Down-Up Algebras#
AUTHORS:
Travis Scrimshaw (2023-4): initial version
- class sage.algebras.down_up_algebra.DownUpAlgebra(alpha, beta, gamma, base_ring)#
Bases:
CombinatorialFreeModule
The down-up algebra.
Let \(R\) be a commutative ring, and let \(\alpha, \beta, \gamma \in R\). The down-up algebra is the associative unital algebra \(DU(\alpha, \beta, \gamma)\) generated by \(d, u\) with relations
\[\begin{split}\begin{aligned} d^2u & = \alpha dud + \beta ud^2 + \gamma d, \\ du^2 & = \alpha udu + \beta u^2d + \gamma u. \end{aligned}\end{split}\]The down-up algebra has a PBW-type basis given by
\[\{ u^i (du)^j d^k \mid i,j,k \in \ZZ_{\geq 0} \}.\]This algebra originates in the study of posets. For a poset \(P\), we define operators acting on \(R[P]\) by
\[d(y) = \sum_x x \qquad\qquad u(y) = \sum_z z,\]where \(y\) covers \(x\) and \(z\) covers \(y\). For \(r\)-differential posets we have \(du - ud = r 1\), and thus it affords a representation of a
Weyl algebra
. This Weyl algebra is obtained as the quotient of \(DU(0, 1, 2r)\) by the ideal generated by \(du - ud - r\). For a \((q, r)\)-differential poset, we have the \(d\) and \(u\) operators satisfying\[\begin{split}\begin{aligned} d^2u & = q(q+1) dud - q^3 ud^2 + r d, \\ du^2 & = q(q+1) udu - q^3 u^2d + r u, \end{aligned}\end{split}\]or \(\alpha = q(q+1)\), \(\beta = -q^3\), and \(\gamma = r\). Specializing \(q = -1\) recovers the \(r\)-differential poset relation.
Two other noteworthy quotients are:
the \(q\)-Weyl algebra from \(DU(0, q^2, q+1)\) by the ideal generated by \(du - qud - 1\), and
the quantum plane \(R_q[d, u]\), where \(du = qud\), from \(DU(2q, -q^2, 0)\) by the ideal generated by \(du - qud\).
EXAMPLES:
We begin by constructing the down-up algebra and perform some basic computations:
sage: R.<a,b,g> = QQ[] sage: DU = algebras.DownUp(a, b, g) sage: d, u = DU.gens() sage: d * u (d*u) sage: u * d u*d sage: d^2 * u b*u*d^2 + a*(d*u)*d + g*d sage: d * u^2 b*u^2*d + a*u*(d*u) + g*u
We verify some examples of Proposition 3.5 in [BR1998], which states that the 0-th degree part is commutative:
sage: DU0 = [u^i * (d*u)^j * d^i for i,j in ....: cartesian_product([range(3), range(3)])] sage: all(x.degree() == 0 for x in DU0) True sage: all(x * y == y * x for x, y in cartesian_product([DU0, DU0])) True
We verify that \(DU(2, -1, \gamma)\) can be described as the universal enveloping algebra of the 3-dimensional Lie algebra spanned by \(x,y,z\) satisfying \(z = [x, y]\), \([x, z] = \gamma x\), and \([z, y] = \gamma y\):
sage: R.<g> = QQ[] sage: L = LieAlgebra(R, {('x','y'): {'z': 1}, ('x','z'): {'x': g}, ('z','y'): {'y': g}}, ....: names='x,y,z') sage: x, y, z = L.basis() sage: (L[x, y], L[x, z], L[z, y]) (z, g*x, g*y) sage: x, y, z = L.pbw_basis().gens() sage: x^2*y - 2*x*y*x + y*x^2 == g*x True sage: x*y^2 - 2*y*x*y + y^2*x == g*y True sage: DU = algebras.DownUp(2, -1, g) sage: d, u = DU.gens() sage: d^2*u - 2*d*u*d + u*d^2 == g*d True sage: d*u^2 - 2*u*d*u + u^2*d == g*u True
Young’s lattice is known to be a differential poset. Thus we can construct a representation of \(DU(0, 1, 2)\) on this poset (which gives a proof that Fomin’s
growth diagrams
are equivalent to edge local rules or shadow lines construction forRSK()
):sage: DU = algebras.DownUp(0, 1, 2) sage: d, u = DU.gens() sage: d^2*u == 0*d*u*d + 1*u*d*d + 2*d True sage: d*u^2 == 0*u*d*u + 1*u*u*d + 2*u True sage: YL = CombinatorialFreeModule(DU.base_ring(), Partitions()) sage: def d_action(la): ....: return YL.sum_of_monomials(la.remove_cell(*c) for c in la.removable_cells()) sage: def u_action(la): ....: return YL.sum_of_monomials(la.add_cell(*c) for c in la.addable_cells()) sage: D = YL.module_morphism(on_basis=d_action, codomain=YL) sage: U = YL.module_morphism(on_basis=u_action, codomain=YL) sage: for la in PartitionsInBox(5, 5): ....: b = YL.basis()[la] ....: assert (D*D*U)(b) == 0*(D*U*D)(b) + 1*(U*D*D)(b) + 2*D(b) ....: assert (D*U*U)(b) == 0*(U*D*U)(la) + 1*(U*U*D)(b) + 2*U(b) ....: assert (D*U)(b) == (U*D)(b) + b # the Weyl algebra relation
Todo
Implement the homogenized version.
REFERENCES:
- algebra_generators()#
Return the algebra generators of
self
.EXAMPLES:
sage: DU = algebras.DownUp(2, 3, 4) sage: dict(DU.algebra_generators()) {'d': d, 'u': u}
- degree_on_basis(m)#
Return the degree of the basis element indexed by
m
.EXAMPLES:
sage: R.<a,b,g> = QQ[] sage: DU = algebras.DownUp(a, b, g) sage: I = DU.indices() sage: DU.degree_on_basis(I([0, 3, 2])) -2 sage: DU.degree_on_basis(I([2, 3, 0])) 2 sage: DU.degree_on_basis(I([2, 0, 3])) -1 sage: DU.degree_on_basis(I([3, 10, 3])) 0
- gens()#
Return the generators of
self
.EXAMPLES:
sage: DU = algebras.DownUp(2, 3, 4) sage: DU.gens() (d, u)
- one_basis()#
Return the index of the basis element of \(1\).
EXAMPLES:
sage: DU = algebras.DownUp(2, 3, 4) sage: DU.one_basis() (0, 0, 0)
- product_on_basis(m1, m2)#
Return the product of the basis elements indexed by
m1
andm2
.EXAMPLES:
sage: R.<a,b,g> = QQ[] sage: DU = algebras.DownUp(a, b, g) sage: I = DU.indices() sage: DU.product_on_basis(I([2,0,0]), I([4,0,0])) u^6 sage: DU.product_on_basis(I([2,0,0]), I([0,4,0])) u^2*(d*u)^4 sage: DU.product_on_basis(I([2,0,0]), I([0,0,4])) u^2*d^4 sage: DU.product_on_basis(I([0,2,0]), I([0,4,0])) (d*u)^6 sage: DU.product_on_basis(I([0,2,0]), I([0,0,4])) (d*u)^2*d^4 sage: DU.product_on_basis(I([0,0,2]), I([0,0,4])) d^6 sage: DU.product_on_basis(I([5,3,1]), I([1,0,4])) u^5*(d*u)^4*d^4 sage: DU.product_on_basis(I([0,1,0]), I([1,0,0])) b*u^2*d + a*u*(d*u) + g*u sage: DU.product_on_basis(I([0,0,2]), I([1,0,0])) b*u*d^2 + a*(d*u)*d + g*d sage: DU.product_on_basis(I([0,0,1]), I([2,0,0])) b*u^2*d + a*u*(d*u) + g*u sage: DU.product_on_basis(I([0,0,1]), I([0,1,0])) b*u*d^2 + a*(d*u)*d + g*d sage: DU.product_on_basis(I([0,1,0]), I([3,0,0])) (a^2*b+b^2)*u^4*d + (a^3+2*a*b)*u^3*(d*u) + (a^2*g+a*g+b*g+g)*u^3 sage: DU.product_on_basis(I([1,1,3]), I([0,1,1])) (a^2*b^2+b^3)*u^3*d^6 + (a^3*b+a*b^2)*u^2*(d*u)*d^5 + (a^2*b*g+b^2*g)*u^2*d^5 + (a^3+2*a*b)*u*(d*u)^2*d^4 + (a^2*g+a*g+b*g+g)*u*(d*u)*d^4
- verma_module(la)#
Return the
Verma module
\(V(\lambda)\) ofself
.EXAMPLES:
sage: R.<a,b,g> = QQ[] sage: DU = algebras.DownUp(a, b, g) sage: DU.verma_module(5) Verma module of weight 5 of Down-Up algebra with parameters (a, b, g) over Multivariate Polynomial Ring in a, b, g over Rational Field
- class sage.algebras.down_up_algebra.VermaModule(DU, la)#
Bases:
CombinatorialFreeModule
The Verma module \(V(\lambda)\) of a down-up algebra.
The Verma module \(V(\lambda)\) for the down-up algebra generated by \(d, u\) is the span of \(\{v_n \mid n \in \ZZ_{\geq 0} \}\) satisfying the relations
\[d \cdot v_n = \lambda_{n-1} v_{n-1}, \qquad\qquad u \cdot v_n = v_{n+1},\]where \(\lambda_n = \alpha \lambda_{n-1} + \beta \lambda_{n-2} + \gamma\) and we set \(\lambda_0 = \lambda\) and \(\lambda_{-1} = 0\).
By Proposition 2.4 in [BR1998], \(V(\lambda)\) is simple if and only if \(\lambda_n \neq 0\) for all \(n \geq 0\). Moreover, a maximal submodule is spanned by \(\{ v_n \mid n > m \}\), where \(m\) is the minimal index such that \(\lambda_m = 0\). Moreover, this is unique unless \(\gamma = \lambda = 0\).
EXAMPLES:
sage: R.<a,b> = QQ[] sage: DU = algebras.DownUp(0, b, 1) sage: d, u = DU.gens() sage: V = DU.verma_module(a) sage: list(V.weights()[:6]) [a, 1, a*b + 1, b + 1, a*b^2 + b + 1, b^2 + b + 1] sage: v = V.basis() sage: d^2 * v[2] a*v[0] sage: d * (d * v[2]) a*v[0]
The weight is computed by looking at the scalars associated to the action of \(du\) and \(ud\):
sage: d*u * v[3] (b+1)*v[3] sage: u*d * v[3] (a*b+1)*v[3] sage: v[3].weight() (b + 1, a*b + 1)
An \(U(\mathfrak{sl}_2)\) example:
sage: DU = algebras.DownUp(2, -1, -2) sage: d, u = DU.gens() sage: V = DU.verma_module(5) sage: list(V.weights()[:10]) [5, 8, 9, 8, 5, 0, -7, -16, -27, -40] sage: v6 = V.basis()[6] sage: d * v6 0 sage: [V.basis()[i].weight() for i in range(6)] [(5, 0), (8, 5), (9, 8), (8, 9), (5, 8), (0, 5)]
Note that these are the same \(\mathfrak{sl}_2\) weights from the usual construction of the irreducible representation \(V(5)\) (but they are different as \(\mathfrak{gl}_2\) weights):
sage: B = crystals.Tableaux(['A',1], shape=[5]) sage: [b.weight() for b in B] [(5, 0), (4, 1), (3, 2), (2, 3), (1, 4), (0, 5)]
An example with periodic weights (see Theorem 2.13 of [BR1998]):
sage: k.<z6> = CyclotomicField(6) sage: al = z6 + 1 sage: (al - 1)^6 == 1 True sage: DU = algebras.DownUp(al, 1-al, 0) sage: V = DU.verma_module(5) sage: list(V.weights()[:8]) [5, 5*z6 + 5, 10*z6, 10*z6 - 5, 5*z6 - 5, 0, 5, 5*z6 + 5]
- class Element#
Bases:
IndexedFreeModuleElement
An element of a Verma module of a down-up algebra.
- is_weight_vector()#
Return if
self
is a weight vector.EXAMPLES:
sage: DU = algebras.DownUp(2, -1, -2) sage: V = DU.verma_module(5) sage: V.zero().is_weight_vector() False sage: B = V.basis() sage: [B[i].weight() for i in range(6)] [(5, 0), (8, 5), (9, 8), (8, 9), (5, 8), (0, 5)] sage: B[5].is_weight_vector() True sage: v = B[0] + B[1] sage: v.is_weight_vector() False sage: DU = algebras.DownUp(2, -1, 0) sage: V = DU.verma_module(0) sage: B = V.basis() sage: v = sum(i*B[i] for i in range(1,5)) sage: v.is_weight_vector() True
- weight()#
Return the weight of
self
.For \(v_n\), this is the vector with the pair \((\lambda_n, \lambda_{n-1})\).
EXAMPLES:
sage: R.<a,b,g> = QQ[] sage: DU = algebras.DownUp(a, b, g) sage: V = DU.verma_module(5) sage: B = V.basis() sage: B[0].weight() (5, 0) sage: B[1].weight() (5*a + g, 5) sage: B[2].weight() (5*a^2 + a*g + 5*b + g, 5*a + g) sage: V.zero().weight() Traceback (most recent call last): ... ValueError: the zero element does not have well-defined weight sage: (B[0] + B[1]).weight() Traceback (most recent call last): ... ValueError: not a weight vector
- highest_weight_vector()#
Return the highest weight vector of
self
that generatesself
as a down-up module.EXAMPLES:
sage: DU = algebras.DownUp(1, 2, 3) sage: V = DU.verma_module(5) sage: V.highest_weight_vector() v[0]
- weights()#
Return the sequence of weights \((\lambda_n)_{n=0}^{\infty}\).
EXAMPLES:
sage: R.<a,b,g> = QQ[] sage: DU = algebras.DownUp(a, b, g) sage: V = DU.verma_module(5) sage: V.weights() lazy list [5, 5*a + g, 5*a^2 + a*g + 5*b + g, ...] sage: V = DU.verma_module(0) sage: DU = algebras.DownUp(a, 1-a, 0) sage: V = DU.verma_module(0) sage: V.weights() lazy list [0, 0, 0, ...]
We reproduce the Fibonacci numbers example from [BR1998]:
sage: R.<la> = QQ[] sage: DU = algebras.DownUp(1, 1, 0, R) sage: V = DU.verma_module(la) sage: list(V.weights()[:11]) [la, la, 2*la, 3*la, 5*la, 8*la, 13*la, 21*la, 34*la, 55*la, 89*la]