Datasets:
Tasks:
Text Generation
Modalities:
Text
Sub-tasks:
language-modeling
Languages:
English
Size:
100K - 1M
License:
File size: 2,506 Bytes
4365a98 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import linear_algebra.pi_tensor_product
import logic.equiv.fin
import algebra.direct_sum.algebra
/-!
# Tensor power of a semimodule over a commutative semirings
We define the `n`th tensor power of `M` as the n-ary tensor product indexed by `fin n` of `M`,
`β¨[R] (i : fin n), M`. This is a special case of `pi_tensor_product`.
This file introduces the notation `β¨[R]^n M` for `tensor_power R n M`, which in turn is an
abbreviation for `β¨[R] i : fin n, M`.
## Main definitions:
* `tensor_power.ghas_one`
* `tensor_power.ghas_mul`
## TODO
Show `direct_sum.galgebra R (Ξ» i, β¨[R]^i M)` and `algebra R (β¨ n : β, β¨[R]^n M)`.
## Implementation notes
In this file we use `β1` and `β*` as local notation for the graded multiplicative structure on
tensor powers. Elsewhere, using `1` and `*` on `graded_monoid` should be preferred.
-/
open_locale tensor_product
/-- Homogenous tensor powers $M^{\otimes n}$. `β¨[R]^n M` is a shorthand for
`β¨[R] (i : fin n), M`. -/
@[reducible] protected def tensor_power (R : Type*) (n : β) (M : Type*)
[comm_semiring R] [add_comm_monoid M] [module R M] : Type* :=
β¨[R] i : fin n, M
variables {R : Type*} {M : Type*} [comm_semiring R] [add_comm_monoid M] [module R M]
localized "notation `β¨[`:100 R `]^`:80 n:max := tensor_power R n"
in tensor_product
namespace tensor_power
open_locale tensor_product direct_sum
open pi_tensor_product
/-- As a graded monoid, `β¨[R]^i M` has a `1 : β¨[R]^0 M`. -/
instance ghas_one : graded_monoid.ghas_one (Ξ» i, β¨[R]^i M) :=
{ one := tprod R fin.elim0 }
local notation `β1` := @graded_monoid.ghas_one.one β (Ξ» i, β¨[R]^i M) _ _
lemma ghas_one_def : β1 = tprod R fin.elim0 := rfl
/-- A variant of `pi_tensor_prod.tmul_equiv` with the result indexed by `fin (n + m)`. -/
def mul_equiv {n m : β} : (β¨[R]^n M) β[R] (β¨[R]^m M) ββ[R] β¨[R]^(n + m) M :=
(tmul_equiv R M).trans (reindex R M fin_sum_fin_equiv)
/-- As a graded monoid, `β¨[R]^i M` has a `(*) : β¨[R]^i M β β¨[R]^j M β β¨[R]^(i + j) M`. -/
instance ghas_mul : graded_monoid.ghas_mul (Ξ» i, β¨[R]^i M) :=
{ mul := Ξ» i j a b, mul_equiv (a ββ b) }
local infix `β*`:70 := @graded_monoid.ghas_mul.mul β (Ξ» i, β¨[R]^i M) _ _ _ _
lemma ghas_mul_def {i j} (a : β¨[R]^i M) (b : β¨[R]^j M) : a β* b = mul_equiv (a ββ b) := rfl
end tensor_power
|