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Let

V

be a vector space over a field

K

. The Tensor Algebra

T(V)

is defined as the direct sum of the

k

-th tensor powers of

V

: \n\n

T(V) = \bigoplus_{k=0}^{\infty} T^k(V) = \bigoplus_{k=0}^{\infty} V^{\bigotimes k}

\n\nwhere

T^0(V) = K

(the base field),

T^1(V) = V

, and for

k \neq 0

, T^k(V) = V \big\otimes_{k} V. \n\n

T(V)

is equipped with a multiplication operation, denoted by

\mu: T(V) \times T(V) \to T(V)

, which is the concatenation of tensor products. Specifically, for

A \in T^k(V)

and

B \in T^m(V)

, the product is defined as:\n\n

A \cdot B = A \otimes B

\n\nThis multiplication makes

T(V)

an associative, graded algebra, i.e.,

T(V) = \bigoplus_{k=0}^{\infty} T^k(V)

, satisfying the associativity axiom:

(A \cdot B) \cdot C = A \cdot (B \cdot C)

for all

A, B, C \in T(V)

. The quotient algebra

T^*/I

(where

I

is the ideal generated by the commutator

[v_1, v_2] = v_1 v_2 - v_2 v_1

) yields the universal enveloping algebra of the Lie algebra associated with

V

Tensor Algebra (Definition)