Bull. Kyushu Inst. Tech.
Math. Natur. Sci. No. 44, 1997, pp. 1-5
NONCOMMUTATIVE POISSON ALGEBRA STRUCTURES ON POSET ALGEBRAS AND MORPHISMS OF LEIBNIZ PAIRS
Fujio KuBo
(Received November 25, 1996)
A non-commutative Poisson algebras A is an algebra over the field C of the complex numbers (we shall take the field C for simplicity) having an associative algebra product, being denoted by xy the associative algebra product of x,y in A and the Lie algebra bracket {-, -} connected with the Leibniz law: {xy, z} == {x, z}y + x{y, z}
for x,y,zEA. For any associative algebra A one can always construct a noncommutative Poisson algebra structure in which we take a Lie product {--, -}
to be the scalar multiple of the ordinary associative commutator, i.e, by setting {x, y}:=1[x, y] =Z(xy-yx),ZGC. Let us denote such a noncommutative Poisson algebra by
Az,
which is called standard or said to have a standard structure. Note that Ai is of the ordinal associative commutator. We shall use the notion of "standard" in this way rather than that of one which we allow a direct sum of noncommutative Poisson algebras Ali,...,A,Z" to be called in [3, 4]. Every finite-dimensional noncommutative Poisson algebra A of a semisimple associative algebra structure must be decomposed into a direct sum of the standard ones Ai',...,Ai' of simple associative algebra
structures ([3, Theorem 7]). Every noncommutative Poisson algebra structure on the algebra M.(C) of all C-endomorphisms of a countable-dimensional C-vector space must be also standard ([4, Proposition]). The associative algebra structure of the noncommutative Poisson algebra structures on the Kac-Moody algebras L of affine type must be almost trivial, that is, L'L' =O for the derived Lie ideal L' of I,-, ([4,
Theorem]). These results were found while facing to our problem of what deforma- tions of Lie algebra structures or associative algebra structures on noncommutative Poisson algebras keep the Leibniz law satisfied, so that one obtains noncommutative Poisson algebras deformed a given one.
In this short article, we first describe the noncommutative Poisson algebra
structures on some associative subalgebras, so called "poset algebras" in the nÅ~n
full matrix algebra M.(C) ([2, g15]). For example, let I={1, 2, 3, 4} a poset on
which a partial ordering is given by 1 -cÅq 2 -Åq 3, 1 -Åq 4. The corresponding poset algebra
A in M4(C) is the associative subalgebra of M4(C) consisting the matrices pictured
below.
N 3
o * * o ****
o o * o o o o *
2
4
1
We can construct a noncommutative Poisson algebra structure on A which is not necessarily standard. Ai == Åqeii, e22, e33, ei2, e23, ei3År, Apa2 = Åqeii, e44, ei4År be the subalgebras spanned by the contents in the triangular brackets and their Lie products {-, -} are defined by {-, -} == A[-, -] on Ai and {-, -} == "[-, -] on A"2,
where eij is the matrix with 1 in the (i,j)th place and O elsewhere, and the bracket [-, -] is the ordinary associative commutator. Then A=Ai+A"2 and AlnA"2 -- Ceii. We will see that every poset algebra in M.(C) having a noncommutative Poisson algebra structure is expressed in the form A=Ai'+•-•+A,Z" such that for ilj, 4•`nA,2•j=O or Cekk for some k (Theorem 1).
A morphism of noncommutative Poisson algebras is simultaneously an associative algebra homomorphism and a Lie algebra homomorphism. Iff is such a morphism of AA into A' then the formula f(A[x, y]) -= [f(x),f(y)] implies (Z - 1)[f(x),f(y)] - O
for any x, yEA. Hence AZ is isomorphic to Ai as noncommutative Poisson algebras only when 2, == 1. There is a nice concept of a morphism to understand a relation between A2 and A', namely a morphism of Leibniz pairs. A Leibniz pair (A, L) consists of an associative algebra A and a Lie algebra L over the common field C of the complex numbers (A Leibniz pair is actually defined over a ring in [1]), connected by a Lie algebra morphism, so called a structural morphism pt: L.DercA, where denoting by DercA the Lie algebra of C-linear derivations of the associative algebra A into itself. When L is identical with A as a C-module, one obtains a noncommutative Poisson algebra A on which the Lie product {-, -} is defined by {x, y}:= pt(x)(y) for x, yGA. Since we do not deform the associative algebra structure here, let us denote by
A{ rm , -}
'
-- a noncommutative Poisson algebra on which a Lie product is taken to be
{-,-}. Note that A'= A[-,-] in our terminology. Then for any poset algebra
A = A{-,-} in M.(C), there exists a morphism (1., W) of the Leibniz pairs (4 A{-,-})
into (A, A[-,-]), as we will see in the section 2.
Noncommutative Poisson algebras and Leibniz pair morphisms 3
1 Noncommutative Poisson algebra structures on poset algebras When a subalgebra A of M.(C) contains all the diagonal matrices, hence, in particular, all eii,i= 1,...,n, A is spanned by those ei,• which it contains. For if an a in A has the fo rm a = ZAij eij then eii aej,• = Ai,- eij, so if Aij 7E O then eij E A. Such algebras A are called poset algebras ([2]). Now define a poset I == I(A) by setting i-Åqj if eij6A, and let I be the poset (without loops) determined by reducing I modulo the equivalence relation defined by the loops, i.e., by identifying to a single element any i andj for which both iKj andjÅqi (hence identifying any ii, i2,...,i, whenever ii -Åq i2 -cÅq •-• -cÅq i,.) Let X :E(A) be the nerve of I(A). This is a finite simplicial complex.
Let A{-,nt} be a poset algebra in M. .(C) having a Lie algebra structure denoted its product by {-, -} satisfying the Leibniz law: {xy, z} == {x, z}y+x{y, z} for
x,y,zEA, Then A must be standard when X(A) is connected and .X(A) has the
property that for any pair of 1-faces there exists a polygon which has these two 1-faces as its edges ([3, Theorem 2]). In particular, the algebra M.(C) and its subalgebra T.(C) of all upper triangular matrices are allowed to have only the standard structures, because the corresponding simplical complex to these algebras is just a (n-1)- simplex. The analysis of the noncommutative Poisson algebra structures on the poset algebra A in M.(C) is based on the formulas {e,i, ej,-}=O, {eii, ejk} = Zi(e,•k) where Ai(ejk) =O if i 7Ej and i 7E k, and
{eij, ekl} = - Ak(eij)eijekl + Ai(ekl)ekleiJ•,
when eij, ekiEA with i 7Ej, k iL l, where of course 2i(ejk)eC. Let i-ÅqA -Åq ••• -cÅqj, be a p-simplex in X = E(A). Since {eii, eij.} == {eii, eij,ej,j,•••ej,-,j.} = {eii, eij,} ej,j,'''eJ•.-,j.
=: Ai (eij ,) eij. for 1 s; k sl p, one gets Zi (eij.) = Ai (e ij ,) for 1 s k s! p. ITIence we can
assign a scalar Z(iÅqJ'iÅq•••Åqj,) to each p-simplex in X. If two simplecies X, Y in .2] have common 1-face then obviously A(X)=: Z(Y). Let .X =: io -t.' i, -Åq •-• -cÅq i, be a p-simplex in . and A(X) a subalgebra of A spanned by {ei,.i.IO f{j t= k sg p}. Then the equality {-, -} = A(X)[-, -] holds on A(X), so that one has a noncommutative Poisson subalgebra A(X)A(X) of A. Now let us introduce some terminology to describe such matters. Assume that X= 2:(A) is connected. By standard component .Ei of X we shall mean a simplical subcomplex maximal respect to the property that each simplex of dimension ;}i2 in Xi has a common 1-face with some other simplex in Xi. Then Z can be described as a sum, say a standard sum, E == Eiu•••uX, of the standard components Xi such that EinEj consists of one vertex or has no element for i#j. If a simplical complex Ei consists of the simplecies Xi,,...,Xi., then we
have a corresponding subalgebra A(E,):=A(2]i,)+-••+A(Xi.) on which the equality
{-, -} = A(.E],)[-, -] holds where A(Ei):= 2(X,,) == •••=2(Xi.), one can then write
A(.E,) = A(X,)A(X`) in our terminology. Therefore we have
standard sum of E. Then there exists a AiEC and a subalgebra Ai for each. isuch that A is expressed as a sum
A-Ati+•••+Air
qf standard noncommutative Poisson subalgebras Ai, where Ai and 2i can be taken to be A(Ei) and Z(Xi) respectively corresponding to a standard component X, of 2-. Here the Ai's sati6fv that ifi7Ej then AinAj is O or equal to a 1-dimensional .s'ubspace Cekk for some k.
2 Morphisms of Leibniz pairs (A,A{-,m}) and (A, A(-.H])
As in the introduction let A{-,-} be a noncommutative Poisson aigebra A whose
Lie product is given by {-,-}. Then we have the structural morphism
": A{-,-}.DercA defined by pa(x)(b):= {x, b} for x, bEA. For the typical standard
noncommutative Poisson algebra A[-,-]=Ai one also has the strucTural morphism pt,: A[-,-].DercA defined similarly by "o(x)(b):= [x, b] for x, bEA. A Leibniz pair
morphism of (A, A{-,-}) into (A, A{-,-]) is a pair (ip, W) consisting an associative
algebra homomorphism di:A-A and a Lie algebra homomorphism ut:A{m,-}- A[-,-] such that the diagram
A-(X)A dii tip
A!E9,E!(IEII4,(ut(X))A
commutes for any xEA.
Now let A be a poset algebra in M.(C) and `". =X(A) the simplical complex associated to A. Suppose that E is connected. As in the theorem 1 let E =XiU•--uE. be a standard sum of X and Ai -- .4 (2]i) a subalgebra corresponding to each standard component Xi of E and Ai:= Z(Xi). Then one has a noncommutative Poisson algebra Ai = Ait. Let us denote simply by Ii a poset I(A). We shall construct a Leibniz pair morphism (l., ut): (A, A{-,-}) --År (4 A[-,-]), so that the condition {x, y}
=:
[V(x),y] holds for x,yEA. Our Lie algebra homomorphism ut is defined as follow. For k, IEIi and k-Åq(l we put
lfr (eki) := Zieki•
