows and columns of matrix
X
T and if we simultaneously interchange P and Q, we obtain an isomorphic
r
a algebra. If P = Q and q = q for i < j and q = q−1 for i > j, then
ij ij
M = M (k) (1-parametric quantized matrix bialgebra). In this paper, we
q
will denote by ti the generators for 1-parametric case.
j
2. (Labels.) It is convenient to consider that the row and column labels
belongtosometotallyorderedset oflabels, notnecessarily theset {1,...,n}.
The main reason is that one often needs to treat some subsets of the set of
labels and the corresponding submatrices of the matrix T = (Ti).
j
3. The quantum Q-space O(kn), quantum P-space O(kn), q- normalized Q-
Q P
spaceS (q,Q), dualq-normalizedQ-spaceS (q,Q),rightP-exterioralgebra
r l
1
Λ , and left Q-exterior algebra Λ , are the algebras defined by generators
P Q
and relations as follows:
O(kn) := khxi,i = 1,...,ni/hxixj −q xjxi,i < ji
Q ij
O(kn) := khy ,i = 1,...,ni/hy y −p y y ,i < jii
P i i j ij j i
S (q,Q) := khr ,...,r i/hr r −qq−1r r , i < ji
r 1 n i j ij j i (2)
S (q,Q) := khl1,...,lni/hlilj −q q−1ljli, i < ji
l ij
Λ := khe1,...,eni/heiej +p−1ejei, i < j, (ei)2i
P ij
Λ := khe ,...,e i/hf f +q−1f f , i < j, (f )2i
Q 1 n i j ij j i i
We note after Manin (e.g. [7, 8]) the simple fact that O(kn),Λ are right
Q Q
and O(kn),Λ are left comodule algebras over M via coactions ρ,ρ ,ρ′,ρ′
P P Λ Λ
which are the unique coactions which are algebra maps and extend formulas
ρ(xi) = xj ⊗Ti, ρ (ei) = ej ⊗Ti, ρ′(y ) = Ti ⊗y and ρ′ (f ) =
Pj j Λ Pj j j Pi j i Λ j
Ti ⊗f .
Pi j i
4. Tuples of labels will be called multilabels. Let L = (l ,...,l ), K =
1 r
(k ,...,k ). The concatenation will be denoted by juxtaposition: LK =
1 s
(l ,...,l ,k ,...,k ). Usually the multilabels will be (ascendingly) ordered
1 r 1 s
to start with and Lˆ denotes the ordered complement of a submultilabel L
(usually in {1,...,n}). By placing the multilabel within the colons, we will
denote its ascendingly ordered version. For example, if K and L are ordered,
then KL is not necessarily ordered, because some labels in L may be smaller
than some labels in K. However : KL : is the multilabel obtained from KL
by permuting the labels until they are ascendingly ordered. We identify the
notation for a single label j and the multilabel (j), and in this vein ˆj = (j)
c
is the same as multilabel (1,2,...,j −1,j +1,...,n).
We use obvious exponent notation: rJ := rj1···rjn and alike.
5. Algebras (2) satisfy the obvious normal basis (“PBW type”) theorems:
fix an order on generators, then the monomials ordered compatibly with this
order form a vector space basis (“PBW basis”), for example xa1xa2 ···xan in
1 2 n
the case of O(kn); however, no higher exponents than 1 appear in the bases
Q
for Λ and Λ because e2 = 0.
P Q i
Passing from arbitrary words in generators to the basis elements clearly
reduces to reordering the generators, and accumulating the proportionality
constants. Let us introduce the notation for those rearrangements factors
which will be needed below. The following are the defining properties of
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coefficient functions ǫ ,ǫ ,ζ ,ζ from sets of multilabels to k:
P Q r l
e := ǫ (J)e , r := ζ (J)r ,
J P :J: J r :J:
f := ǫ (J)f , l := ζ (J)l ,
J Q :J: J l :J:
where ǫ (J),ǫ (J) are defined only when J has no repeted labels inside, but
P Q
ζ and ζ are defined even for multilabels with repetition. The following is
r l
obvious:
ζr(k1,...,ks) = Y q−1qkikj
i<j,ki>kj
ζl(k1,...,ks) = Y qqk−i1kj = ζr−1(k1,...,ks)
i<j,ki>kj
Clearly, if J is ascendingly ordered multilabel with out repetitions, then
ǫ (σJ) = (−q)−l(σ)ζ (σJ) = (−p ),
P r Qk<l,σ(k)>σ(l) jσ(k)jσ(l)
ǫ (σJ) = (−q)l(σ)ζ (σJ) = (−q ),
Q l Qk<l,σ(k)>σ(l) jσ(k)jσ(l)
(recall that (q )−1 = q ).
ij ji
6. (Gradings.) Let I be the set of labels of generators of S = S (q,Q) (they
r r
label rows!). Let J be the set of labels of generators of S = S (q,Q) (they
l l
label columns!). Both sets are bijective to {1,...,n}. Thus the free Abelian
group Z[I] is isomorphic to Zn (and could be naturally identified with the
weight lattice for SL ). Now we assign Z[I]−Z[J] bigrading to algebras
n
S (q,Q),S (q,Q)andM (k). Ifi ,...,i ,j ,...,j aretheelementsofI and
l r q 1 n 1 n
J, thenabidegree isaformalsumoftheforma i +...+a i +b j +...b j ,
1 1 n n 1 1 n n
e.g. −i +2i +j , and we may separate the I and J grading with comma
3 4 3
for clarity, e.g. (−i + 2i , j ) ≡ −i + 2i + j . We assign the bidegree
3 4 3 3 4 3
(−i,0) to the generator r of S (notice the negative sign!) ZhI∗i-degree to
i l
zero and similarly the dual prescription (0,−j) to lj of S . We also assign the
r
bidegree (+i,+j) to each generator ti of the 1-parametric algebra M (k).
j q
The defining ideals are bihomogeneous hence we extend this prescription
multiplicatively to a bigrading on the algebras S (q,Q), S (q,Q) and M (k).
l r q
7. Notation. Consider the tensor product of bigraded algebras
M := S (q,Q)⊗M (k)⊗S (q,Q). (3)
l q r
c
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8. Lemma. Any (bi)homogenous element in a tensor product of (bi)graded
algebras is a sum of tensor products of (bi)homogenous elements in tensor
factors. If one chooses a set of homogeneous generators in each tensor factor
than the summands can be chosen as tensor products of monomials in those
generators.
9. Lemma. lj ⊗ ti ⊗ r generate the subalgebra of all elements of bidegree
j i
(0,0) in M.
c
Proof. If J = (j ,...,j ) is some ordered s-tuple of labels (repetitions
1 s
of labels possible) denote lJ = lj1···ljs and we adopt obvious extension of
this multilabel notation for r-s and t-s. It is clear that any tensor product
of monomials which is of bidegree (0,0) is of the form lτI ⊗tJ ⊗r where σ
I σJ
and τ are permutations on |I| = |J| letters. Then by lemma 8 it is enough
to show that any such tensor product lτI ⊗tJ ⊗r may be written as sum
I σJ
of products of the form lj ⊗ ti ⊗ r . But lJ and lτJ are proportional in S ,
j i l
and similarly r and r are proportional in S . Hence, up to accounting
I σI r
for a scalar factor, we may assume that σ and τ are trivial. But then the
expression is manifestly the product of elements of the required form.
10. Theorem. Suppose (1) holds. Let ti and Ti denote the generators of
j j
q-deformed and P,Q-deformed quantum matrix algebras respectively. Then
(i) the rule
ι : Ti 7→ lj ⊗ti ⊗r (4)
q,Q j j i
extends to a unique algebra homomorphism ι : M(P,Q;k) → M.
q,Q
c
(ii) This homomorphism is injective and its image is the subalgebra of all
elements of (0,0)-bidegree in M.
(iii) Similarly, rescaling ejcby lj produces the relations in Λ from the
P
relations in Λ .
q
Proof. (i) One needs to show that ι sends the ideal of relations (in free
q,Q
algebra on T-s) to zero. For example, omitting the tensor product notation,
we calculate, for i < j and k < l,
ι(q TiTj) = q lltir ·lktjr
kl l k kl l i k j
= q (lllk)(titj)(r r )
kl l k i j
= q (qq−1lkll)(tjti)(qq−1r r )
kl kl k l ij j i
= q2q−1lklltjtir r
ij k l j i
= ι(p TjTi).
ij k l
The other cases are left to the reader.
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(ii) Forinjectivity onecanuse e.g. thenormalbasis theoremforthequan-
tum matrix algebras: monomials of the form (T1)α11(T1)α12 ···(T1)αnn make
1 1 1
a basis of M(P,Q;k). It is clear that the images are linearly independent
because the middle tensor factors of the images are such (by the normal basis
theorem for 1-parametric case) and the other two tensor factors are nonzero.
The description of the image of ι follows from 9.
q,Q
(iii) Easy.
11. Remarks. This isomorphism will be very useful for our purpose. Essen-
tially this proposition is a mechanism essentially equivalent to the cocycle-
twisting of [1]. Namely, in both approaches, the difference between the al-
gebra relations for ti-s and for Ti-s is reflected in rescaling factors for each
j j
monomial, which may be expressed in terms of a bicharacter and depends
only on the bidegree of the monomial.
However, thereisanimportant difference inusing our isomorphism ιfrom
theusageoftwisting in[1]. Namely, itisshown in[1]thatthecorrespondence
ti 7→ Ti which they use, extends multiplicatively on monomials to an isomor-
j j
phisms of vector spaces, and even of coalgebras; whereas it does not respect
the algebra structure. On the other hand, our map ι is a monomorphism of
algebras, as stated above, but it does not respect the coalgebra structure!
12. (Quantum minors.) If J and K are ascendingly ordered row and col-
umn multilabels of the same cardinality m without repetitions, then the
corresponding quantum minor DK is the element of M satisfying
L
DLK = X ǫP(σK)Tlk1σ1···Tlkmσm = X ǫP(σK)ǫ−P1(τL)Tlkτσ11···Tlkτσmm
σ∈Σ(m) σ∈Σ(m)
DLK = X ǫQ(σL)Tlkσ11 ···Tklσmm = X ǫQ(σL)ǫ−Q1(τK)Tlkστ11···Tlkστmm
σ∈Σ(m) σ∈Σ(m)
where τ ∈ Σ(m) is a fixed permutation.
13. Proposition. Let ι be the monomorphism (4). Then ι(DK) = lJdKr
J J K
where dK denotes the quantum minor in 1-parametric case.
J
Proof.
ι(DK) = (−q)l(σ)ζ (σK)ι(Tkσ(1))···ι(Tkσ(m))
J Pσ∈Σ(m) r j1 jm
= (−q)l(σ)lj1lj2···ljm ⊗tkσ(1)···tkσ(m) ⊗ζ (σK)r ···r
Pσ∈Σ(m) j1 jm r k1 km
= (−q)l(σ)lj1lj2···ljm ⊗tkσ(1)···tkσ(m) ⊗r ···r
Pσ∈Σ(m) j1 jm kσ(1) kσ(m)
= lJDKr .
J K
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This proposition reminds but is different to the statement of Lemma 5
in [1] which asserts that the twisting considered as an identity map but
changing its algebra structure, interchanges the quantum determinants. No-
tice that the DK and dK on the two sides are given by different formulas in
L L
terms of generators Tk and tk respectively.
l l
14. Nowoneneeds tosee what happens when oneconsiders monomialsinD-
s,forexampleι(DKDM). Bythesame,method,onegets(lLlM)(dKdM)(r r ).
L N L N K M
One knows that the relations in M are homogeneous in the sense that the
total row multilabel and column multilabel are the same up to the ordering.
Every relation is a sum of homogeneous. Now if we take different monomials
d∗ = dK1···dKm in usual quantum minors then in order to make them
Q ∗ L1 Lm
manifestly in the image of ι on some monomial in multiparametric quantum
minors, we need to homogenize expression by multiplying it by lS and r
V
where S and V are the ascendingly ordered column and row total multilabel
of d∗, that is S =: K ···K : and V =: L ···L :. Thus the S and V are
Q ∗ 1 m 1 m
the same for all monomials in the identity (this is more or less the definition
of a homogeneous identity). Then we reorder the multilabels in l and in r
separately to get the same ordering, but this involves introducing inverse of
ζ and ζ corresponding to the ordering on the column and row multilabels
l r
seperately. For a homogeneous identity the multiplier lS and r will be the
V
same, however the reordering factors will be clearly different. Thus we get, in
terms of D-s the same identity up to different homogeneous factors in front
of different monomials will be ζ−1(L L ···L )ζ−1(K K ···K ) where K -s
l 1 2 m r 1 2 m i
is the row multilabel of i-th row and L of i-th column.
i
Theorem. This procedure induces the 1-1 correspondence between the
quantum minor identities for the 1-parametric and the minor identities for
multiparametric minors.
Notice that despite the fact that ι is not an algebra homomorphism,
essentially ι and extracting the proprotionality constants from reorderings of
r-s and l-s do the job. If one would use the original twisting of [1] one has
there a coalgebra map, hence it sends identities to something what are not,
hence it is not clear how to directly use it for the same result.
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