# Tensor Algebras¶

AUTHORS:

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

Todo

• Coerce to/from free algebra.

class sage.algebras.tensor_algebra.BaseRingLift

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)

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:

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: 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: 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)
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)

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 $$R$$