Tensor Algebras

AUTHORS:

• Travis Scrimshaw (2014-01-24): Initial version

Todo

• Coerce to/from free algebra.

class sage.algebras.tensor_algebra.BaseRingLift

Bases: Morphism

Morphism \(R \to T(M)\) which identifies the base ring \(R\) of a tensor algebra \(T(M)\) with the \(0\)-th graded part of \(T(M)\).

class sage.algebras.tensor_algebra.TensorAlgebra(M, prefix='T', category=None, **options)

Bases: CombinatorialFreeModule

The tensor algebra \(T(M)\) of a module \(M\).

Let \(\{ b_i \}_{i \in I}\) be a basis of the \(R\)-module \(M\). Then the tensor algebra \(T(M)\) of \(M\) is an associative \(R\)-algebra, with a basis consisting of all tensors of the form \(b_{i_1} \otimes b_{i_2} \otimes \cdots \otimes b_{i_n}\) for nonnegative integers \(n\) and \(n\)-tuples \((i_1, i_2, \ldots, i_n) \in I^n\). The product of \(T(M)\) is given by

\[(b_{i_1} \otimes \cdots \otimes b_{i_m}) \cdot (b_{j_1} \otimes \cdots \otimes b_{j_n}) = b_{i_1} \otimes \cdots \otimes b_{i_m} \otimes b_{j_1} \otimes \cdots \otimes b_{j_n}.\]

As an algebra, it is generated by the basis vectors \(b_i\) of \(M\). It is an \(\NN\)-graded \(R\)-algebra, with the degree of each \(b_i\) being \(1\).

It also has a Hopf algebra structure: The comultiplication is the unique algebra morphism \(\delta : T(M) \to T(M) \otimes T(M)\) defined by:

\[\delta(b_i) = b_i \otimes 1 + 1 \otimes b_i\]

(where the \(\otimes\) symbol here forms tensors in \(T(M) \otimes T(M)\), not inside \(T(M)\) itself). The counit is the unique algebra morphism \(T(M) \to R\) sending each \(b_i\) to \(0\). Its antipode \(S\) satisfies

\[S(b_{i_1} \otimes \cdots \otimes b_{i_m}) = (-1)^m (b_{i_m} \otimes \cdots \otimes b_{i_1}).\]

This is a connected graded cocommutative Hopf algebra.

REFERENCES:

• Wikipedia article Tensor_algebra

See also

TensorAlgebra

EXAMPLES:

sage: C = CombinatorialFreeModule(QQ, ['a','b','c'])
sage: TA = TensorAlgebra(C)
sage: TA.dimension()
+Infinity
sage: TA.base_ring()
Rational Field
sage: TA.algebra_generators()
Finite family {'a': B['a'], 'b': B['b'], 'c': B['c']}

algebra_generators()

Return the generators of this algebra.

EXAMPLES:

sage: C = CombinatorialFreeModule(QQ, ['a','b','c'])
sage: TA = TensorAlgebra(C)
sage: TA.algebra_generators()
Finite family {'a': B['a'], 'b': B['b'], 'c': B['c']}
sage: m = SymmetricFunctions(QQ).m()
sage: Tm = TensorAlgebra(m)
sage: Tm.algebra_generators()
Lazy family (generator(i))_{i in Partitions}

antipode_on_basis(m)

Return the antipode of the simple tensor indexed by m.

EXAMPLES:

sage: C = CombinatorialFreeModule(QQ, ['a','b','c'])
sage: TA = TensorAlgebra(C)
sage: s = TA(['a','b','c']).leading_support()
sage: TA.antipode_on_basis(s)
-B['c'] # B['b'] # B['a']
sage: t = TA(['a', 'b', 'b', 'b']).leading_support()
sage: TA.antipode_on_basis(t)
B['b'] # B['b'] # B['b'] # B['a']

base_module()

Return the base module of self.

EXAMPLES:

sage: C = CombinatorialFreeModule(QQ, ['a','b','c'])
sage: TA = TensorAlgebra(C)
sage: TA.base_module() is C
True

construction()

Return the functorial construction of self.

EXAMPLES:

sage: C = CombinatorialFreeModule(ZZ, ['a','b','c'])
sage: TA = TensorAlgebra(C)
sage: f, M = TA.construction()
sage: M == C
True
sage: f(M) == TA
True

coproduct_on_basis(m)

Return the coproduct of the simple tensor indexed by m.

EXAMPLES:

sage: C = CombinatorialFreeModule(QQ, ['a','b','c'])
sage: TA = TensorAlgebra(C, tensor_symbol="(X)")
sage: TA.coproduct_on_basis(TA.one_basis())
1 # 1
sage: I = TA.indices()
sage: ca = TA.coproduct_on_basis(I.gen('a')); ca
1 # B['a'] + B['a'] # 1
sage: s = TA(['a','b','c']).leading_support()
sage: cp = TA.coproduct_on_basis(s); cp
1 # B['a'](X)B['b'](X)B['c'] + B['a'] # B['b'](X)B['c']
 + B['a'](X)B['b'] # B['c'] + B['a'](X)B['b'](X)B['c'] # 1
 + B['a'](X)B['c'] # B['b'] + B['b'] # B['a'](X)B['c']
 + B['b'](X)B['c'] # B['a'] + B['c'] # B['a'](X)B['b']

We check that \(\Delta(a \otimes b \otimes c) = \Delta(a) \Delta(b) \Delta(c)\):

sage: cb = TA.coproduct_on_basis(I.gen('b'))
sage: cc = TA.coproduct_on_basis(I.gen('c'))
sage: cp == ca * cb * cc
True

counit(x)

Return the counit of x.

INPUT:

• x – an element of self

EXAMPLES:

sage: C = CombinatorialFreeModule(QQ, ['a','b','c'])
sage: TA = TensorAlgebra(C)
sage: x = TA(['a','b','c'])
sage: TA.counit(x)
0
sage: TA.counit(x + 3)
3

degree_on_basis(m)

Return the degree of the simple tensor m, which is its length (thought of as an element in the free monoid).

EXAMPLES:

sage: C = CombinatorialFreeModule(QQ, ['a','b','c'])
sage: TA = TensorAlgebra(C)
sage: s = TA(['a','b','c']).leading_support(); s
F['a']*F['b']*F['c']
sage: TA.degree_on_basis(s)
3

gens()

Return the generators of this algebra.

EXAMPLES:

sage: C = CombinatorialFreeModule(QQ, ['a','b','c'])
sage: TA = TensorAlgebra(C)
sage: TA.algebra_generators()
Finite family {'a': B['a'], 'b': B['b'], 'c': B['c']}
sage: m = SymmetricFunctions(QQ).m()
sage: Tm = TensorAlgebra(m)
sage: Tm.algebra_generators()
Lazy family (generator(i))_{i in Partitions}

one_basis()

Return the empty word, which indexes the \(1\) of this algebra.

EXAMPLES:

sage: C = CombinatorialFreeModule(QQ, ['a','b','c'])
sage: TA = TensorAlgebra(C)
sage: TA.one_basis()
1
sage: TA.one_basis().parent()
Free monoid indexed by {'a', 'b', 'c'}
sage: m = SymmetricFunctions(QQ).m()
sage: Tm = TensorAlgebra(m)
sage: Tm.one_basis()
1
sage: Tm.one_basis().parent()
Free monoid indexed by Partitions

product_on_basis(a, b)

Return the product of the basis elements indexed by a and b, as per AlgebrasWithBasis.ParentMethods.product_on_basis().

INPUT:

• a, b – basis indices

EXAMPLES:

sage: C = CombinatorialFreeModule(QQ, ['a','b','c'])
sage: TA = TensorAlgebra(C)
sage: I = TA.indices()
sage: g = I.gens()
sage: TA.product_on_basis(g['a']*g['b'], g['a']*g['c'])
B['a'] # B['b'] # B['a'] # B['c']

class sage.algebras.tensor_algebra.TensorAlgebraFunctor(base)

Bases: ConstructionFunctor

The tensor algebra functor.

Let \(R\) be a unital ring. Let \(V_R\) and \(A_R\) be the categories of \(R\)-modules and \(R\)-algebras respectively. The functor \(T : V_R \to A_R\) sends an \(R\)-module \(M\) to the tensor algebra \(T(M)\). The functor \(T\) is left-adjoint to the forgetful functor \(F : A_R \to V_R\).

INPUT:

• base – the base \(R\)

rank = 20